Tom Rocks Maths
Sir Michael Atiyah Riemann Hypothesis Proof Lecture
updated
*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.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:
beautifulequations.net/collections/tom-rocks-maths
Part 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.pdf
The questions covered in the video are as follows:
1:26 – Q16: Force Diagram
20:47 – Q18: Projectile Motion
49:44 – Multiple choice section: Q1, Q2, Q3, Q4, Q5, Q10, Q13
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 to
Physics Online
Lewis Matheson
St Edmund Hall
University of Oxford
Nicoguaro: en.wikipedia.org/wiki/Stress%E2%80%93strain_curve#/media/File:Stress_strain_ductile.svg
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=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
Steven 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.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 to
Noga Alon
Hong Kong Laureate
The Shaw Prize
Numberphile
3Blue1Brown: youtube.com/watch?v=yuVqxCSsE7c
Eurovision Song Contest
Byle do Maja: youtube.com/watch?v=e-qliUeynjY
Filip Frantál: youtube.com/watch?v=KB4TL6gIJMc
*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
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 to
Steven Balbus
Hong Kong Laureate Forum
Dan Addison
ESA/Hubble
Tudor
ESO
EHT Collaboration
NASA’s Goddard Space Flight Center/Jeremy Schnittman
Gary Settles
Josh Estey/AUSaid
WHOI
Simons Centre
Peter Mercator
MVLAN
Berkeley
The Wire
Rogelio Bernal Andreo
ESO/P. Kervella
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.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 to
Alvaro Gonzalez Hernandez
Txoki
Oxford University Micro-Internship Scheme
Science Museum London
Vysotsky (Wikipedia)
Rob Koopman
Ag2gaeh (Wikipedia)
Sam Derbyshire
Stephan Heiss
Sign-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/@singingbanana
Thanks to @isaacnewtoninstitute for giving us access to the lecture room.
With thanks to:
James Grime
The Isaac Newton Institute
Maplesoft
Queens' College
Centre for Mathematical Sciences
Trinity College
Cullinan Studio
It's No Game
CMGlee
Dake
Joe Double
Physics Classroom
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
Recorded at the 2023 Heidelberg Laureate Forum. Find out more about the event at heidelberg-laureate-forum.org
The 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=6q8LkyBaPTc
Where 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 to
Efim Zlemanov
Heidelberg Laureate Forum
Heidelberg Laureate Forum Foundation
Math Genealogy
IMU
Links to the other videos mentioned:
Inner Product Space youtube.com/watch?v=ywjA25xFuew
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.pdf
You 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/vid26063
As 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/wdOFi8aUNp0
Dimension Formula: youtu.be/lGo9E5dZ4f8
Direct Sum: youtube.com/watch?v=b1VZm4Oecq4
Linear Transformations: youtu.be/UF59Mok4fsQ
Rank Nullity Theorem: youtu.be/yZ52iqxaiCc
Inner 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.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:
beautifulequations.net/collections/tom-rocks-maths
Check out Proprep with a 30-day free trial here: proprep.uk/info/TOM-Crawford
University 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
Sign-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=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
Links to Daniel's talks at the 2023 HLF.
Lightning Talks: youtube.com/watch?v=euWWcuGdCwM
Sparks Session: youtube.com/watch?v=IHEfrVrYQSI
Links to the two publications mentioned in the video.
