Oxford Calculus: Solving Homogeneous First Order Differential Equations Tom Rocks Maths 2021-12-14 | University of Oxford mathematician Dr Tom Crawford explains how to solve homogeneous first order differential equations with a worked through example involving partial fractions and implicit differentiation.Test yourself with some exercises on solving homogeneous first order differential equations with this FREE worksheet in Maple Learn: learn.maplesoft.com/doc/mq9nbn9nls/trm-homogeneous-1st-order-des-worksheetWe start by introducing the general form of a homogeneous first order differential equation and the substitution required to solve problems of this type. The resulting equation can then be solved using the method of ‘separation of variables’. The method of solution is then demonstrated with a fully worked example. Once the change of variables has been made, the resulting integral is solved using partial fractions and the reverse chain rule. The answer is then checked by implicitly differentiating and rearranging to get the required form seen in the original equation. Check your working using the Maple Calculator App – available for free on Google Play and the App Store. Android: play.google.com/store/apps/details?id=com.maplesoft.companion&hl=en Apple: apps.apple.com/us/app/maple-companion/id1466659419 Other videos in the Oxford Calculus series can be found here: youtube.com/playlist?list=PLMCRxGutHqflZoTY8JCm1GRzCdGXvZ3_S Finding critical points for functions of several variables: youtu.be/Leomuu82-u8 Classifying critical points using the method of the discriminant: youtube.com/watch?v=5M_ts8Q2LEM Partial differentiation explained: youtu.be/RVwcBGzQcT8 Second order linear differential equations: youtu.be/F54yhRB9qDI Integrating factors explained: youtube.com/watch?v=ftqKuOfOX3E Solving simple PDEs: youtu.be/uztjxrGY6Jw Jacobians explained: youtube.com/watch?v=YqMelRryG8U Separation of variables integration technique explained: youtu.be/zk41c0vs9XQ Find out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel: youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmathstwitter.com/tomrocksmathsinstagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Oxford University Mathematician REACTS to Animation vs. Physics Tom Rocks Maths 2023-12-28 | Oxford Mathematician Dr Tom Crawford watches @alanbecker's "Animation vs. Physics" for the very first time. Watch the original video here: youtube.com/watch?v=ErMSHiQRnc8*The copyright of the original video is the property of Alan Becker. The footage is shown here under a fair usage policy.Watch Tom take a variety of high school maths exams from around the world on the designated playlist here: youtube.com/playlist?list=PLMCRxGutHqfm3t0IVJabEab6OasV9WLrl Watch Tom on 'Numberphile' here: youtube.com/watch?v=oSXVmuNIfRI&list=PLt5AfwLFPxWJ2TNTA79reKQ6S88kTvDR- Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.ukFor more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here: beautifulequations.net/collections/tom-rocks-maths
Can an Oxford University Mathematician solve a High School Physics Exam? (with @PhysicsOnline) Tom Rocks Maths 2023-12-20 | Oxford Mathematician Dr Tom Crawford is challenged by Lewis from @PhysicsOnline to try some questions from an A-level Physics exam. Find the accompanying Maple Learn worksheet with some Physics questions to try for yourself here: learn.maplesoft.com/doc/v9ten9mzuaPart 2 where Lewis tries some Maths exam questions set by Tom is here: youtube.com/watch?v=6Y3cj0Us45M Sign-up for Maple Learn Premium using the code TOMROCKSMATHS for a discounted subscription. Head to getlearn.maplesoft.com for more information.Find out more about Maple Learn on the Maplesoft YouTube channel: youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ The questions featured in the video are taken from the OCR 2020 A Level Physics exam paper entitled “Modelling Physics”. You can download the exam paper for yourself here: tomrocksmaths.files.wordpress.com/2023/12/643598-question-paper-modelling-physics.pdf The Mark Scheme is here: tomrocksmaths.files.wordpress.com/2023/12/643603-mark-scheme-modelling-physics.pdfThe questions covered in the video are as follows:1:26 – Q16: Force Diagram20:47 – Q18: Projectile Motion49:44 – Multiple choice section: Q1, Q2, Q3, Q4, Q5, Q10, Q13Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.ukFor more maths content check out Tom's website tomrocksmaths.comYou can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmathstwitter.com/tomrocksmathsinstagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequation.com/collections/tom-rocks-maths With thanks toPhysics OnlineLewis MathesonSt Edmund HallUniversity of OxfordNicoguaro: en.wikipedia.org/wiki/Stress%E2%80%93strain_curve#/media/File:Stress_strain_ductile.svg
The Mathematician with 600 Publications - Noga Alon (2022 Shaw Prize) Tom Rocks Maths 2023-12-10 | Professor Noga Alon (Shaw Prize 2022) discusses his career in Mathematics which has led to over 600 publications so far. Interview with University of Oxford mathematician Dr Tom Crawford recorded at the Hong Kong Laureate Forum 2023. Noga begins with his recollection of being awarded the Shaw Prize in Mathematics, and the recent award ceremony in Hong Kong. He then explains how the interdisciplinary nature of his work has led to over 600 publications, including work in Biology, Economics and Neuroscience. Noga also discusses how he decides which questions are worth his time, and some of the great unsolved problems he has thought about in the past (eg. Collatz, Goldbach, Riemann, P vs NP).The second part of the video looks at some of his work in more detail, including his work on ‘necklace splitting’ and the Borsuk-Ulam Theorem which he covers in a recent Numberhpile video here: youtube.com/watch?v=rwiEiGqgetU Finally, Noga shares a story from his childhood involving the Eurovision Song Contest which convinced him of the objectivity of the subject of Mathematics.Links to Tom's other interviews with Laureates in Maths and Computer Science. Robert Langlands: youtube.com/watch?v=XNh0cd09hxQWhitfield Diffie: youtube.com/watch?v=UaanzpCkc8cLesley Lamport: youtube.com/watch?v=DPVvReKyhmwAlessio Figalli: youtube.com/watch?v=Oob466Ia9f4Martin Hairer: youtube.com/watch?v=Z6XP3n-SjiwMichael Atiyah: youtube.com/watch?v=alujy8SVIDMDaniel Spielman: youtu.be/SVfabuPRkYgEfim Zlemanov: youtube.com/watch?v=Nz3_RzZzuO8Steven Balbus: youtu.be/W1teZXKqdCw Full list of publications by N. Alon: https://web.math.princeton.edu/~nalon/PDFS/publications.html Shaw Prize 2022 acceptance speech by Noga Alon: youtube.com/watch?v=bU7X9TI-cVg Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.uk All video footage is shown under a fair use policy for educational purposes via a Creative Commons licence. The copyright remains the property of the owner of the footage. For more maths content check out Tom's website tomrocksmaths.comYou can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmathstwitter.com/tomrocksmathsinstagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here: beautifulequation.com/collections/tom-rocks-mathsWith thanks toNoga AlonHong Kong LaureateThe Shaw PrizeNumberphile3Blue1Brown: youtube.com/watch?v=yuVqxCSsE7c Eurovision Song ContestByle do Maja: youtube.com/watch?v=e-qliUeynjY Filip Frantál: youtube.com/watch?v=KB4TL6gIJMc
Oxford University Mathematician takes Admissions Interview (with @AnotherRoof) Tom Rocks Maths 2023-11-26 | Oxford University Mathematician Dr Tom Crawford gets a taste of his own medicine as he is asked some admissions interview questions by Alex from @AnotherRoof. Part 1 where Tom interviews Alex is here: youtube.com/watch?v=xrfLDuehzog*CORRECTION: For the 1st question l'hopital's rule can only be used when the derivatives are continuous and technically we are not told this in the question.Interview questions covered in the video:1. If f(x+y)=f(x)f(y) and f'(0)=3, what is f(x)?2. How many zeroes does 1000! have?Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.uk For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here:beautifulequation.com/collections/tom-rocks-maths
Prize Winning Astrophysicist Explains Black Hole Formation Tom Rocks Maths 2023-11-19 | Professor Steven Balbus (Shaw Prize 2013) explains his work on black hole formation. Interview with University of Oxford mathematician Dr Tom Crawford recorded at the Hong Kong Laureate Forum 2023.Steven discusses his reaction at being awarded the prize in 2013, before a detailed explanation of his influential work on accretion disks, and how the magnetic field around a black hole can be understood by considering a spring between rotating masses. We also discuss his position at the University of Oxford as the Savilian Professor of Astronomy, and why he decided to work in astrophysics. Finally, Steven answers some quick-fire questions includng "blackboards versus whitebaords", "pi vs tau", "angles versus radians", "favourite number", "favourite star" and "favourite mathematical result".Links to Tom's other interviews with Laureates in Maths and Computer Science. Robert Langlands: youtube.com/watch?v=XNh0cd09hxQ Whitfield Diffie: youtube.com/watch?v=UaanzpCkc8c Lesley Lamport: youtube.com/watch?v=DPVvReKyhmw Alessio Figalli: youtube.com/watch?v=Oob466Ia9f4 Martin Hairer: youtube.com/watch?v=Z6XP3n-Sjiw Michael Atiyah: youtube.com/watch?v=alujy8SVIDM Daniel Spielman: youtu.be/SVfabuPRkYg Efim Zlemanov: youtube.com/watch?v=Nz3_RzZzuO8 Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.uk For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here:beautifulequation.com/collections/tom-rocks-maths With thanks toSteven BalbusHong Kong Laureate ForumDan AddisonESA/HubbleTudorESOEHT CollaborationNASA’s Goddard Space Flight Center/Jeremy SchnittmanGary SettlesJosh Estey/AUSaidWHOISimons CentrePeter MercatorMVLANBerkeleyThe WireRogelio Bernal AndreoESO/P. Kervella
