Computing Improper Integrals using the Residue Theorem | Cauchy Principal Value @FacultyofKhan

Faculty of Khan Welcome back to a new school year! In this video, I begin by defining the Cauchy Principal Value and proving a couple of theorems about it. Then, I use those theorems to establish a technique which may be used to evaluate improper integrals of certain rational functions, after which I conclude by applying this technique to a simple example.

If you would like me to do more complicated examples (since some of you may like them), let me know in the comments! For now though, I'm going to temporarily move away from Complex Variables and focus on my other playlists (though I do plan on returning for videos on topics like conformal mapping *wink*).

Questions/requests? Let me know in the comments below!

How to find Residues: youtube.com/watch?v=sSj7z-pz-yY
Prereqs: The videos before this one on the playlist: youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB

Lecture Notes: drive.google.com/file/d/0BzC45hep01Q4SWFiaW8wMzFqekk/view?usp=sharing&resourcekey=0-k4crBBk2gN-IMT2TZHcXfQ
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

updated 7 years ago

Computing Improper Integrals using the Residue Theorem | Cauchy Principal Value

Faculty of Khan 2017-08-28 | Welcome back to a new school year! In this video, I begin by defining the Cauchy Principal Value and proving a couple of theorems about it. Then, I use those theorems to establish a technique which may be used to evaluate improper integrals of certain rational functions, after which I conclude by applying this technique to a simple example.

If you would like me to do more complicated examples (since some of you may like them), let me know in the comments! For now though, I'm going to temporarily move away from Complex Variables and focus on my other playlists (though I do plan on returning for videos on topics like conformal mapping *wink*).

Questions/requests? Let me know in the comments below!

How to find Residues: youtube.com/watch?v=sSj7z-pz-yY
Prereqs: The videos before this one on the playlist: youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB

Lecture Notes: drive.google.com/file/d/0BzC45hep01Q4SWFiaW8wMzFqekk/view?usp=sharing&resourcekey=0-k4crBBk2gN-IMT2TZHcXfQ
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Real Analysis: Introduction to Functions

Faculty of Khan 2023-03-30 | To try everything Brilliant has to offer—free—for a full 30 days, visit http://brilliant.org/FacultyofKhan/. The first 200 of you will get 20% off Brilliant’s annual premium subscription.

This video introduces #Functions in #realanalysis. I discuss inverse functions, composite functions, and bijective/injective/surjective functions. Questions/requests? Let me know in the comments!

Lecture Notes: drive.google.com/file/d/1n9eZeRn55rKwlGGe-42n5GwjMhM9z4NZ/view?usp=share_link
Pre-reqs: The previous videos in this playlist: youtube.com/playlist?list=PLdgVBOaXkb9DOwrQqCGNQQwKuRXkj9pIu
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
Tchsurvives
Patapom
Alec Dinerstein
Eugene Bulkin
Andy Johnston
chosenthomas
Lina Stritt

Chapters:
0:00 - Special thanks to Brilliant!
0:50 - Real Analysis: Introduction to Functions

Duhamels Principle for Partial Differential Equations

Faculty of Khan 2023-03-14 | To try everything Brilliant has to offer—free—for a full 30 days, visit http://brilliant.org/FacultyofKhan/. The first 200 of you will get 20% off Brilliant’s annual premium subscription.

This video introduces #DuhamelsPrinciple, and solves a #PDE problem using this principle alongside the #LaplaceTransform.

Questions/requests? Let me know in the comments! This video was sponsored by Brilliant.

Lecture Notes: drive.google.com/file/d/1JRBB-D64PoK5bFmwNL8jJY9z9fANwNYP/view?usp=share_link
Pre-reqs: The previous videos in this playlist: youtube.com/playlist?list=PLdgVBOaXkb9Ab7UM8sCfQWgdbzxkXTNVD
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
Tchsurvives
Patapom
Alec Dinerstein
Eugene Bulkin
Andy Johnston
chosenthomas
Lina Stritt

Chapters:
0:00 - Special thanks to Brilliant!
0:47 - Duhamel's Principle for Partial Differential Equations

Solving the Schrodinger Equation for the Quantum Harmonic Oscillator | Part 1

Faculty of Khan 2023-02-14 | To try everything Brilliant has to offer—free—for a full 30 days, visit http://brilliant.org/FacultyofKhan/. The first 200 of you will get 20% off Brilliant’s annual premium subscription.