Spielman: arxiv.org/abs/1911.03071
Spencer: jstor.org/stable/2000258
Link to interview with Daniel at HLF5: youtube.com/watch?v=SoVRmfFrnhs
With thanks to:
Daniel Spielman
Heidelberg Laureate Forum Foundation
Photograph of Commodore PET by Rama, Wikimedia Commons, Cc-by-sa-2.0-fr
Photograph of Notebooks by Brandon Schulman for Quanta Magazine
Tom Geller
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
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 Societies
bbno$ - 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.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:
beautifulequations.net/collections/tom-rocks-maths
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.pdf
You 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/vid10133
As 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/wdOFi8aUNp0
Dimension Formula: youtu.be/lGo9E5dZ4f8
Direct Sum: youtube.com/watch?v=b1VZm4Oecq4
Linear Transformations: youtu.be/UF59Mok4fsQ
Rank Nullity Theorem: youtu.be/yZ52iqxaiCc
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:
beautifulequations.net/collections/tom-rocks-maths
Check out Proprep with a 30-day free trial here: proprep.uk/info/TOM-Crawford
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.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:
beautifulequations.net/collections/tom-rocks-maths
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:
beautifulequations.net/collections/tom-rocks-maths
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/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
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.com
YouTube: youtube.com/tomrocksmaths
Twitter: twitter.com/tomrocksmaths
Facebook: facebook.com/tomrocksmaths
Instagram: instagram.com/tomrocksmaths
Follow JFM on:
Website: cambridge.org/core/journals/journal-of-fluid-mechanics
Youtube: youtube.com/user/CambridgeUPAcPro
Twitter: twitter.com/JFluidMech
Follow the UK Fluids Network on:
Website: fluids.ac.uk
Twitter: twitter.com/UKFluidsNetwork
--------
Featuring "The effect of seam orientation on the flight of a baseball": doi.org/10.1103/APS.DFD.2019.GFM.V0038
The 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=NuDMCgTpg8U
Authors:
Andrew Smith, Utah State University
Rob Friedman, Pitching Ninja
Nazmus Sakib, Utah State University
John Garrett, Utah State University
Barton Smith, Utah State University
Publication:
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.jpg
Trevor Bauer image copyright Erik Dros (Creative Commons 2.0 licence): commons.wikimedia.org/wiki/File:Trevor_Bauer_(47903072211).jpg
With thanks to:
Barton Smith
Andrew Smith
AugustineMLB
Fangraphs Prospects
Horia Varlan
Erik Drost
JFM
UK Fluid Network
APS DFD 2019
All MLB footage is presented under a fair usage policy for the purpose of education. It remains copyright of MLB.
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.pdf
You 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/vid10197
As 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/wdOFi8aUNp0
Dimension Formula: youtu.be/lGo9E5dZ4f8
Direct 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/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
Check out Proprep with a 30-day free trial here: proprep.uk/info/TOM-Crawford
Original song "Total Eclipse of the Hearth - Bonnie Tyler": youtube.com/watch?v=lcOxhH8N3Bo
Original 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 identities
every 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
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.pdf
You 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/vid10223
As 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/wdOFi8aUNp0
Dimension Formula: youtu.be/lGo9E5dZ4f8
Direct 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/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
*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
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.pdf
A 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.pdf
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
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=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
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 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 Theorem
Q3. Holomorphic Functions, Cauchy-Riemann Equations, Weierstrass M-test
Q4. Contour Integration, Residue Theorem, Laurent's Theorem
Q5. Cauchy Integral Formula
Q6. Stereographic Projection, Conformal Maps
Filmed 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
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
You + me
Is all I’d ever dream to be
You see
You’re my -7x^(-8) + 3
And equally
I’m your x^(-7) + 3x + c
Woah
My love’s exponential
Its domain is boundless and it reaches into infinity
While at first incremental
The rate at which it’s growing is increasing gradually
They say I resemble
e^(iπ), beautiful in all I do
But I am just helpless
‘Cos I still feel like a -1 without
You + Me
Is all I’d ever dream to be
You see
You’re my -7x^(-8) + 3
And equally
I’m your x^(-7) + 3x + c
Woah
Your heart’s logarithmic
It inverts every positive thing that my love might try
It’s just tough luck, isn’t it?
My love will only touch your heart in an imaginary place like
I could try to envisage
Any other number from infinity to 1 or 2
And their sum you’d outdistance
While I’d still feel like a -1/12 without
You + Me
Is all I’d ever dream to be
You see,
You’re my -7x^(-8) + 3
And equally
I’m your x^(-7) + 3x + c
Can we
Be we
Instead of just you + me?
We are not defined when x is 0
And there are no real solutions here
My favourite number’s 2
But nothing’s really greater than you
The 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/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
Links to the other videos mentioned:
Subspace Test - youtu.be/3_MxBlWQsgs
Dimension Formula: youtu.be/lGo9E5dZ4f8
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-%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.
Direct Sum: proprep.com/courses/all/linear-algebra/general-vector-spaces/subspaces/vid30219
Sum of Subspaces: proprep.com/university-of-warwick/warwick-university/ma106/general-vector-spaces/subspaces/vid30220
As 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/wdOFi8aUNp0
Dimension Formula: youtu.be/lGo9E5dZ4f8
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
#BigNumberNatter #NationalNumeracyDay
All questions with timestamps below.