Can Dogs Do Math? Tom Rocks Maths 2023-11-12 | TRM intern Alvaro Gonzalez Hernandez and his dog Txoki explain the maths behind involutes - the curve formed when a dog unwraps its lead from around a tree trunk.Involutes have been studied throughout history, most notably by Christian Huygens when trying to construct an accurate clock using a pendulum. They are also found in gear systems where the shape of the gears follows that of a circular involute to reduce friction and increase torque. Calculation of the parametric form of the circular involute by Dr Tom Crawford.Produced by TRM intern Alvaro Gonzalez Hernandez with assistance from Dr Tom Crawford. Alvaro is a fourth year mathematics undergraduate at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.ukFor more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequation.com/collections/tom-rocks-mathsWith thanks toAlvaro Gonzalez HernandezTxokiOxford University Micro-Internship SchemeScience Museum LondonVysotsky (Wikipedia)Rob KoopmanAg2gaeh (Wikipedia)Sam DerbyshireStephan Heiss
Two Truths and a Lie: Isaac Newton in Cambridge with @singingbanana Tom Rocks Maths 2023-11-05 | Oxford Mathematician Dr Tom Crawford joins Dr James Grime @singingbanana and @numberphile to determine which stories about Isaac Newton in Cambridge are true and which are false... Learn more about the work of Isaac Newton with the Maple Learn worksheet here: learn.maplesoft.com/doc/urkp0izyjs/trm-newton-worksheetSign-up for Maple Learn Premium using the code TOMROCKSMATHS for a discounted subscription. Head to getlearn.maplesoft.com for more information.Subscribe to James' channel here: youtube.com/@singingbananaThanks to @isaacnewtoninstitute for giving us access to the lecture room.With thanks to:James GrimeThe Isaac Newton InstituteMaplesoftQueens' CollegeCentre for Mathematical SciencesTrinity CollegeCullinan StudioIt's No GameCMGleeDakeJoe DoublePhysics ClassroomProduced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.ukFor more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequation.com/collections/tom-rocks-maths
Why Do People Hate Mathematics? Efim Zelmanov (Fields Medal 1994) Tom Rocks Maths 2023-10-29 | Efim Zelmanov (1994 Fields Medal) talks to University of Oxford Mathematician Dr Tom Crawford about why people hate maths, how he solved the Burnside problem, and what makes math beautiful. Full list of questions below.Recorded at the 2023 Heidelberg Laureate Forum. Find out more about the event at heidelberg-laureate-forum.orgThe Fields Medal is awarded every four years during the opening ceremony of the International Congress of Mathematicians (ICM). It recognizes outstanding mathematical achievement for existing work and for the promise of future achievement. Two to four medals are awarded to mathematicians who have to be of age less than forty years on January 1 of the Congress year. The Fields Medal, established in 1936 and named after the Canadian mathematician J. C. Fields, is one of the most prestigious awards in the field of mathematics and often described as the "Nobel Prize of Mathematics".Efim Zelmanov was awarded the Fields Medal in 1994 “for the solution of the restricted Burnside problem in group theory”. Efim is the Chair Professor at the Southern University of Science and Technology (SUSTech).Full list of questions covered in the interview:Why do people hate mathematics? Is maths a science or an art? How did you feel when you solved the Burnside problem and were awarded the Fields medal?Will the remainder of the Burnside problem be solved anytime soon?What area of maths do you find the most beautiful?Should maths be split into pure and applied subjects?What is your mathematical genealogy?Do you prefer Newton or Leibniz notation for derivatives?Is zero a natural number?Was maths discovered or invented?Do you have a favourite number?Do you prefer blackboards or whiteboards?Links to Tom's other interviews with Laureates in Maths and Computer Science. Whitfield Diffie: youtube.com/watch?v=UaanzpCkc8c Lesley Lamport: youtube.com/watch?v=DPVvReKyhmw Alessio Figalli: youtube.com/watch?v=Oob466Ia9f4 Martin Hairer: youtube.com/watch?v=Z6XP3n-Sjiw Michael Atiyah: youtube.com/watch?v=alujy8SVIDM Daniel Spielman: youtu.be/SVfabuPRkYg Links to the other events/talks mentioned in the video:Why do people hate mathematics? youtube.com/watch?v=_rHwxqsETFw Laszlo Lovasz lecture at HLF 2023 youtube.com/watch?v=hg1OH0PofKw Efim Zelmanov: is math a science or an art? youtube.com/watch?v=6q8LkyBaPTcWhere can mathematics and computer science interact fruitfully? youtube.com/watch?v=aeSYEaP0N_E Find your own Math Genealogy here: mathgenealogy.org Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.uk For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here:beautifulequation.com/collections/tom-rocks-maths With thanks toEfim ZlemanovHeidelberg Laureate ForumHeidelberg Laureate Forum FoundationMath Genealogy IMU
Oxford Linear Algebra: Gram-Schmidt Process Tom Rocks Maths 2023-10-22 | University of Oxford Mathematician Dr Tom Crawford introduces the steps of the Gram-Schmidt Process and explains why the algorithm gives you an orthonormal set of vectors. Check out ProPrep with a 30-day free trial: proprep.uk/info/TOM-Crawford Links to the other videos mentioned: Inner Product Space youtube.com/watch?v=ywjA25xFuewTest your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20Inner%20Product%20Spaces%20-%20workbook.pdfYou can also find several video lectures from ProPrep explaining the content covered in the video here: proprep.com/courses/all/linear-algebra/inner-product-spaces/gram-schmitt-process/vid26063As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises. The video begins with a reminder of the definition of an orthonormal set, before introducing the 3 steps of the Gram-Schmidt Process. Step 1: normalise the first vector from a linearly independent set. Step 2: subtract the projection of the first orthonormal vector from the second vector in the linearly independent set, then normalise. Step 3: repeat step 2 for each of the remaining vectors.Step 2 is explored in more detail through a direct calculation of the inner product and an explicit example in the 2D plane, including a visualisation of the projection map. The video ends with a fully worked example of computing an orthonormal set in the polynomial inner product space where the inner product is defined via an integral. Watch the other videos from the Oxford Linear Algebra series at the links below. Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): youtu.be/9pF__coVyEE Calculating the inverse of 2x2, 3x3 and 4x4 matrices: youtu.be/VKOaG3Ogf9Q What is the Determinant Function: youtube.com/watch?v=bLsBWVYSg0A The Easiest Method to Calculate Determinants: youtu.be/qniUv4EZB0w Eigenvalues and Eigenvectors Explained: youtu.be/8uISh6xyW7w Spectral Theorem Proof: youtu.be/ADwsk9G5s_8 Vector Space Axioms: youtu.be/draqOOUoWQM Subspace Test: youtu.be/3_MxBlWQsgs Basis, Spanning and Linear Independence: youtu.be/wdOFi8aUNp0Dimension Formula: youtu.be/lGo9E5dZ4f8Direct Sum: youtube.com/watch?v=b1VZm4Oecq4 Linear Transformations: youtu.be/UF59Mok4fsQ Rank Nullity Theorem: youtu.be/yZ52iqxaiCcInner Product Space: youtube.com/watch?v=ywjA25xFuew Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.ukFor more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-mathsCheck out Proprep with a 30-day free trial here: proprep.uk/info/TOM-Crawford
Oxford University Mathematician takes High School IB Maths Exam Tom Rocks Maths 2023-10-12 | Subscribe to Curiosity Stream now using the code TOMROCKS for 25% off and start exploring the world around you! curiositystream.com/TomRocksUniversity of Oxford Mathematician Dr Tom Crawford sits the IB Maths Exam taken by High School students around the world. The test is usually taken at the end of school by students aged 17-18. The exam taken by Tom is the ‘Mathematics: Applications and Interpretation Higher Level Specimen Paper’. You can download the test (and mark scheme) for yourself here: tomrocksmaths.files.wordpress.com/2023/10/dp-mathematics-applications-and-interpretation-specimen-papers-en.pdf The Specimen Paper and all questions contained within are property of the International Baccalaureate Organization. Material is used under a 'fair use' policy for educational purposes. Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.uk Watch Tom take other Exams on the designated playlist here: youtube.com/playlist?list=PLMCRxGutHqfm3t0IVJabEab6OasV9WLrl A-level Maths: youtube.com/watch?v=uupjxztr2q8 A-level Further Maths: youtube.com/watch?v=C8o0vfQe_6A GCSE Maths: youtube.com/watch?v=hQVcv-T7IiY GCSE Further Maths: youtube.com/watch?v=KDn6N7eo9uo SAT Maths: youtube.com/watch?v=u0se7iWdNZw Cambridge University Admissions Test (STEP Paper) Part 1: youtube.com/watch?v=Qsthjv-Ygng Cambridge University Admissions Test (STEP Paper) Part 2: youtube.com/watch?v=f0Fm97oM608 German Maths Abitur: youtu.be/ZjCzWmRn6UM For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here: beautifulequation.com/collections/tom-rocks-maths
Solving the Hardest Problems with Daniel Spielman (Nevanlinna Prize 2010) Tom Rocks Maths 2023-10-01 | University of Oxford Mathematician Dr Tom Crawford talks to Professor Daniel Spielman of Yale University about his career so far, including what it was like to win the 2010 Nevanlinna Prize. Learn more about Discrepancy Theory with the Maple Learn worksheet here: learn.maplesoft.com/doc/8n4hf5eiaw/discrepancy-theorySign-up for Maple Learn Premium using the code TOMROCKSMATHS for a discounted subscription. Head to getlearn.maplesoft.com for more information.Recorded at the 2023 Heidelberg Laureate Forum. Find out more about the event at heidelberg-laureate-forum.org The Nevalinna Prize (now renamed the Abacus Medal) is awarded by the International Mathematical Union (IMU) every 4 years for outstanding contributions in Mathematical Aspects of Information Sciences. Daniel received his prize in 2010 "for smoothed analysis of Linear Programming, algorithms for graph-based codes and applications of graph theory to Numerical Computing."Links to Tom's other interviews with Laureates in Maths and Computer Science.Whitfield Diffie: youtube.com/watch?v=UaanzpCkc8cLesley Lamport: youtube.com/watch?v=DPVvReKyhmwAlessio Figalli: youtube.com/watch?v=Oob466Ia9f4Martin Hairer: youtube.com/watch?v=Z6XP3n-SjiwMichael Atiyah: youtube.com/watch?v=alujy8SVIDMLinks to Daniel's talks at the 2023 HLF.Lightning Talks: youtube.com/watch?v=euWWcuGdCwMSparks Session: youtube.com/watch?v=IHEfrVrYQSILinks to the two publications mentioned in the video.Spielman: arxiv.org/abs/1911.03071Spencer: jstor.org/stable/2000258Link to interview with Daniel at HLF5: youtube.com/watch?v=SoVRmfFrnhsWith thanks to:Daniel SpielmanHeidelberg Laureate Forum FoundationPhotograph of Commodore PET by Rama, Wikimedia Commons, Cc-by-sa-2.0-frPhotograph of Notebooks by Brandon Schulman for Quanta MagazineTom GellerProduced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.ukFor more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequation.com/collections/tom-rocks-maths