In this video, I introduce the #QuantumHarmonicOscillator and begin to find the solution to the time-independent #SchrodingerEquation for this system!

Questions/requests? Let me know in the comments!

Thanks to Brilliant for sponsoring the video!

Pre-reqs: The previous videos in this playlist: youtube.com/playlist?list=PLdgVBOaXkb9AtG88OsK_c8FDEBDLCC6_9
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
Tchsurvives
Patapom
Alec Dinerstein
Eugene Bulkin
Andy Johnston
K2c2321
chosenthomas
Lina Stritt

Chapters:
0:00 - Special thanks to Brilliant!
0:44 - The Quantum Harmonic Oscillator

Solving the Infinite Square Well Problem | Quantum Mechanics

Faculty of Khan 2022-11-14 | This video derives and discusses the solution to the #InfiniteSquareWell problem in #QuantumMechanics.

Questions/requests? Let me know in the comments!

Pre-reqs: The previous videos in this playlist: youtube.com/playlist?list=PLdgVBOaXkb9AtG88OsK_c8FDEBDLCC6_9
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
Patapom
Alec Dinerstein
Eugene Bulkin
Kelvin Xie
Andy Johnston
Ike J.
K2c2321
Thomas Rossiter
Lina Stritt

Solving the Heat Equation with Convection | Partial Differential Equations

Faculty of Khan 2022-09-12 | Back again with my second video in the last week: here, I talk about the Heat Equation with Convection. Here are some relevant timestamps:
a. Intuition: 0:00
b. Solved Problem: 3:40

Questions? Let me know in the comments!

PDE Playlist: youtube.com/playlist?list=PLdgVBOaXkb9Ab7UM8sCfQWgdbzxkXTNVD

Lecture Notes: drive.google.com/file/d/1rJato1GtQNHKhK9HVyn2ZVjH4mkqL1MO/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
Patapom
Alec Dinerstein
Eugene Bulkin
Kelvin Xie
Andy Johnston
Ike J.
K2c2321
Danielle Shin
Thomas Rossiter
Lina Stritt

Fourier Transforms in Partial Differential Equations

Faculty of Khan 2022-09-07 | After a 6-month hiatus (sorry guys, I've been rather busy with residency of late), I'm finally back with a video: this time, I talk about using Fourier Transforms to solve PDEs, particularly those with an infinite spatial domain. Here are some relevant timestamps:
a. Intro: 0:00
b. Solved Problem: 8:36

Questions? Let me know in the comments!

PDE Playlist (you'll find the Laplace transform video here): youtube.com/playlist?list=PLdgVBOaXkb9Ab7UM8sCfQWgdbzxkXTNVD
Convolution Video: youtube.com/watch?v=N-zd-T17uiE
Green's Function Video: youtu.be/xNqLZnM-PPY

Lecture Notes: drive.google.com/file/d/1C-Ja9QFHKr51EZMWcwjG3ETVjIrVF15J/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
Patapom
Alec Dinerstein
Eugene Bulkin
Kelvin Xie
Andy Johnston
Ike J.
K2c2321
Danielle Shin
Thomas Rossiter
Lina Stritt

Relativity of Velocity | Special Relativity

Faculty of Khan 2022-03-09 | Finally, I'm back after a rather busy 3 month hiatus! Here's a video deriving and explaining #RelativityofVelocity in #SpecialRelativity, using the #LorentzTransformations discussed in the previous video.

Questions/requests? Let me know in the comments!

Pre-reqs: Previous videos in this playlist - youtube.com/playlist?list=PLdgVBOaXkb9CkdpdSWT9oPDSvQ8BjjNTq
Lecture Notes: drive.google.com/file/d/1SSiQqi4kvZNtVgxhnNsS4gjVCRtpQ7Mo/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special Thanks to my Patrons:
Patapom
Michael Mark
Alec Dinerstein
Eugene Bulkin
Rene Gastelumendi
Kelvin Xie
Andy Johnston
Ike J.
K2c2321
Danielle Shin
Thomas Rossiter
Eleonora

The Lorentz Transformation | Special Relativity

Faculty of Khan 2021-11-28 | Video deriving and explaining #LorentzTransformations in #SpecialRelativity, using the principles of time dilation and length contraction discussed in previous videos.