0:00 - Introduction
2:50 – 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-EE0M
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
All 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 Riley
National Numeracy
Hoxey Roo
ASMRebel
Heels2000
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=true
And the mark scheme is here: tomrocksmaths.files.wordpress.com/2023/05/websolutions21_0.pdf?force_download=true
Watch Tom take more exams via 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
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
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/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
Filmed for @duolingo DuoCon 2022 - a free global event at the intersection of language, learning, and technology: duolingo.com/duocon
Production by West Bound Filmworks
Katy Romdall - Executive Producer
Susan Kirsch - Creative Director
Tom Crawford - Writer, Video Co-Director, Researcher
Ryan Claypool - Producer and Video Co-Director
Ilaria Ghattas - Program Manager
Matt Andrews - Aerials and Camera Operator
Leo Brake & Juanru Zhao - Production Assistants
Dr 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-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
Links to the other videos mentioned:
Subspace Test - youtu.be/3_MxBlWQsgs
Basis, Spanning and 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-%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.
Dimension Formula: proprep.com/university-of-warwick/warwick-university/ma106/general-vector-spaces/subspaces/vid30223
Sum of Subspaces: proprep.com/university-of-warwick/warwick-university/ma106/general-vector-spaces/subspaces/vid30220
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
Basis, Spanning and Linear Independence: youtu.be/wdOFi8aUNp0
The 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/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
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=UOAylfclg14
04:01 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=W0vqiWir0-0
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
Maple Learn worksheet on Parallelograms here: learn.maplesoft.com/doc/6p665268db/trm-parallelograms
Sign-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,1690
9 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=en
Apple: 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/uqvqjbAcbL8
Taylor’s Theorem explained with examples and derivation: youtube.com/watch?v=DULzJmUHN5g
Heat Equation derivation: youtu.be/rz3cdzZXQms
Separable Solutions to PDEs: youtube.com/watch?v=hcm-CgHFbwI
How to solve the Heat Equation: youtu.be/l6spigOZCOs
Fourier 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/tomrocksmaths
twitter.com/tomrocksmaths
instagram.com/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
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/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
Marking begins from 1:06:28
The 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.pdf
The 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/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
The 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.pdf
The 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/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
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/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
Filmed for @duolingo DuoCon 2022 - a free global event at the intersection of language, learning, and technology: duolingo.com/duocon
Production by West Bound Filmworks
Katy Romdall - Executive Producer
Susan Kirsch - Creative Director
Tom Crawford - Writer, Video Co-Director, Researcher
Ryan Claypool - Producer and Video Co-Director
Ilaria Ghattas - Program Manager
Matt Andrews - Aerials and Camera Operator
Leo Brake & Juanru Zhao - Production Assistants
Dr 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-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
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=en
Apple: 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/uqvqjbAcbL8
Taylor’s Theorem explained with examples and derivation: youtube.com/watch?v=DULzJmUHN5g
Heat Equation derivation: youtu.be/rz3cdzZXQms
Separable Solutions to PDEs: youtube.com/watch?v=hcm-CgHFbwI
How to solve the Heat Equation: youtu.be/l6spigOZCOs
Fourier 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/tomrocksmaths
twitter.com/tomrocksmaths
instagram.com/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
Buy the book for yourself here: https://amzn.eu/d/f7sxwPM
Part I: The algebraic and order properties of the real numbers
Part II (16:59): Absolute value of the real line
Part III (34:43): The completeness property of the real numbers
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
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
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/contact
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
All copyright footage used for educational purposes under a fair-usage policy.
Breaking Bad: Copyright AMC / Sony Pictures
The Simpsons: Copyright 20th Century Animations
The 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=DVyRgQHHoMw
IceCreeper28 - 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
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
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
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
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
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 to
Nintendo
The Pokémon Company
Gamefreak
Darkmurkrow
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_kM0
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
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
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/uqvqjbAcbL8
Taylor’s Theorem explained with examples and derivation: youtube.com/watch?v=DULzJmUHN5g
Heat Equation derivation: youtu.be/rz3cdzZXQms
Separable Solutions to PDEs: youtube.com/watch?v=hcm-CgHFbwI
How to solve the Heat Equation: youtu.be/l6spigOZCOs
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/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
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
Content based on the original article by TRM intern Sam Flower. Read it here: tomrocksmaths.com/2021/12/10/maths-of-the-pokedex
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.
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-maths
With thanks to
Matt Parker
Sam Flower
Nintendo
Gamefreak
The Pokémon Company
DaddyGamer Fred
Bilovitskiy
Andrew Whale
Mars Inc.
NASA
Britax
Chris Vids
U.S. Navy Seal and SWCC
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