Oxford University Mathematician REACTS to Math Rap - Lil Mabu and bbno$ Tom Rocks Maths 2023-09-24 | University of Oxford Mathematician Dr Tom Crawford watches 2 music videos described as 'Math Rap' - @LilMabu 'MATHEMATICAL DISRESPECT' and @bbnomoney 'mathematics'. Watch the original videos at the links below.Lil Mabu - MATHEMATICAL DISRESPECT: youtube.com/watch?v=zXV4GX6Whyw [Merlin] Too Lost (on behalf of Lil Mabu); BMI - Broadcast Music Inc., SOLAR Music Rights Management, Reservoir Media (Publishing), LatinAutorPerf, and 5 Music Rights Societiesbbno$ - mathematics: youtube.com/watch?v=tH_i76JVTDg (on behalf of bbno$); Sony Music Publishing, Abramus Digital, LatinAutor, SOLAR Music Rights Management, CMRRA, LatinAutor - PeerMusic, Polaris Hub AB, LatinAutorPerf, Ultra Publishing, Pulse Recording (music publishing), BMI - Broadcast Music Inc., UNIAO BRASILEIRA DE EDITORAS DE MUSICA - UBEM, LatinAutor - SonyATV, and 10 Music Rights Societies*The copyright of the original video is the property of the artists and respective record labels. The footage is shown here under a fair usage policy.Application of Information Theory to 'Wordle' from @3blue1brown: youtube.com/watch?v=v68zYyaEmEA Watch Tom take a variety of high school maths exams from around the world here: youtube.com/playlist?list=PLMCRxGutHqfm3t0IVJabEab6OasV9WLrl Watch Tom on @numberphile here: youtube.com/watch?v=oSXVmuNIfRI&list=PLt5AfwLFPxWJ2TNTA79reKQ6S88kTvDR- Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.ukFor more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Oxford Linear Algebra: Inner Product Space Tom Rocks Maths 2023-09-17 | University of Oxford Mathematician Dr Tom Crawford introduces the concept of a Bilinear Form, Inner Product, Sesquilinear Form and Inner Product Space. Check out ProPrep with a 30-day free trial: proprep.uk/info/TOM-Crawford Links to the other videos mentioned: Linear Transformations: youtu.be/UF59Mok4fsQ Spanning, Basis & Linear Independence: youtu.be/wdOFi8aUNp0 Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20Inner%20Product%20Spaces%20-%20workbook.pdfYou can also find several video lectures from ProPrep explaining the content covered in the video here: proprep.com/courses/all/linear-algebra/inner-product-spaces/inner-product-spaces/vid10133As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises. The video begins with the definition of a Bilinear Form with a concrete example of the dot product on R^n. This is shown to also satisfy the criteria to be symmetric and positive definite, thus making it an Inner Product.The concept of an Inner Product Space is then introduced as a Real Vector Space equipped with an Inner Product. A second example involving an integral over the space of real polynomials is then explored.In the second part of the video Orthonormal Sets are introduced via a definition and then the proof of a lemma stating that any Orthonormal Set in an Inner Product Space is Linearly Independent. The video concludes with a final definition of a Sesquilinear Form and a discussion of a Complex Inner Product Space. Watch the other videos from the Oxford Linear Algebra series at the links below. Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): youtu.be/9pF__coVyEE Calculating the inverse of 2x2, 3x3 and 4x4 matrices: youtu.be/VKOaG3Ogf9Q What is the Determinant Function: youtube.com/watch?v=bLsBWVYSg0A The Easiest Method to Calculate Determinants: youtu.be/qniUv4EZB0w Eigenvalues and Eigenvectors Explained: youtu.be/8uISh6xyW7w Spectral Theorem Proof: youtu.be/ADwsk9G5s_8 Vector Space Axioms: youtu.be/draqOOUoWQM Subspace Test: youtu.be/3_MxBlWQsgs Basis, Spanning and Linear Independence: youtu.be/wdOFi8aUNp0Dimension Formula: youtu.be/lGo9E5dZ4f8Direct Sum: youtube.com/watch?v=b1VZm4Oecq4 Linear Transformations: youtu.be/UF59Mok4fsQ Rank Nullity Theorem: youtu.be/yZ52iqxaiCcProduced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.ukFor more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-mathsCheck out Proprep with a 30-day free trial here: proprep.uk/info/TOM-Crawford
Climbing Mount Kilimanjaro 2023 - Machame Route Tom Rocks Maths 2023-09-10 | Something a little different to my usual content, but I wanted to share the experience with all of you as a thank you for supporting the channel. I climbed Kilimanjaro via the Machame Route in August 2023 and this is what happened...Day 1: Machame Gate (1640m) to Machame Camp (2850m)Day 2: Machame Camp (2850m) to Shira Camp (3810m)Day 3 Part I: Shira Camp (3810m) to Lava Tower (4630m)Day 3 Part II: Lava Tower (4630m) to Barranco Camp (3976m)Day 4 Part I: Barranco Camp (3976m) to Karanga Camp (3995m)Day 4 Part II: Karanga Camp (3995m) to Barafu Camp (4673m)Day 5 Part I: Barafu Camp (4673m) to Uhuru Peak (5895m)Day 5 Part II: Uhuru Peak (5895m) to Mweka Camp (3068m)Day 6: Mweka Camp (3068m) to Mweka Gate (1640m)Thanks to our lead guide David, assistant guide Raymond, Cubs Expeditions and the whole team of porters that helped us on the journey. And thanks to George and Jon for coming with me - and starring in most of the footage as I was the one holding the camera...Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.ukFor more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Tom Rocks Maths World Tour 2024 Tom Rocks Maths 2023-09-03 | University of Oxford Mathematician Dr Tom Crawford will be travelling to Australia, New Zealand, Malaysia, Singapore, China and surrounding countries from May - October 2024. Request a visit from Tom here: tomrocksmaths.com/contactProduced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.ukFor more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
TEDx: Using Maths to Save Our Planet Tom Rocks Maths 2023-08-27 | TEDx talk by University of Oxford Mathematician Dr Tom Crawford. Recorded at TEDxUHasselt April 2023. Original video @TEDx here: youtube.com/watch?v=PW0F485f9M4 By creating a mathematical model which identifies where pollution goes when it enters the ocean, we can stay one step ahead of the waste and help to save our planet. Dr Tom Crawford is a mathematician at the University of Oxford with a mission to make maths fun and accessible for all. In this talk, he discusses his research in the field of fluid dynamics, and how he has been able to construct a model which can predict where the pollution from any river, anywhere in the world will end up, using only three pieces of information widely available on the internet. For many people, mathematical formulae and calculations can be a source of anxiety, but Tom's experience and expertise will have you hanging on his every word from beginning to end, as he explains not only how we can use maths to save the planet, but how YOU can use the power of mathematical modelling to help to solve any problem you might encounter in your life. Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Baseball Aerodynamics Tom Rocks Maths 2023-08-13 | Professor Barton Smith of Utah State University explains his work on the fluid dynamics of a baseball pitch. For more information check out his website: baseballaero.com Research by Barton Smith and Andrew Smith at Utah State University.Interview with University of Oxford Mathematician Dr Tom Crawford. Tom is an Early-Career Teaching and Outreach Fellow in Mathematics at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford This video is part of a collaboration with the Journal of Fluid Mechanics and the UK Fluids Network featuring a series of interviews with researchers from the APS DFD 2019 conference. Sponsored by the Journal of Fluid Mechanics and the UK Fluids Network. Produced by Tom Crawford.For more maths related fun check out Tom's website tomrocksmaths.com Get your Tom Rocks Maths merchandise here: beautifulequations.net/collections/tom-rocks-maths-------Follow Tom:Website: tomrocksmaths.comYouTube: youtube.com/tomrocksmathsTwitter: twitter.com/tomrocksmathsFacebook: facebook.com/tomrocksmathsInstagram: instagram.com/tomrocksmathsFollow JFM on:Website: cambridge.org/core/journals/journal-of-fluid-mechanicsYoutube: youtube.com/user/CambridgeUPAcProTwitter: twitter.com/JFluidMechFollow the UK Fluids Network on:Website: fluids.ac.ukTwitter: twitter.com/UKFluidsNetwork--------Featuring "The effect of seam orientation on the flight of a baseball": doi.org/10.1103/APS.DFD.2019.GFM.V0038The understanding of aerodynamics in baseball has, over the history of the game, been continuously evolving. This video demonstrates some of the complex behaviour inherent in baseball flight due to the irregular seam locations on the ball and how additional forces, beyond those due to spin and gravity, can play a major role in determining the baseball's flight path.Gallery of Fluid Motion entry:youtube.com/watch?v=NuDMCgTpg8UAuthors:Andrew Smith, Utah State UniversityRob Friedman, Pitching NinjaNazmus Sakib, Utah State UniversityJohn Garrett, Utah State UniversityBarton Smith, Utah State UniversityPublication:Smith AW, Smith BL. Using baseball seams to alter a pitch direction: The seam shifted wake. Proceedings of the Institution of Mechanical Engineers, Part P: Journal of Sports Engineering and Technology. 2021;235(1):21-28. doi:10.1177/1754337120961609 Golf ball image copyright Horia Varlan (Creative Commons 3.0 licence): commons.wikimedia.org/wiki/Category:Golf_balls#/media/File:Golf_ball_4.jpgTrevor Bauer image copyright Erik Dros (Creative Commons 2.0 licence): commons.wikimedia.org/wiki/File:Trevor_Bauer_(47903072211).jpgWith thanks to:Barton SmithAndrew SmithAugustineMLBFangraphs ProspectsHoria VarlanErik DrostJFMUK Fluid NetworkAPS DFD 2019All MLB footage is presented under a fair usage policy for the purpose of education. It remains copyright of MLB.