Questions/requests? Let me know in the comments!

Pre-reqs: Previous videos in this playlist - youtube.com/playlist?list=PLdgVBOaXkb9CkdpdSWT9oPDSvQ8BjjNTq
Lecture Notes: drive.google.com/file/d/1l8j0od5AQvmnk3XLYUpPHPtoZnSnHRhe/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special Thanks to my Patrons:
Patapom
Michael Mark
Alec Dinerstein
Mattheus Reischl
Eugene Bulkin
Rene Gastelumendi
Jose Antonio Sanchez-Migallon
Kelvin Xie
Andy Johnston
Ike J.
K2c2321
Danielle Shin
Thomas Rossiter
Alexander Taylor-Lash

Length Contraction in Special Relativity

Faculty of Khan 2021-09-23 | Video deriving and explaining the phenomenon of #LengthContraction in #SpecialRelativity, followed by an example problem.

Questions/requests? Let me know in the comments!

Pre-reqs: Previous videos in this playlist - youtube.com/playlist?list=PLdgVBOaXkb9CkdpdSWT9oPDSvQ8BjjNTq
Lecture Notes: drive.google.com/file/d/1l8j0od5AQvmnk3XLYUpPHPtoZnSnHRhe/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special Thanks to my Patrons:
Patapom
Charles Twardy
Michael Mark
Alec Dinerstein
Michael I Golub
Mattheus Reischl
Eugene Bulkin
Rene Gastelumendi
Jose Antonio Sanchez-Migallon
Kelvin Xie
Nidhi Rathi
Andy Johnston
Ike J.
K2c2321
Danielle Shin
Alexander Taylor-Lash

Deriving the Second Variation | Calculus of Variations

Faculty of Khan 2021-08-05 | Derivation of the Second Variation of Variational Calculus. This is basically the analog to the second derivative in ordinary calculus, in that it allows you to determine the nature of your function for a particular functional (e.g. whether it's a minimum like a straight line minimizing distance on a plane, maximum etc.).

As you'll see in this video: the derivation is more involved than my derivation for the regular Euler-Lagrange equation. Unlike Euler-Lagrange, the second variation is also much harder to apply as it's not a simple matter of solving a differential equation.

Questions/requests? Let me know in the comments!

Pre-reqs: The previous videos in this playlist: youtube.com/playlist?list=PLdgVBOaXkb9CD8igcUr9Fmn5WXLpE8ZE_

Lecture Notes: drive.google.com/file/d/13K1c7vQG1975ISsL-8rWymf0sK--OYV6/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
Patapom
Charles Twardy
Michael Mark
Alec Dinerstein
Lisa Bouchard
Eugene Bulkin
Rene Gastelumendi
Borgeth
Jacob Soares
Kelvin Xie
Nidhi Rathi
Andy Johnston
Ike J.
K2c2321

Hermite Differential Equation and Hermite Polynomials

Faculty of Khan 2021-06-21 | In this video, I demonstrate how to solve the #HermiteODE using the #SeriesSolution method to obtain the #HermitePolynomials.

EDIT: At 1:40, I say that the derivative of the sum is the sum of the derivatives - this only applies for an infinite series if we're working within the radius of convergence. In this situation though, the radius of convergence can be infinite since the ODE doesn't have any singularities. Thanks to Quinn Kolt in the comments for pointing this out!

Questions/requests? Let me know in the comments!

Pre-Reqs: This video in my 'Topics in ODEs playlist': youtube.com/watch?v=c3XtwTsE7QY&list=PLdgVBOaXkb9ATVsK2Q84ghjBgIk5faHNc&index=1
Lecture Notes: drive.google.com/file/d/1LAztdvb4J3dxEX2Bgwa-9-G6vZkrtTrY/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
Daigo Saito
Patapom
Charles Twardy
Michael Mark
Alec Dinerstein
Yenyo Pal
Lisa Bouchard
Eugene Bulkin
Rene Gastelumendi
Borgeth
Jacob Soares
Kelvin Xie
Nidhi Rathi
Andy Johnston
Ike J.
K2c2321

Time Dilation in Special Relativity: Derivation + Example

Faculty of Khan 2021-06-03 | Video deriving and explaining the phenomenon of #TimeDilation in #SpecialRelativity, followed by an example problem inspired by a masterpiece of #Gaming (huehuehuehue).

Questions/requests? Let me know in the comments!

Pre-reqs: Previous videos in this playlist - youtube.com/playlist?list=PLdgVBOaXkb9CkdpdSWT9oPDSvQ8BjjNTq
Lecture Notes: drive.google.com/file/d/1NrNGui-m6QRxalSpV_L7Xi7ZX3dWoeNO/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
Daigo Saito
Patapom
Charles Twardy
Michael Mark
Alec Dinerstein
Yenyo Pal
Lisa Bouchard
Eugene Bulkin
Rene Gastelumendi
Borgeth
Jacob Soares
Kelvin Xie
Nidhi Rathi
Andy Johnston
Ike J.
K2c2321

Theorems on Stationary States | Quantum Mechanics

Faculty of Khan 2021-03-08 | The first 1000 people to use the link will get a free trial of Skillshare Premium Membership: https://skl.sh/facultyofkhan03211

Continuing from the previous lesson, this video further discusses #StationaryStates as solutions to the time-independent #SchrodingerEquation, by proving a number of theorems related to stationary states.

Questions/requests? Let me know in the comments!

Pre-reqs: The previous videos in this playlist: youtube.com/playlist?list=PLdgVBOaXkb9AtG88OsK_c8FDEBDLCC6_9

Lecture Notes: drive.google.com/file/d/1suD-f7K4z_LQ9rp0J59G07A3U0zFg7s_/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
Daigo Saito
Patapom
Charles Twardy
Yenyo Pal
Lisa Bouchard
Eugene Bulkin
Rene Gastelumendi
Borgeth
Jose Antonio Sanchez-Migallon
Celso Carranza
Kelvin Xie
Nidhi Rathi
Andy Johnston
Ike J.

This video was sponsored by Skillshare.

How to Solve a 4th Order Partial Differential Equation | Vibrating Beam Part 1/2

Faculty of Khan 2021-02-15 | The first 1000 people to use the link will get a free trial of Skillshare Premium Membership: https://skl.sh/facultyofkhan02211

This video is the first of two parts on a mini-series discussing the solution to 4th order partial differential equations, using the vibrating beam as an example.

Questions/requests? Let me know in the comments!

Pre-reqs: The previous videos in this playlist - youtube.com/playlist?list=PLdgVBOaXkb9Ab7UM8sCfQWgdbzxkXTNVD

Lecture Notes: drive.google.com/file/d/1K7tSRaBBAWaUvZsG3D9yqfCHnc5kvvkr/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
Daigo Saito
Patapom
Yenyo Pal
Lisa Bouchard
Eugene Bulkin
Rene Gastelumendi
Borgeth
Jose Antonio Sanchez-Migallon
Celso Carranza
Kelvin Xie
Nidhi Rathi
Andy Johnston

This video was sponsored by Skillshare.

Isoperimetric Problems | Calculus of Variations

Faculty of Khan 2021-01-11 | The first 1000 people to use the link will get a free trial of Skillshare Premium Membership: https://skl.sh/facultyofkhan01211

Happy New Year! This video introduces #IsoperimetricProblems in #CalculusofVariations. These are constrained variation problems in which the length of the stationary function to be determined is a fixed constant. Here, we solve an example involving finding the equation of the curve of fixed length between two points such that the area under the curve is maximized. It turns out to be an arc on a circle connecting those two points!

Questions/requests? Let me know in the comments!

Pre-reqs: The previous videos in this playlist (esp. the constrained variation video): youtube.com/playlist?list=PLdgVBOaXkb9CD8igcUr9Fmn5WXLpE8ZE_

Lecture Notes: drive.google.com/file/d/1VzX3KcIpnN5qoy7LKluUxESFl9PfU4V1/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
Daigo Saito
Patapom
Jacob Best
Yenyo Pal
Lisa Bouchard
Eugene Bulkin
Rene Gastelumendi
Borgeth
Jose Antonio Sanchez-Migallon
Celso Carranza
Kelvin Xie

This video was sponsored by Skillshare.

Stationary States in Quantum Mechanics

Faculty of Khan 2020-12-15 | The first 1000 people to use the link will get a free trial of Skillshare Premium Membership: https://skl.sh/facultyofkhan12201

This video introduces #StationaryStates as solutions to the time-independent #SchrodingerEquation discussed in the previous video.

Questions/requests? Let me know in the comments!