Oxford Linear Algebra: Rank Nullity Theorem Tom Rocks Maths 2023-08-06 | University of Oxford mathematician Dr Tom Crawford introduces the concepts of rank and nullity for a linear transformation, before going through a full step-by-step proof of the Rank Nullity Theorem. Check out ProPrep with a 30-day free trial: proprep.uk/info/TOM-Crawford Links to the other videos mentioned: Linear Transformations: youtu.be/UF59Mok4fsQ Dimension Formula: youtu.be/lGo9E5dZ4f8 Spanning, Basis & Linear Independence: youtu.be/wdOFi8aUNp0 Subspace Test: youtu.be/3_MxBlWQsgs Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20Linear%20Transformations%20-%20workbook.pdfYou can also find several video lectures from ProPrep explaining the content covered in the video here: proprep.com/courses/all/linear-algebra/linear-transformations/image-and-kernel/vid10197As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises. The video begins with the definition of the kernel and image of a linear transformation, as well as their dimensions as the nullity and rank of the linear map. Next is a fully worked example of a map in R^2 with an explicit calculation of the kernel, image, nullity and rank.In the second part of the video the Rank Nullity Theorem is presented: the dimension of the starting vector space of a linear map is equal to the rank of the linear map (dimension of its image) plus the nullity of the map (dimension of its kernel). A full step-by-step proof is then presented by constructing a basis for each component.Watch the other videos from the Oxford Linear Algebra series at the links below. Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): youtu.be/9pF__coVyEE Calculating the inverse of 2x2, 3x3 and 4x4 matrices: youtu.be/VKOaG3Ogf9Q What is the Determinant Function: youtube.com/watch?v=bLsBWVYSg0A The Easiest Method to Calculate Determinants: youtu.be/qniUv4EZB0w Eigenvalues and Eigenvectors Explained: youtu.be/8uISh6xyW7w Spectral Theorem Proof: youtu.be/ADwsk9G5s_8 Vector Space Axioms: youtu.be/draqOOUoWQM Subspace Test: youtu.be/3_MxBlWQsgs Basis, Spanning and Linear Independence: youtu.be/wdOFi8aUNp0Dimension Formula: youtu.be/lGo9E5dZ4f8Direct Sum: youtube.com/watch?v=b1VZm4Oecq4 Linear Transformations: youtu.be/UF59Mok4fsQ Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-mathsCheck out Proprep with a 30-day free trial here: proprep.uk/info/TOM-Crawford
Integrate By Parts - Total Eclipse of the Heart Maths Parody Tom Rocks Maths 2023-07-30 | Tom Rocks Maths intern Fred Tyrrell sings a mathematical version of the song "Total Eclipse of the Heart" by Bonnie Tyler. Full lyrics on screen and below.Original song "Total Eclipse of the Hearth - Bonnie Tyler": youtube.com/watch?v=lcOxhH8N3BoOriginal song copyright: SME; Abramus Digital, LatinAutor, Hexacorp (music publishing), LatinAutorPerf, CMRRA, MINT_BMG, BMI - Broadcast Music Inc., ARESA, and 13 Music Rights Societies.Inspired by Jimmy and Eric: youtu.be/-reFBJ4R9iA Lyrics:Integrate...Every now and then I get a little bit sad because my maths all comes out wrong.Integrate...Every now and then I get a little bit tired of integrals for so many years.Integrate...Every now and then I get a little bit nervous that the function can't be solved for y.Integrate…Every now and then I get a little bit terrified and I forget to add on a pi.Substitute function,every now and then that doesn't work.Use the trig identitiesevery now and then that doesn't work.And I need another way,and I need it more than ever.And I cry into the night,when they're multiplied together.And I just can't seem to do this right,'cause it just comes out wrong.Together these two terms are causing all of my plight.This problem is a shadow on me all of the time.I don't know what to do should I integrate by parts?But picking u and v I don't know where to start...I need to integrate,I'm gonna see this problem through,I’m gonna see this problem through. Once upon a time I thought maths was so dumb,but now I see that it takes some smarts.Dumb, dumb, dumb, dumb...There's nothing left to do, so I integrate by parts.Once upon a time there was light in my life,but now I integrate in the dark.One thing left to do so I integrate by parts.I integrate by parts.How to do integrals?Every now and then I go with parts.How to do integrals?Every now and then I go with parts.And I hope I get it right, and I hope I get the answer.And if I only use my mind,I can do two functions together.But when substitution doesn't work,and trig makes it worse.Together these two functions make it seem too hard,but integrate by parts as your trump card.As your trump card.I don't know what to do so I'll integrate by parts,u dv and v du make math look like art.I really needed to know when the integral is just too hard,the integral is just too hard.Once upon a time there was light in my life but now I integrate in the dark.Nothing left to do so I integrate by parts.I integrate by parts.Integrate... by... parts...Integrate... by... parts...Integrate by parts.Produced by TRM intern Fred Tyrrell with assistance from Dr Tom Crawford. Fred is a fourth year mathematics undergraduate at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford Thanks to the University of Oxford Micro-Internship scheme. If you would like to take part in the Tom Rocks Maths intern scheme, please get in touch using the contact form on the TRM website: tomrocksmaths.com/contact For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Oxford Linear Algebra: Linear Transformations Explained Tom Rocks Maths 2023-07-23 | University of Oxford mathematician Dr Tom Crawford introduces the concept of a Linear Transformation with a motivation for the definition and several worked examples. Check out ProPrep with a 30-day free trial: proprep.uk/info/TOM-Crawford Links to the other videos mentioned:Vector Space Axioms - youtu.be/3_MxBlWQsgs Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20Linear%20Transformations%20-%20workbook.pdfYou can also find several video lectures from ProPrep explaining the content covered in the video here: proprep.com/courses/all/linear-algebra/linear-transformations/linear-transformation-definition/vid10223As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises. The video begins with the formal mathematical definition of a linear transformation and how their properties relate to those of a vector space. There is also a discussion as to how linear maps ensure that the structure of a vector space is preserved. An alternative definition for a linear transformation is then introduced, which is often used to check whether or not a given map is indeed linear. We also see how a linear transformation must always map the zero vector in the starting space to the zero vector of the final space. A generalised application to a basis of a vector space is also briefly mentioned (this will be discussed further in the next video on the Rank-Nullity Theorem.)Several examples of linear maps are looked at in detail, including multiplication by a matrix, differentiation, and three explicit maps in R3. The final example re-introduces the concept of a direct sum (as seen in the previous video) and how it can be used to define a projection map. Watch the other videos from the Oxford Linear Algebra series at the links below. Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): youtu.be/9pF__coVyEE Calculating the inverse of 2x2, 3x3 and 4x4 matrices: youtu.be/VKOaG3Ogf9Q What is the Determinant Function: youtube.com/watch?v=bLsBWVYSg0A The Easiest Method to Calculate Determinants: youtu.be/qniUv4EZB0w Eigenvalues and Eigenvectors Explained: youtu.be/8uISh6xyW7w Spectral Theorem Proof: youtu.be/ADwsk9G5s_8 Vector Space Axioms: youtu.be/draqOOUoWQM Subspace Test: youtu.be/3_MxBlWQsgs Basis, Spanning and Linear Independence: youtu.be/wdOFi8aUNp0Dimension Formula: youtu.be/lGo9E5dZ4f8Direct Sum: youtube.com/watch?v=b1VZm4Oecq4 Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Oxford University Mathematician REACTS to Animation vs. Math Tom Rocks Maths 2023-07-16 | Oxford Mathematician Dr Tom Crawford watches @alanbecker's "Animation vs. Math" for the very first time. Watch the original video here: youtube.com/watch?v=B1J6Ou4q8vE *The copyright of the original video is the property of Alan Becker. The footage is shown here under a fair usage policy.The "Gamma Function" video can be found here: youtube.com/watch?v=7y-XTrfNvCs Watch Tom take a variety of high school maths exams from around the world on the designated playlist here: youtube.com/playlist?list=PLMCRxGutHqfm3t0IVJabEab6OasV9WLrl Watch Tom on 'Numberphile' here: youtube.com/watch?v=oSXVmuNIfRI&list=PLt5AfwLFPxWJ2TNTA79reKQ6S88kTvDR- Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow in Mathematics at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here: beautifulequations.net/collections/tom-rocks-maths
Oxford University Mathematician takes German High School Maths Exam Tom Rocks Maths 2023-07-09 | Oxford Mathematician Dr Tom Crawford sits the 'Abitur Mathematik' - the maths exam taken by students in Germany at the end of high school. Test yourself with the Maple Learn worksheet here: learn.maplesoft.com/doc/iefswls387/trm---abitur-geometry-q1 The exam taken by Tom is the 2019 Abitur Mathematik from the German region of Bavaria. The paper is taken when leaving school at the age of 17 or 18. You can download the test for yourself here: tomrocksmaths.files.wordpress.com/2023/07/abiturpruefung_mathematik_2019_pruefungsteil_a.pdfA version translated into English (using Google translate) can be found here: tomrocksmaths.files.wordpress.com/2023/07/translated-copy-of-abiturpruefung_mathematik_2019_pruefungsteil_a-google-docs.pdfProduced by Dr Tom Crawford at the University of Oxford.Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford Thanks to Miles Simmons for the Maple Learn worksheet and visualisations. Sign-up for Maple Learn Premium using the code TOMROCKSMATHS for a discounted subscription. Head to getlearn.maplesoft.com for more information.Watch Tom take other Exams on the designated playlist here: youtube.com/playlist?list=PLMCRxGutHqfm3t0IVJabEab6OasV9WLrlA-level Maths: youtube.com/watch?v=uupjxztr2q8A-level Further Maths: youtube.com/watch?v=C8o0vfQe_6A GCSE Maths: youtube.com/watch?v=hQVcv-T7IiYGCSE Further Maths: youtube.com/watch?v=KDn6N7eo9uo SAT Maths: youtube.com/watch?v=u0se7iWdNZw Cambridge University Admissions Test (STEP Paper) Part 1: youtube.com/watch?v=Qsthjv-YgngCambridge University Admissions Test (STEP Paper) Part 2: youtube.com/watch?v=f0Fm97oM608For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Watch a Real Oxford University Lesson || Undergraduate Maths Tutorial Tom Rocks Maths 2023-06-25 | Real-life recording of a tutorial at the University of Oxford with mathematician Dr Tom Crawford. The lesson goes over a past exam paper as part of a revision tutorial covering the second year undergraduate Metric Spaces and Complex Analysis course.The exam covered in the tutorial is from the second year undergraduate course in Metric Spaces and Complex Analysis. More information on the course can be found here: courses.maths.ox.ac.uk/course/view.php?id=1045 Students would be expected to answer 4 out of 6 questions in a 3-hour closed book exam at the end of the year. The course is consists of 32 hours of lectures and 8 hours of tutorials. Topics covered in the video include: compact sets, continuous functions, complete metric spaces, Contraction Mapping Theorem, multifunctions, Cauchy-Riemann equations, complex logarithm, Weierstrass M-test, contour integrals, indentation lemma, Residue Theorem, Cauchy's Integral Formula, conformal maps, and stereographic projection. For a question by question breakdown please see below.Q2. Metric Spaces, Compact Sets, Contraction Mapping TheoremQ3. Holomorphic Functions, Cauchy-Riemann Equations, Weierstrass M-testQ4. Contour Integration, Residue Theorem, Laurent's TheoremQ5. Cauchy Integral FormulaQ6. Stereographic Projection, Conformal MapsFilmed at St Edmund Hall, University of Oxford. Thanks to Luke and Will for taking part (both current second year undergraduate maths students). Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow in Mathematics at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here: beautifulequations.net/collections/tom-rocks-maths