Pre-reqs: The previous videos in this playlist: youtube.com/playlist?list=PLdgVBOaXkb9AtG88OsK_c8FDEBDLCC6_9

Lecture Notes: drive.google.com/file/d/1_RdzODnHLbtltvZ0-UmajCNHWOs0HCxz/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
Cesar Garza
Daigo Saito
Patapom
Jacob Best
Yenyo Pal
Lisa Bouchard
Eugene Bulkin
Rene Gastelumendi
Borgeth
Jose Antonio Sanchez-Migallon
Celso Carranza
David Lee

This video was sponsored by Skillshare.

The Material Derivative | Fluid Mechanics

Faculty of Khan 2020-11-17 | The first 1000 people to use the link will get a free trial of Skillshare Premium Membership: https://skl.sh/facultyofkhan11201

This video introduces the #MaterialDerivative and provides some intuition with derived examples using temperature and velocity. Essentially, the material derivative of a quantity carried by the particle in a substance in which that quantity varies is the combination of the intrinsic time variation of the system and the movement of the particle across the spatial gradient of the quantity.

Questions/requests? Let me know in the comments!

Pre-reqs: Basic understanding of ordinary and partial derivatives.

Lecture Notes: drive.google.com/file/d/1hqF32UGpa4lc0NR0K36O4BwbW_7NYwoe/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
Cesar Garza
Daigo Saito
Alvin Barnabas
Patapom
Damjan
Yenyo Pal
Lisa Bouchard
Eugene Bulkin
Rene Gastelumendi
Borgeth
Jose Antonio Sanchez-Migallon
Celso Carranza
David Lee

This video was sponsored by Skillshare.

Conformal Mapping in Complex Variables

Faculty of Khan 2020-10-15 | The first 1000 people to use this link will get a free trial of Skillshare Premium Membership: https://skl.sh/facultyofkhan10201

Introduction to #ConformalMapping with the definition and conformal mapping theorem (i.e. a conformal map is a function that preserves angles when it is used to transform one complex variable to another). I use the fact that a conformal map has preservation of angles to derive the requisite condition for a complex function f(z) to be a conformal map: that is to say, f(z) must be analytic and have a non-zero derivative in the area of interest.

Questions/requests? Let me know in the comments!

Pre-reqs: My basic #ComplexVariables playlist: youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB

Lecture Notes: drive.google.com/file/d/1NKQ9VHriqionsQFJd_snDUFVIM-yy9et/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
Cesar Garza
Daigo Saito
Alvin Barnabas
Patapom
Yenyo Pal
Lisa Bouchard
Eugene Bulkin
Rene Gastelumendi
Borgeth
Jose Antonio Sanchez-Migallon

This video was sponsored by Skillshare.

Introducing Weird Differential Equations: Delay, Fractional, Integro, Stochastic!

Faculty of Khan 2020-08-03 | A brief standalone video that introduces weird types of differential equations, where 'weird' means differential equations that aren't conventionally taught unless you get to super specialized areas. Here, they include delay differential equations, integro-differential equations, stochastic differential equations, and fractional differential equations.

Questions/requests? Let me know in the comments!

Pre-requisites: know what a differential equation is (so basically 1st year undergrad calculus should be more than enough).

Lecture Notes: drive.google.com/file/d/1zufFLUeUVR6NZqb0MHCQWlnN_6SpOqIb/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
Aaron Kaw
Cesar Garza
Daigo Saito
Alvin Barnabas
Patapom
Damjan
Yenyo Pal
Lisa Bouchard
Gabriel Sommer
Eugene Bulkin
Rene Gastelumendi
Nicholas Chowdhury
Borgeth
Jose Antonio Sanchez-Migallon

Mathematical model of epidemics: Development and Analysis (2/2)

Faculty of Khan 2020-07-25 | A topical video on the development and simplification of a typical mathematical model for an epidemic: the SIR model. In fact, with daily new cases rising, it might even be more topical! Part 2 of 2.

Questions/requests? Let me know in the comments!

Part 1: youtu.be/hx2moxHC-bk
Video on the separate plot technique (i.e. where I plot the line and exponential separately): youtu.be/5oPSHfKRKvI
Prerequisites: Previous videos on Nonlinear Dynamics: youtube.com/playlist?list=PLdgVBOaXkb9C8iPDD5xW0jT-c3dtP4TR5

Lecture Notes: drive.google.com/file/d/1h5s2Fh410Vmzs877_wkJzBl-i6RfEUeQ/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
Cesar Garza
Daigo Saito
Alvin Barnabas
Patapom
Damjan
Yenyo Pal
Lisa Bouchard
Gabriel Sommer
Eugene Bulkin
Rene Gastelumendi
Nicholas Chowdhury
Borgeth
Jose Antonio Sanchez-Migallon

Mathematical model of epidemics: Development and Analysis (1/2)

Faculty of Khan 2020-04-26 | A topical video on the development and simplification of a typical mathematical model for an epidemic: the SIR model. Part 1 of 2.

Questions/requests? Let me know in the comments!

Prerequisites: Knowing about nondimensionalization: youtube.com/watch?v=Fa28ifA7H9o and previous videos on Nonlinear Dynamics, though the latter are more relevant in part 2: youtube.com/playlist?list=PLdgVBOaXkb9C8iPDD5xW0jT-c3dtP4TR5

Lecture Notes: drive.google.com/open?id=1GURkwn--Z4zbQ7JO-SADz_s-YlApMJgA
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
Cesar Garza
Daigo Saito
Odissei
Patapom
Alvin Barnabas
Damjan
Yenyo Pal
Lisa Bouchard
Gabriel Sommer
Eugene Bulkin
Adam Pesl
Yiu Chong
René Gastelumendi
Nicholas Chowdhury

Stationary Phase Approximation

Faculty of Khan 2020-04-20 | #StationaryPhaseApproximation from #AsymptoticAnalysis used to determine the value of integrals involving a complex exponential. An example of a Bessel function is used to apply this approximation.

Questions/requests? Let me know in the comments!

Prerequisites: Video on #LaplaceMethod - youtube.com/watch?v=7PuZQhqkWxk

Lecture Notes: drive.google.com/file/d/1_RdzODnHLbtltvZ0-UmajCNHWOs0HCxz/view?usp=sharing
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
Cesar Garza
Daigo Saito
Odissei
Patapom
Alvin Barnabas
Damjan
Yenyo Pal
Lisa Bouchard
Gabriel Sommer
Eugene Bulkin
Adam Pesl
Yiu Chong
René Gastelumendi
Nicholas Chowdhury

The Time-Independent Schrodinger Equation

Faculty of Khan 2020-04-12 | Explaining and deriving the time-independent #SchrodingerEquation using separation of variables to break up the full Schrodinger equation.

Questions/requests? Let me know in the comments!

Video on separation of variables in PDEs: youtube.com/watch?v=aq2DAkJIA2w&list=PLdgVBOaXkb9Ab7UM8sCfQWgdbzxkXTNVD&index=4
Prerequisites: The videos before this one in my #QuantumMechanics playlist: youtube.com/playlist?list=PLdgVBOaXkb9AtG88OsK_c8FDEBDLCC6_9

Lecture Notes: drive.google.com/open?id=12poejGIqrDLEZHdDoBdvOpCl6VHXLoYO
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
Cesar Garza
Daigo Saito
Alvin Barnabas
Damjan
Yenyo Pal
Lisa Bouchard
Patapom
Gabriel Sommer
Eugene Bulkin
Adam Pesl
Yiu Chong
René Gastelumendi
Nicholas Chowdhury

Complex Integration Using Branch Cuts

Faculty of Khan 2020-04-03 | This video covers #ComplexIntegration of a natural logarithm using #BranchCuts to convert the natural log from a multiple-valued function to a single-valued function. This integration involved using an elaborately constructed contour, the #ResidueTheorem, and the ML inequality (link: youtube.com/watch?v=sEyVa_W1Syo&list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB&index=5&t=0s) to arrive at the final answer.

Questions/requests? Let me know in the comments!

Pre-reqs: Every video before this one in my #ComplexVariables playlist, especially the video on Branch Points and Branch Cuts right before this: youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB

Lecture Notes: drive.google.com/open?id=1JHiTM9r6liwQDz9XTE_lUmhe3najPJ1h
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
Cesar Garza
Daigo Saito
Alvin Barnabas
Damjan
Yenyo Pal
Lisa Bouchard
Patapom
Gabriel Sommer
Eugene Bulkin
Adam Pesl
Yiu Chong
René Gastelumendi

Introducing Branch Points and Branch Cuts | Complex Variables

Faculty of Khan 2020-04-02 | The video many of you have requested is finally here! In this lesson, I introduce #BranchPoints and #BranchCuts in the context of multiple-valued functions of #ComplexVariables.