Building the Multiverse Library with @henryseg Tom Rocks Maths 2023-06-15 | Watch me and @henryseg explore the Multiverse Library here: youtube.com/watch?v=f8C8hlAmdVc. This video explains the maths used to create the Multiverse Library, namely branch cuts and multifunctions. Featuring Oxford Mathematician Dr Tom Crawford and Dr Henry Segerman of Oklahoma State University.The main maths content of the video focuses on defining a holomorphic branch of the complex logarithm, which is exemplified by looking at the complex square root function. Multifunctions are usually covered in a Complex Analysis course at university. Watch “Peace for Triple Piano” with @Vihart here: youtube.com/watch?v=HcRW3FMuttY Library footage filmed at St Edmund Hall, University of Oxford. Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow in Mathematics at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here: beautifulequations.net/collections/tom-rocks-maths
Maths Love Song: You + Me (with Mathematical Explanation) Tom Rocks Maths 2023-06-02 | Song written and performed by Tom Rocks Maths intern Siddiq Islam - a second year undergraduate Maths student @oxforduniversity. Explanation by Oxford Mathematician Dr Tom Crawford. FULL LYRICS BELOW.You + meIs all I’d ever dream to beYou seeYou’re my -7x^(-8) + 3And equallyI’m your x^(-7) + 3x + cWoahMy love’s exponentialIts domain is boundless and it reaches into infinityWhile at first incrementalThe rate at which it’s growing is increasing graduallyThey say I resemblee^(iπ), beautiful in all I doBut I am just helpless‘Cos I still feel like a -1 withoutYou + MeIs all I’d ever dream to beYou seeYou’re my -7x^(-8) + 3And equallyI’m your x^(-7) + 3x + cWoahYour heart’s logarithmicIt inverts every positive thing that my love might tryIt’s just tough luck, isn’t it?My love will only touch your heart in an imaginary place likeI could try to envisageAny other number from infinity to 1 or 2And their sum you’d outdistanceWhile I’d still feel like a -1/12 withoutYou + MeIs all I’d ever dream to beYou see,You’re my -7x^(-8) + 3And equallyI’m your x^(-7) + 3x + cCan weBe weInstead of just you + me?We are not defined when x is 0And there are no real solutions hereMy favourite number’s 2But nothing’s really greater than youThe show was first performed at IF Oxford - the Oxford Science and Ideas Festival. Thanks to the London Mathematical Society (LMS) for financial support for the event.Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow in Mathematics at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Oxford Linear Algebra: Direct Sum of Vector Spaces Tom Rocks Maths 2023-05-26 | University of Oxford mathematician Dr Tom Crawford explains the concept of the direct sum of vector subspaces with several fully-worked examples. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: proprep.uk/info/TOM-Crawford Links to the other videos mentioned:Subspace Test - youtu.be/3_MxBlWQsgs Dimension Formula: youtu.be/lGo9E5dZ4f8Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20General%20Vector%20Spaces%20-%20workbook.pdfYou can also find several video lectures from ProPrep explaining the content covered in the video at the links below. Direct Sum: proprep.com/courses/all/linear-algebra/general-vector-spaces/subspaces/vid30219Sum of Subspaces: proprep.com/university-of-warwick/warwick-university/ma106/general-vector-spaces/subspaces/vid30220As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises. The video begins with the formal mathematical definition of a direct sum via the sum of vector spaces which only have the zero vector in their intersection. This is then shown to be equivalent to an alternative definition which says the representation via the sum of vectors is unique. Next we look at 3 fully-worked examples where we are asked to check if the given sum of vector spaces is a true direct sum or not. By checking the intersection we see that two of them are not a direct sum as their intersection contains a non-zero vector. For the third example the intersection is shown to be zero, so we then check whether the two subspaces span the larger vector space. By constructing a basis for each subspace, we see that this is indeed the case and thus conclude it is a true direct sum.Watch the other videos from the Oxford Linear Algebra series at the links below. Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): youtu.be/9pF__coVyEE Calculating the inverse of 2x2, 3x3 and 4x4 matrices: youtu.be/VKOaG3Ogf9Q What is the Determinant Function: youtube.com/watch?v=bLsBWVYSg0A The Easiest Method to Calculate Determinants: youtu.be/qniUv4EZB0w Eigenvalues and Eigenvectors Explained: youtu.be/8uISh6xyW7w Spectral Theorem Proof: youtu.be/ADwsk9G5s_8 Vector Space Axioms: youtu.be/draqOOUoWQM Subspace Test: youtu.be/3_MxBlWQsgs Basis, Spanning and Linear Independence: youtu.be/wdOFi8aUNp0Dimension Formula: youtu.be/lGo9E5dZ4f8Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Maths Speed Dating with Countdowns Rachel Riley Tom Rocks Maths 2023-05-17 | Oxford Mathematician Dr Tom Crawford asks TV superstar Rachel Riley randomly selected questions taken from speed dating websites. A special episode for National Numeracy Day 2023. Take the National Numeracy Challenge here: nationalnumeracy.org.uk/challenge/bnn #BigNumberNatter #NationalNumeracyDayAll questions with timestamps below.0:00 - Introduction2:50 – Why do you enjoy maths?4:48 – If you could invite anyone dead or alive to dinner who would it be?7:08 – What was your Oxford admissions interview like?10:49 – If you could commit one crime without being caught, what would it be?12:05 – What modules did you take during your degree?17:15 – What is one of your favourite maths facts? 21:19 – Do you have any advice for people who want to improve at maths?23:56 – Do you have any maths hacks that you can share with us? 25:23 – DIY or call someone?26:53 – What is the most adventurous thing you have ever done?29:18 – Who is your favourite band or musician?31:53 – Can you tell me something mathematical about Oxford?33:56 – Can you remember when you fell in love with maths?35:45 – Do you have a favourite number?38:08 - What is your favourite sports team?38:57 – Can you describe a recent time you’ve used maths in your daily life?42:41 – Honest or sparing someone’s feelings?More in the Maths Speed Dating series.@3blue1brown : youtube.com/watch?v=Xh2LVy7B5q0@MikeBoyd : youtu.be/BQnQsa-VDeg@DorFuchs : youtube.com/watch?v=nKr7NQJ_kM0@SimonClark : youtube.com/watch?v=W0vqiWir0-0@mathemaniac : youtu.be/RrMkTF-EE0MProduced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow in Mathematics at St Edmund Hall: seh.ox.ac.uk/people/tom-crawfordFor more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmathstwitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-mathsAll footage is used under a Fair Usage Policy for educational purposes, or where appropriate a Creative Commons Licence. Countdown footage copyright Channel 4.It's Not Rocket Science footage copyright ITV.Celebrity Mastermind footage copyright BBC One.Original links to clips used in the video.Countdown: youtube.com/watch?v=ZjCbWg4ZUAY It's Not Rocket Science: youtube.com/watch?v=zl5gf1D6MYo Celebrity Mastermind: youtube.com/watch?v=BlDuNQK-33I With thanks to:Rachel RileyNational NumeracyHoxey RooASMRebelHeels2000
Can ChatGPT Pass the Oxford University Admissions Test? Tom Rocks Maths 2023-05-12 | Oxford Mathematician Dr Tom Crawford puts ChatGPT through its paces with the Oxford Maths Admissions Test. Is the AI chatbot clever enough to pass the entrance exam?Accompanying Maple Learn worksheet: learn.maplesoft.com/doc/z4tp3rbsw6 Sign-up for Maple Learn Premium using the code TOMROCKSMATHS for a discounted subscription. Head to getlearn.maplesoft.com for more information.The exam is the 2021 Maths Admissions Test (MAT) which is taken by candidates applying to study Undergraduate Maths at the University of Oxford. The syllabus is based on material from the penultimate year of high school, which in the UK would mean the first year of A-level Maths. You can download the exam paper here: tomrocksmaths.files.wordpress.com/2023/05/test21.pdf?force_download=trueAnd the mark scheme is here: tomrocksmaths.files.wordpress.com/2023/05/websolutions21_0.pdf?force_download=trueWatch Tom take more exams via the designated playlist here: youtube.com/playlist?list=PLMCRxGutHqfm3t0IVJabEab6OasV9WLrl A-level Maths: youtube.com/watch?v=uupjxztr2q8A-level Further Maths: youtube.com/watch?v=C8o0vfQe_6A GCSE Maths: youtube.com/watch?v=hQVcv-T7IiYGCSE Further Maths: youtube.com/watch?v=KDn6N7eo9uo SAT Maths: youtube.com/watch?v=u0se7iWdNZw Cambridge University Admissions Test (STEP Paper) Part 1: youtube.com/watch?v=Qsthjv-YgngCambridge University Admissions Test (STEP Paper) Part 2: youtube.com/watch?v=f0Fm97oM608Check your working using the Maple Calculator App – available for free on Google Play and the App Store. Android: play.google.com/store/apps/details?id=com.maplesoft.companion&hl=enApple: apps.apple.com/us/app/maple-companion/id1466659419 Find out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel: youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ Produced by Dr Tom Crawford at the University of Oxford.Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Raising the Titanic with Ping-Pong Balls Tom Rocks Maths 2023-04-30 | Oxford Mathematician Dr Tom Crawford calculates how many ping-pong balls it would take to raise the Titanic. Full video here: youtu.be/54EmUfF1-0cFilmed for @duolingo DuoCon 2022 - a free global event at the intersection of language, learning, and technology: duolingo.com/duoconProduction by West Bound Filmworks Katy Romdall - Executive Producer Susan Kirsch - Creative DirectorTom Crawford - Writer, Video Co-Director, ResearcherRyan Claypool - Producer and Video Co-DirectorIlaria Ghattas - Program ManagerMatt Andrews - Aerials and Camera Operator Leo Brake & Juanru Zhao - Production AssistantsDr Tom Crawford is an Early-Career Teaching and Outreach Fellow in Mathematics at St Edmund Hall, at the University of Oxford: seh.ox.ac.uk/people/tom-crawfordFor more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmathstwitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Oxford Linear Algebra: Dimension Formula for Vector Spaces Tom Rocks Maths 2023-04-23 | University of Oxford mathematician Dr Tom Crawford introduces the dimension formula for vector spaces via a worked example before going through a complete proof. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: proprep.uk/info/TOM-Crawford Links to the other videos mentioned:Subspace Test - youtu.be/3_MxBlWQsgs Basis, Spanning and Linear Independence - youtu.be/wdOFi8aUNp0Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20General%20Vector%20Spaces%20-%20workbook.pdfYou can also find several video lectures from ProPrep explaining the content covered in the video at the links below. Dimension Formula: proprep.com/university-of-warwick/warwick-university/ma106/general-vector-spaces/subspaces/vid30223Sum of Subspaces: proprep.com/university-of-warwick/warwick-university/ma106/general-vector-spaces/subspaces/vid30220As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises. Watch the other videos from the Oxford Linear Algebra series at the links below. Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): youtu.be/9pF__coVyEE Calculating the inverse of 2x2, 3x3 and 4x4 matrices: youtu.be/VKOaG3Ogf9Q What is the Determinant Function: youtube.com/watch?v=bLsBWVYSg0A The Easiest Method to Calculate Determinants: youtu.be/qniUv4EZB0w Eigenvalues and Eigenvectors Explained: youtu.be/8uISh6xyW7w Spectral Theorem Proof: youtu.be/ADwsk9G5s_8 Vector Space Axioms: youtu.be/draqOOUoWQM Subspace Test: youtu.be/3_MxBlWQsgs Basis, Spanning and Linear Independence: youtu.be/wdOFi8aUNp0The video begins by defining the dimension of a vector space as the number of elements in its basis. This is then exemplified by looking at the vector space of polynomials up to degree n, which has a dimension of n+1. We then go through a fully worked example in R^4 by calculating explicitly the dimensions of the subspaces X and Y, the subspace X+Y, and the intersection of X and Y. This is used as motivation for the dimension formula: dim(X+Y) = dim(X) + dim(Y) - dim(X and Y).Finally, a complete proof of the dimension formula is presented where we construct a basis of the space X+Y which is shown to be both spanning and linearly independent.Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Maths Speed Dating with @mathemaniac Tom Rocks Maths 2023-04-16 | Trevor from Maths YouTube channel @mathemaniac answers randomly selected speed-dating-style questions with University of Oxford Mathematician Dr Tom Crawford. Full list of questions with timestamps below. More information on the UK Schooling Cambridge Summer Camps here: ukschooling.co.uk Videos mentioned in the interview:How hard was my Cambridge interview? youtube.com/watch?v=2n2XGNHsKjA From Miller to entering Cambridge at age 40 - George Green youtube.com/watch?v=o4ELd69ZgcI Two opposite games involving the golden ratio youtube.com/watch?v=UOAylfclg1404:01 Did you have any pets growing up?05:54 What is your dream job?08:29 Would you rather be forgotten or remembered for all the wrong reasons?11:13 What was the biggest trouble that you got into at school?14:23 What is your favourite movie?16:54 Can you tell me a joke?18:20 What was your best weekend this year?21:07 What are other YouTube channels that you watch?25:10 What are you currently watching?26:32 Do you have any nicknames?29:12 What is the most adventurous thing you have ever done?33:02 What time in history would you have liked to be born in, and why?35:50 What are your favourite places you have visited?39:52 What are you currently reading?43:36 Do you play video games?46:46 Can you describe your favourite theorem?52:00 What is your most embarrassing moment?Check out Trevor's channel @mathemaniac here: youtube.com/mathemaniac More in the Maths Speed Dating series:@3blue1brown : youtube.com/watch?v=Xh2LVy7B5q0@MikeBoyd : youtu.be/BQnQsa-VDeg@DorFuchs : youtube.com/watch?v=nKr7NQJ_kM0@SimonClark : youtube.com/watch?v=W0vqiWir0-0Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow in Mathematics at St Edmund Hall: seh.ox.ac.uk/people/tom-crawfordFor more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmathstwitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
The OLDEST Maths Books in Oxford Tom Rocks Maths 2023-04-06 | Oxford Mathematician Dr Tom Crawford looks through some of the oldest maths textbooks at the University of Oxford with assistance from the @StEdmundHall Librarian James Howarth. Link for Maple Learn worksheet below.Maple Learn worksheet on Parallelograms here: learn.maplesoft.com/doc/6p665268db/trm-parallelogramsSign-up for Maple Learn Premium using the code TOMROCKSMATHS for a discounted subscription. Head to getlearn.maplesoft.com for more information.List of books covered in the video:1. Euclid's "Elements"2. William Leybourne's "Cursus Mathematicus"3. Galileo's "System of the World"4. Kepler's "On The Six-Angled Snowflake"5. Robert Hooke's "Micrographia"6. Isaac Newton's "Principia Mathematica"7. Lewis Carroll's "The Game Of Logic"Highlights include: Galileo's drawing of the heliocentric solar system with planets orbiting the sun; Robert Hooke's drawings of snowflakes seen under a microscope for the first time; and Lewis Carroll's game of logic meant to be played by children. Some notes from James on the books.Fol. C 8 Mathematical collections and translations: the first tome. London: Printed by William Leybourn, MD CLXI [1661].Translations of Galileo, Kepler etc. First volume only, almost all copies of vol. 2 destroyed in the Great fire of London.Fol. O 13 William Leybourn Cursus Mathematicus, London: Printed for Thomas Basset, Benjamin Tooke, Thomas Sawbridge, Awnsham and John Churchill,16909 book, 900 page course in mathematics – tons of diagrams including practical ones for calculating field area or height of buildings etc. Full title is fun: Cursus mathematicus. Mathematical sciences, in nine books. : Comprehending arithmetick, vulgar, decimal, instrumental, algebraical. Geometry, plain, solid. Cosmography, cœlestial, terrestrial. Astronomy, theorical, practical. Navigation, plain, spherical.4° G 18 (1-7) Sammelband of 7 mathematical pamphlets including Kepler on Snowflakes.Two of the tracts we have the only copy in Oxford, two where it’s only us and the Box. Given by Timothy Goodwyn who transferred to Oxford from Leyden and became Bishop of Kildare.JJ94, John Newton, The scale of interest: or The use of decimal fractions, and the table of logarithmes, in the most easie and exact resolving all questions in anatocism, or compound interest; with tables of simple interest also at 6. per cent. per annum. Together with their use in the measuring of board, timber, stone, and gauging of cask, &c. very necesary for all carpenters, joyners, masons, glasiers, and all tradesmen whatsoever.Check your working using the Maple Calculator App – available for free on Google Play and the App Store. Android: play.google.com/store/apps/details?id=com.maplesoft.companion&hl=enApple: apps.apple.com/us/app/maple-companion/id1466659419 Don’t forget to check out the other videos in the ‘Oxford Calculus’ series – all links below.Full playlist: youtube.com/playlist?list=PLMCRxGutHqflZoTY8JCm1GRzCdGXvZ3_S Finding critical points for functions of several variables: youtu.be/Leomuu82-u8 Classifying critical points using the method of the discriminant: youtube.com/watch?v=5M_ts8Q2LEM Partial differentiation explained: youtu.be/RVwcBGzQcT8 Second order linear differential equations: youtu.be/F54yhRB9qDI Integrating factors explained: youtube.com/watch?v=ftqKuOfOX3E Solving simple PDEs: youtu.be/uztjxrGY6Jw Jacobians explained: youtube.com/watch?v=YqMelRryG8U Separation of variables integration technique explained: youtu.be/zk41c0vs9XQ Solving homogeneous first order differential equations: youtu.be/uqvqjbAcbL8Taylor’s Theorem explained with examples and derivation: youtube.com/watch?v=DULzJmUHN5gHeat Equation derivation: youtu.be/rz3cdzZXQmsSeparable Solutions to PDEs: youtube.com/watch?v=hcm-CgHFbwIHow to solve the Heat Equation: youtu.be/l6spigOZCOsFourier Series Derivation: youtu.be/LEonZYoW6sk Find out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel: youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmathstwitter.com/tomrocksmathsinstagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Oxford Linear Algebra: Basis, Spanning and Linear Independence Tom Rocks Maths 2023-03-31 | University of Oxford mathematician Dr Tom Crawford explains the terms basis, spanning and linear independence in the context of vectors and vector spaces. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: proprep.uk/info/TOM-Crawford Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20Vector%20Spaces%20over%20R%20(Rn)%20-%20workbook.pdf And here: api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20General%20Vector%20Spaces%20-%20workbook.pdf You can also find several video lectures from ProPrep explaining the content covered in the video at the links below.Basis for R^n: proprep.com/courses/all/linear-algebra/vector-spaces-over-r-(rn)/basis-for-rn/vid9962 Basis: proprep.com/courses/all/linear-algebra/general-vector-spaces/vector-basis/vid25730 Spanning: proprep.com/courses/all/linear-algebra/general-vector-spaces/linear-combination,-dependence-and-span/vid25722 Linear Independence: proprep.com/courses/all/linear-algebra/vector-spaces-over-r-(rn)/linear-combination,-dependence-and-span/vid10012 As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises. Watch the other videos from the Oxford Linear Algebra series at the links below. Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): youtu.be/9pF__coVyEE Calculating the inverse of 2x2, 3x3 and 4x4 matrices: youtu.be/VKOaG3Ogf9Q What is the Determinant Function: youtube.com/watch?v=bLsBWVYSg0A The Easiest Method to Calculate Determinants: youtu.be/qniUv4EZB0w Eigenvalues and Eigenvectors Explained: youtu.be/8uISh6xyW7w Spectral Theorem Proof: youtu.be/ADwsk9G5s_8 Vector Space Axioms: youtu.be/draqOOUoWQM Subspace Test: youtu.be/3_MxBlWQsgs The video begins with an intuitive example of a basis via the vector space of polynomials up to degree n. We then give the formal definition of a basis as a spanning set of linearly independent vectors. The terms spanning and linear independence are then formally defined with examples given for each. We also show the definition of linear independence is equivalent to showing that the only solution to a linear combination of the vectors being equal to zero is for all of the coefficients to be zero. Linear dependence is defined as the lack of linear independence, or when a vector in a set can be written as a linear combination of the other vectors in the set. Finally, we move on to a series of worked examples, beginning with several possible bases for the Cartesian plane R^2. We then look at examples of linearly independent and linearly dependent sets of vectors, and how to show this is the case. Finally, we construct two possible bases for 3D space R^3.Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Oxford University Mathematician sits Cambridge Entrance Exam (STEP Paper) PART 2 Tom Rocks Maths 2023-03-22 | Oxford Mathematician Dr Tom Crawford completes the STEP Exam which is used by Cambridge University as part of the admissions process to study Undergraduate Mathematics. Part 1 is here: youtu.be/Qsthjv-YgngMarking begins from 1:06:28The exam taken by Tom is the STEP Paper 2 from 2021. The exam forms part of the entrance requirements for admission to the University of Cambridge to study Undergraduate Maths. You can download the test for yourself here: tomrocksmaths.files.wordpress.com/2023/03/step-2-2021-question-paper.pdf And the mark scheme is here: tomrocksmaths.files.wordpress.com/2023/03/step-2-2021-examiners-report-and-mark-scheme.pdfThe exam is based on material covered in A-level Maths and AS-level Further Maths which are taken by 17-18 year old students in the UK as part of their high school education.Produced by Dr Tom Crawford at the University of Oxford.Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Oxford University Mathematician takes Cambridge Entrance Exam (STEP Paper) PART 1 Tom Rocks Maths 2023-03-09 | Oxford Mathematician Dr Tom Crawford completes the STEP Exam which is used by Cambridge University as part of the admissions process to study Undergraduate Mathematics. Part 2 is here: youtube.com/watch?v=f0Fm97oM608The exam taken by Tom is the STEP Paper 2 from 2021. The exam forms part of the entrance requirements for admission to the University of Cambridge to study Undergraduate Maths. You can download the test for yourself here: tomrocksmaths.files.wordpress.com/2023/03/step-2-2021-question-paper.pdf And the mark scheme is here: tomrocksmaths.files.wordpress.com/2023/03/step-2-2021-examiners-report-and-mark-scheme.pdfThe exam is based on material covered in A-level Maths and AS-level Further Maths which are taken by 17-18 year old students in the UK as part of their high school education.Produced by Dr Tom Crawford at the University of Oxford.Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Oxford University Mathematician takes Cambridge Entrance Exam (STEP Paper) - Trailer Tom Rocks Maths 2023-03-07 | University of Oxford Mathematician Dr Tom Crawford takes the Cambridge University Maths Entrance Exam (STEP Paper) with no preparation... Full video premieres 1pm (GMT) Thursday 9th March.Who will triumph in the battle of Oxford vs Cambridge? Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
How a Mathematician Takes a Penalty Kick Tom Rocks Maths 2023-03-03 | Oxford Mathematician Dr Tom Crawford calculates the perfect placement of a penalty kick and then tries to hit the target... Full video here: youtu.be/54EmUfF1-0cFilmed for @duolingo DuoCon 2022 - a free global event at the intersection of language, learning, and technology: duolingo.com/duoconProduction by West Bound Filmworks Katy Romdall - Executive Producer Susan Kirsch - Creative DirectorTom Crawford - Writer, Video Co-Director, ResearcherRyan Claypool - Producer and Video Co-DirectorIlaria Ghattas - Program ManagerMatt Andrews - Aerials and Camera Operator Leo Brake & Juanru Zhao - Production AssistantsDr Tom Crawford is an Early-Career Teaching and Outreach Fellow in Mathematics at St Edmund Hall, at the University of Oxford: seh.ox.ac.uk/people/tom-crawfordFor more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmathstwitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Oxford Calculus: Gradient (Grad) and Divergence (Div) Explained Tom Rocks Maths 2023-02-21 | University of Oxford Mathematician Dr Tom Crawford explains the gradient vector (Grad) and the divergence (Div) for scalar and vector functions. Test yourself with this accompanying FREE worksheet in Maple Learn: learn.maplesoft.com/doc/ipubec5cip/trm-grad-and-div-worksheet Sign-up for Maple Learn Premium using the code TOMROCKSMATHS for a discounted subscription. Head to getlearn.maplesoft.com for more information.Check your working using the Maple Calculator App – available for free on Google Play and the App Store. Android: play.google.com/store/apps/details?id=com.maplesoft.companion&hl=enApple: apps.apple.com/us/app/maple-companion/id1466659419 We begin with the formal definition of the gradient vector (Grad) and a visualisation of what it represents for a multivariable function. We then look at some examples with explicit calculation and 3D plots.The Divergence (Div) of a vector function is then introduced - both as an equation and via the physical interpretation of what it represents. We calculate the divergence for several vector fields and then show where the notation 'Grad Dot F' comes from with a derivation.Finally, the link between Grad, Div and the Laplacian is explored. Don’t forget to check out the other videos in the ‘Oxford Calculus’ series – all links below.Full playlist: youtube.com/playlist?list=PLMCRxGutHqflZoTY8JCm1GRzCdGXvZ3_S Finding critical points for functions of several variables: youtu.be/Leomuu82-u8 Classifying critical points using the method of the discriminant: youtube.com/watch?v=5M_ts8Q2LEM Partial differentiation explained: youtu.be/RVwcBGzQcT8 Second order linear differential equations: youtu.be/F54yhRB9qDI Integrating factors explained: youtube.com/watch?v=ftqKuOfOX3E Solving simple PDEs: youtu.be/uztjxrGY6Jw Jacobians explained: youtube.com/watch?v=YqMelRryG8U Separation of variables integration technique explained: youtu.be/zk41c0vs9XQ Solving homogeneous first order differential equations: youtu.be/uqvqjbAcbL8Taylor’s Theorem explained with examples and derivation: youtube.com/watch?v=DULzJmUHN5gHeat Equation derivation: youtu.be/rz3cdzZXQmsSeparable Solutions to PDEs: youtube.com/watch?v=hcm-CgHFbwIHow to solve the Heat Equation: youtu.be/l6spigOZCOsFourier Series Derivation: youtu.be/LEonZYoW6sk Find out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel: youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmathstwitter.com/tomrocksmathsinstagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Maths ASMR: Fireside Textbook Reading Tom Rocks Maths 2023-02-10 | Oxford mathematician Dr Tom Crawford reads from the textbook "Introduction to Real Analysis" by Donald Sherbert and Robert Bartle at fireside. Buy the book for yourself here: https://amzn.eu/d/f7sxwPM Part I: The algebraic and order properties of the real numbersPart II (16:59): Absolute value of the real linePart III (34:43): The completeness property of the real numbersProduced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Oxford Linear Algebra: Subspace Test Tom Rocks Maths 2023-02-05 | University of Oxford mathematician Dr Tom Crawford explains the subspace test for vector spaces. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: proprep.uk/info/TOM-Crawford Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: proprep.uk/Academic/DownloadBook?file=Vector%20Spaces%20over%20R%20(Rn)%20-%20workbook%20uk.pdf And here: proprep.uk/Academic/DownloadBook?file=Proprep%20-%20Linear%20Algebra%20-%20General%20Vector%20Spaces%20-%20workbook%20uk.pdf You can also find several video lectures from ProPrep explaining subspaces here: proprep.uk/general-modules/all/linear-algebra/vector-spaces-over-r-(rn)/subspaces/vid9982 And further videos explaining subspaces for more general vector spaces here: proprep.uk/general-modules/all/linear-algebra/general-vector-spaces/subspaces?ucid=0 As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises. Watch other videos from the Oxford Linear Algebra series at the links below. Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): youtu.be/9pF__coVyEE Calculating the inverse of 2x2, 3x3 and 4x4 matrices: youtu.be/VKOaG3Ogf9Q What is the Determinant Function: youtube.com/watch?v=bLsBWVYSg0A The Easiest Method to Calculate Determinants: youtu.be/qniUv4EZB0w Eigenvalues and Eigenvectors Explained: youtu.be/8uISh6xyW7w Spectral Theorem Proof: youtu.be/ADwsk9G5s_8 Vector Space Axioms: youtu.be/draqOOUoWQM The video begins with the definition of a subspace U contained in a vector space V, and some trivial examples for U = V and U = 0. The subspace test is then introduced and shown to be equivalent to the definition. The subspace test requires the zero vector to be contained in U, and any linear combination of vectors in U to also be contained in U. Finally, 3 fully worked examples are shown. First, we show that the x-y plane is a subspace of 3-dimensional coordinate space. Second, we show that for U and W subspaces of a vector space V, the intersection of U and W is always a subspace. Third, we show that the subspace of differentiable functions from the real numbers to the real numbers is a subspace of the vector space of all functions from R to R. Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here: beautifulequations.net/collections/tom-rocks-maths
Enter the Oxford University Essay Competition Teddy Rocks Maths Tom Rocks Maths 2023-01-27 | The 2023 Teddy Rocks Maths Essay Competition is open for entries! Everyone is eligible - submit your essay here: https://seh.ac/teddyrocksmaths This is your chance to write a short article about your favourite mathematical topic which could win you a cash prize of up to £100. All entries will be showcased at tomrocksmaths.com with the winners published on the St Edmund Hall website. St Edmund Hall (or Teddy Hall as it is affectionately known) is the college where Tom is based at the University of Oxford. Entries should be between 1000-2000 words and must be submitted as Microsoft Word documents or PDF files using the online form here: seh.ox.ac.uk/study/outreach/teddy-rocks-maths The closing date is Saturday 1st April 2023 and the winners will be announced in May/June. Two prizes of £50 are available for the overall winner and for the best essay from a high school student (defined as someone who is still at school when submitting their essay). There are no eligibility requirements – all you need is a passion for Maths and a flair for writing to participate! The winners will be selected by Dr Tom Crawford, Early Career Teaching and Outreach Fellow in Mathematics St Edmund Hall and the creator of the ‘Tom Rocks Maths’ outreach programme. All entries, including previous winners from the 2022, 2021 and 2020 editions can be found here: tomrocksmaths.com/teddy-rocks-maths If you have any questions or would like more information please get in touch with Tom using the contact form on his website: tomrocksmaths.com/contactProduced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow in Mathematics at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford All copyright footage used for educational purposes under a fair-usage policy. Breaking Bad: Copyright AMC / Sony PicturesThe Simpsons: Copyright 20th Century AnimationsThe following clips are used under a Creative Commons licence.Ethereal Sound - Oxford Aerial View: youtube.com/watch?v=PI7XaeaMCtQ Dheeraj Sahu - Math Formula: youtube.com/watch?v=DVyRgQHHoMwIceCreeper28 - Dragon Curve: youtube.com/watch?v=xeBcRsHguOY For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
How to Solve Every Tech Interview Measuring Question Tom Rocks Maths 2023-01-22 | Can you measure 1ml of water using unmarked beakers of size 5ml and 7ml? Learn how to solve tech interview "measuring questions" using Bezout's Lemma.We begin by filling and emptying the beakers until we arrive at a solution. By analysing what we have done, we are able to convert the problem into an algebraic equation known as a Linear Diophantine Equation. The solvability of Linear Diophantine Equations is then discussed by introducing the concept of "greatest common divisor" (GCD) and the definition of "co-prime". A further application of Bezout's Lemma to the Chinese Zodiac Calendar is explored by introducing the concepts of "leat common multiple" (LCM) and the 'Chinese Remainder Theorem". Produced by TRM intern Shucheng Li with assistance from Dr Tom Crawford. Shucheng is a second year mathematics undergraduate at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford If you would like to take part in the Tom Rocks Maths intern scheme, please get in touch using the contact form on the TRM website: tomrocksmaths.com/contact For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
How does RSA Cryptography work? Tom Rocks Maths 2023-01-13 | Oxford Sedleian Professor of Natural Philosophy Jon Keating explains the RSA Cryptography Algorithm. Get 25% off Blinkist premium and enjoy 2 memberships for the price of 1! Start your 7-day free trial by clicking here blinkist.com/tomrocksmaths RSA encryption is used everyday to secure information online, but how does it work? And why is it referred to as a type of public key cryptography? Professor Jon Keating worked alongside the UK intelligence agency GCHQ for many years, and therefore knows a thing or two about encrypting secret messages. Here, he explains how the RSA algorithm works in general, and goes through 2 worked examples with small prime numbers. The algorithm relies on the idea that whilst it is very easy to multiply two prime numbers together, it is extremely difficult to break up a large number (with several hundred digits) back into its prime factors. Using some clever results from Number Theory - including Fermat's Little Theorem and the Euler Totient Function - the message can be decrypted only if you know the original prime factors. This means advertising the product of the primes, or 'public key', enables people to send you a message without compromising the security of the encryption system. Even if the message is intercepted, it can only be decoded with knowledge of the prime factors - and these are incredibly difficult to obtain. This video is sponsored by Blinkist.Additional images and footage are used under a creative commons licence – links below.Blockchain Travel: youtube.com/watch?v=_Ui3s-Vp8_k Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Oxford Linear Algebra: What is a Vector Space? Tom Rocks Maths 2023-01-05 | University of Oxford mathematician Dr Tom Crawford explains the vector space axioms with concrete examples. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: proprep.uk/info/TOM-Crawford Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: proprep.uk/Academic/DownloadBook?file=Vector%20Spaces%20over%20R%20(Rn)%20-%20workbook%20uk.pdf And here: proprep.uk/Academic/DownloadBook?file=Proprep%20-%20Linear%20Algebra%20-%20General%20Vector%20Spaces%20-%20workbook%20uk.pdf You can also find several video lectures from ProPrep explaining the vector space R^n here: proprep.uk/general-modules/all/linear-algebra/vector-spaces-over-r-(rn)/the-vector-space-rn?ucid=0 And further videos explaining more general vector spaces here: proprep.uk/general-modules/all/linear-algebra/general-vector-spaces/vector-spaces?ucid=0 As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises. Watch other videos from the Oxford Linear Algebra series at the links below. Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): youtu.be/9pF__coVyEE Calculating the inverse of 2x2, 3x3 and 4x4 matrices: youtu.be/VKOaG3Ogf9Q What is the Determinant Function: youtube.com/watch?v=bLsBWVYSg0A The Easiest Method to Calculate Determinants: youtu.be/qniUv4EZB0w Eigenvalues and Eigenvectors Explained: youtu.be/8uISh6xyW7w Spectral Theorem Proof: youtu.be/ADwsk9G5s_8 The video begins by introducing the vector space axioms. We first define the addition and scalar multiplication maps, before listing the 8 axioms that must be satisfied: commutativity of addition, associativity of addition, the existence of an identity element, the existence of additive inverses, distributivity of scalar multiplication over addition, distributivity of scalar multiplication over field addition, interaction of scalar multiplication and field multiplication, and the existence of an identity for scalar multiplication.Each axiom is then verified for 3D coordinate vectors as a canonical example. Finally, further properties of vector spaces are discussed, such as the uniqueness of identity elements and inverses. A full proof using the axioms is provided to show the additive identity is unique. Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Pokémon Catch Rate Formula Explained Tom Rocks Maths 2022-12-29 | University of Oxford Mathematician Dr Tom Crawford breaks down the Pokémon Catch Rate Formula for the Generation I video games. Explore the formula for yourself using this FREE interactive worksheet from Maple Learn: learn.maplesoft.com/d/AKFPLNCOIOEUKQFKDNOHASGUGOJRMJGQIKJOFOLTFPKKMRAGGKGLCFBSJFBUORJLITJLCJGSNRJMBIOSCIAMBQFILTOJEOARIMJN Check your working using the Maple Calculator App – available for free on Google Play and the App Store. Android: play.google.com/store/apps/details?id=com.maplesoft.companion&hl=enApple: apps.apple.com/us/app/maple-companion/id1466659419 Find out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel: youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ The video begins with an explanation of the full catch rate formula, featuring maxima, minima and floor functions. This is then simplified to arrive at the more commonly used formula for the approximate probability of catching a wild Pokémon in the Generation I video games. Each variable is explained in turn, and the effects of status, HP, and ball type are discussed. Next, the formula is derived using the catch algorithm for the game. This is outlined in detail on ‘Bulbapedia’ here: bulbapedia.bulbagarden.net/wiki/Catch_rate Finally, using some visualisations from Maple Learn, the probability of success when using a great ball is compared to that for an ultra ball for a Pokémon with catch rate 45 (such as Butterfree) and a Pokémon with catch rate 3 (such as Zapdos). You can explore the formulae for yourself for FREE in the accompanying interactive Maple Learn worksheet here: learn.maplesoft.com/d/AKFPLNCOIOEUKQFKDNOHASGUGOJRMJGQIKJOFOLTFPKKMRAGGKGLCFBSJFBUORJLITJLCJGSNRJMBIOSCIAMBQFILTOJEOARIMJN Full list of Pokémon catch rates on ‘Bulbapedia’ here: bulbapedia.bulbagarden.net/wiki/List_of_Pok%C3%A9mon_by_catch_rate All in-game footage and images are copyright The Pokémon Company, Gamefreak and Nintendo. They are shown under a fair-use policy for the purposes of education and review. Darkmurkrow video footage used under a Creative Commons licence. Kangaskhan: youtube.com/watch?v=6tcL47jn_5o Zapdos: youtube.com/watch?v=_zoyyYlja70 Snorlax: youtube.com/watch?v=WX7zwUps1dg Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths With thanks toNintendoThe Pokémon CompanyGamefreakDarkmurkrow
Maths Speed Dating with @SimonClark Tom Rocks Maths 2022-12-22 | @SimonClark answers randomly selected speed-dating-style questions with University of Oxford Mathematician Dr Tom Crawford. The video is best summarised by two nerds having way too much fun. Featuring warhammer, life as a full-time YouTuber, Dad jokes, and something or other maths related... Full list of questions with timestamps below. 3:04 - Do you have any nicknames?4:13 - What are you currently watching?8:47 - What is your favourite millennium problem?12:33 - What are your favourite places you have visited?15:39 - If you could commit one crime and get away with it, what would it be?17:36 - What was the biggest trouble you got into at school?20:31 - What is your favourite number?23:29 - Honesty or sparing someone's feelings?25:18 - What is the most adventurous thing you have ever done?29:22 - What would be the title of your biography?31:14 - What is your favourite equation?34:44 - What is your dream job?38:24 - What colour best describes your personality?40:20 - What is your best joke?41:32 - What are other YouTube channels you watch?45:44 - Exploring or lazing on the beach?48:00 - Would you rather be forgotten or remembered for all the wrong reasons?49:56 - What was your most embarrassing moment?Check out Simon's channel here: youtube.com/c/simonoxfphys More in the Maths Speed Dating series:@3blue1brown: youtube.com/watch?v=Xh2LVy7B5q0@MikeBoyd: youtu.be/BQnQsa-VDeg@DorFuchs: youtube.com/watch?v=nKr7NQJ_kM0Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow in Mathematics at St Edmund Hall: seh.ox.ac.uk/people/tom-crawfordFor more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmathstwitter.com/tomrocksmaths instagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Oxford Calculus: Fourier Series Derivation Tom Rocks Maths 2022-12-13 | University of Oxford Mathematician Dr Tom Crawford explains how to derive the Fourier Series coefficients for any periodic function. Accompanying FREE worksheet courtesy of Maple Learn here: learn.maplesoft.com/d/DLEJJJNPPUILFLPNCTLRGSJHCTMHCRDJCTAUIRKKGSGPBSFHJNIFNRPODNPFBLJROIHMNIPIOUKLPHNIILISJMFLKULTNLGPHGNS Check your working using the Maple Calculator App – available for free on Google Play and the App Store. Android: play.google.com/store/apps/details?id=com.maplesoft.companion&hl=enApple: apps.apple.com/us/app/maple-companion/id1466659419 We start by deriving the orthogonality relations for sine and cosine, which are essential for the derivations of the Fourier Series coefficients. The integral relations rely on the trigonometric ‘product-to-sum formulae’ which enable the product of two sine or cosine terms to be separated and thus integrated directly. The delta function is also introduced to help to simplify the notation.We then assume that a Fourier Series of the required form exists, with as yet unknown coefficients a0, an and bn. These are derived by first integrating the entire equation from -L to L to get a0; then multiplying by cosine and integrating to get the an coefficients for each n; and finally multiplying by sine and integrating to get the bn coefficients for each n. The integrals are evaluated using the previously derived orthogonality relations.Finally, the interchanging of the summation and integral signs is addressed with a very brief discussion of uniform convergence and what this means in the context of a series. Don’t forget to check out the other videos in the ‘Oxford Calculus’ series – all links below.Full playlist: youtube.com/playlist?list=PLMCRxGutHqflZoTY8JCm1GRzCdGXvZ3_S Finding critical points for functions of several variables: youtu.be/Leomuu82-u8 Classifying critical points using the method of the discriminant: youtube.com/watch?v=5M_ts8Q2LEM Partial differentiation explained: youtu.be/RVwcBGzQcT8 Second order linear differential equations: youtu.be/F54yhRB9qDI Integrating factors explained: youtube.com/watch?v=ftqKuOfOX3E Solving simple PDEs: youtu.be/uztjxrGY6Jw Jacobians explained: youtube.com/watch?v=YqMelRryG8U Separation of variables integration technique explained: youtu.be/zk41c0vs9XQ Solving homogeneous first order differential equations: youtu.be/uqvqjbAcbL8Taylor’s Theorem explained with examples and derivation: youtube.com/watch?v=DULzJmUHN5gHeat Equation derivation: youtu.be/rz3cdzZXQmsSeparable Solutions to PDEs: youtube.com/watch?v=hcm-CgHFbwIHow to solve the Heat Equation: youtu.be/l6spigOZCOsFind out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel: youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmathstwitter.com/tomrocksmathsinstagram.com/tomrocksmathsGet your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-maths
Korean SAT Exam Question with @DrPKMath Tom Rocks Maths 2022-12-03 | University of Oxford Mathematician Dr Tom Crawford works through his solution to a killer question from the 2022 Korean Suneung CSAT exam with some help from @DrPKMath. Use my link gauthmath.onelink.me/SUq5/d58dv886 to download Gauthmath and don't forget to use code 4SBW8X to get a 1 month discount now!Check out PK's channel here: youtube.com/c/DrPKMath The question is taken from the Korean SAT exam taken in November 2022. This was apparently a 'killer' question which hardly any students were able to solve within the time limit. The exam is taken at the end of high school as part of the college admissions process in the Republic of Korea (South Korea).PK begins by explaining a little about the exam before sharing the question. We are asked to work out the value of a cubic function at the point x=2, but in order to determine the coefficients of the cubic we must solve a series of equations involving exponential functions, trigonometric functions, derivatives and composite functions. The full question is as follows: A cubic function f(x) with positive leading coefficient, the function g(x) = exp(sin(pi*x)) - 1, and the composite function h(x) = g(f(x)) on the domain of all real numbers, satisfy the following conditions:1) Function h(x) has the maximum value of 0 when x is 0.2) In the open interval (0, 3), there are 7 distinct real solutions to h(x) = 1.3). f(3)= 1/2, f’(3)=0, and f(2)= p/q. What is p+q? (where p and q are coprime integers).This video is sponsored by Gauthmath.Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here: beautifulequations.net/collections/tom-rocks-maths
The Pokédex is WEIRD (with @standupmaths) Tom Rocks Maths 2022-11-23 | Oxford University Mathematician Dr Tom Crawford explores the maths behind some of the strangest Pokédex entries… with a cameo from Matt Parker @standupmaths. Check out his video on the Pokémon Calculator here: youtu.be/65WoL6e728Q Content based on the original article by TRM intern Sam Flower. Read it here: tomrocksmaths.com/2021/12/10/maths-of-the-pokedexAll in-game footage and images are copyright The Pokémon Company, Gamefreak and Nintendo. They are shown under a fair-use policy for the purposes of education and review. Here are the Pokémon considered and why their Pokédex entries are ridiculous...Wailord: has a density lighter than air, despite living underwater.Cosmoem: has a density higher than a black hole.Magcargo: 50 of them could power the UK.Bewear: creates more force than a car crash at 30mph.Rhyhorn: has more power than 1000 lbs of explosive.Blaziken: has a standing jump of over 100m.Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here:beautifulequations.net/collections/tom-rocks-mathsWith thanks to Matt ParkerSam FlowerNintendoGamefreakThe Pokémon CompanyDaddyGamer FredBilovitskiyAndrew WhaleMars Inc.NASABritaxChris VidsU.S. Navy Seal and SWCC
Oxford Linear Algebra: Spectral Theorem Proof Tom Rocks Maths 2022-11-18 | University of Oxford mathematician Dr Tom Crawford goes through a full proof of the Spectral Theorem. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: proprep.uk/info/TOM-Crawford Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: proprep.uk/Academic/DownloadBook?file=Proprep%20-%20Linear%20Algebra%20-%20Inner%20Product%20Spaces%20-%20workbook%20uk.pdf You can also find several video lectures from ProPrep explaining the Spectral Theorem here: proprep.uk/general-modules/all/linear-algebra/inner-product-spaces/the-spectral-theorem/vid30947 And further videos explaining the Gram-Schmidt process are here: proprep.uk/general-modules/all/linear-algebra/inner-product-spaces/gram-schmitt-process/vid27109 Finally, fully worked video solutions from ProPrep instructors are here: proprep.uk/general-modules/all/linear-algebra/inner-product-spaces/the-spectral-theorem/vid30951 Watch other videos from the Oxford Linear Algebra series at the links below. Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): youtu.be/9pF__coVyEE Calculating the inverse of 2x2, 3x3 and 4x4 matrices: youtu.be/VKOaG3Ogf9Q What is the Determinant Function: youtube.com/watch?v=bLsBWVYSg0A The Easiest Method to Calculate Determinants: youtu.be/qniUv4EZB0w Eigenvalues and Eigenvectors Explained: youtu.be/8uISh6xyW7w The video goes through a full proof of the Spectral Theorem, which states that every real, symmetric matrix, has real eigenvalues, and can be diagonalised using a basis of its eigenvectors. The first part of the proof uses the eigenvalue equation to show that any eigenvalue is in fact equal to its complex conjugate, and thus is real. The second part of the proof shows that a matrix similarity transformation using an orthogonal matrix exists, and results in a diagonal matrix. We first construct an orthonormal basis (where the first vector is an eigenvector) using the Gram-Schmidt process, and then use these vectors as the columns of our orthogonal matrix. Next, we show that the resulting similarity matrix is also symmetric. This then allows us to conclude that the first row and first column are diagonal as required. The final step is to use induction on the size of the matrix. Assuming the result is true for a (n-1) x (n-1) matrix, we use our earlier calculation to construct the final orthogonal matrix, and show that when it is used as a change of basis matrix the result is diagonal, as we wanted.Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.facebook.com/tomrocksmaths twitter.com/tomrocksmaths instagram.com/tomrocksmaths Get your Tom Rocks Maths merchandise here: beautifulequations.net/collections/tom-rocks-maths