Specifically, I describe the natural log function as an example of a multiple-valued function requiring us to 'cut it up' to form a single-valued function which is more amenable to integration. The process of 'cutting it up' involves the construction of a branch cut, which will come in handy when we get to integrating these functions in the next video.

Questions/requests? Let me know in the comments!

Pre-reqs: Every video before this one in my #ComplexVariables playlist: youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB

Lecture Notes: drive.google.com/open?id=1RMcdOb5zeqUL2Lasvy7LC84PTHNg8z2c
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
Cesar Garza
Daigo Saito
Alvin Barnabas
Damjan
Yenyo Pal
Lisa Bouchard
Patapom
Gabriel Sommer
Eugene Bulkin
Adam Pesl
Yiu Chong
René Gastelumendi

Winding Numbers and Meromorphic Functions Explained! | Complex Variables

Faculty of Khan 2020-04-02 | In this video, I explain the concepts of #WindingNumber and #MeromorphicFunctions. I begin the video by defining the argument of a complex function, which leads nicely to a definition and explanation of meromorphic functions.

I describe meromorphic functions as lying in the middle of the spectrum of continuity (between frankly discontinuous functions and perfectly continuous functions). I then define and give the intuition of the winding number, both for typical curves and for functions.

Questions/requests? Let me know in the comments!

Pre-reqs: Every video before this one in my #ComplexVariables playlist: youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB

Lecture Notes: drive.google.com/open?id=1vr6WIptIgHJH9ATLwxIMTAy6ANwLc5_g
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
Cesar Garza
Daigo Saito
Alvin Barnabas
Damjan
Yenyo Pal
Lisa Bouchard
Patapom
Gabriel Sommer
Eugene Bulkin
Adam Pesl
Yiu Chong
René Gastelumendi

Curvature: Intuition and Derivation | Differential Geometry

Faculty of Khan 2020-03-25 | In my 5th video on #DifferentialGeometry, I define the #Curvature for both a unit speed curve reparametrized with respect to arc length and a regular curve parameterized by t.

I describe the intuition behind the curvature as the extent to which a curve deviates from a straight line (zero curvature). I then derive the expression for curvature for both a unit speed curve and a regular curve, using the #UnitNormal.

Questions/feedback? Let me know in the comments!

Pre-reqs: the previous videos in my playlist youtube.com/playlist?list=PLdgVBOaXkb9DJjk8V0-RkXTnD4ZXUOFsc
Lecture Notes: drive.google.com/open?id=1_40zI8E2r81zOmb_nkhWbN7q2gSqUc7x
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special Thanks to my Patrons:
Cesar Garza
Daigo Saito
Alvin Barnabas
Damjan
Yenyo Pal
Lisa Bouchard
Patapom
Gabriel Sommer
Eugene Bulkin
Yiu Chong
René Gastelumendi

How to Use Perturbation Methods for Differential Equations

Faculty of Khan 2020-01-20 | Click here to explore your creativity and get 2 free months of Premium Membership: https://skl.sh/facultyofkhan

In this video, I discuss perturbation methods in ODEs (ordinary differential equations). Perturbation methods become necessary in differential equations which are otherwise linear, but are 'perturbed' by a small complicated (usually nonlinear) term. I discuss the steps behind performing perturbation methods, and finish the video with an example problem of using perturbation methods to solve a second-order ODE.

Questions/requests? Let me know in the comments!

Prereqs: my playlist on topics in differential equations (youtube.com/playlist?list=PLdgVBOaXkb9ATVsK2Q84ghjBgIk5faHNc) is helpful, but ultimately, knowing how to solve linear ODEs with constant coefficients is enough.
Lecture Notes: drive.google.com/open?id=1FzDATKN-hhyjKPOJXOCx7BOV0fW4dXQM
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special Thanks to my Patrons:
Cesar Garza
Daigo Saito
Alvin Barnabas
Yenyo Pal
Lisa Bouchard
Patapom
Gabriel Sommer
Eugene Bulkin
Jiexian Ma

This video was sponsored by Skillshare.

Closed Curves and Periodic Curves | Differential Geometry 4

Faculty of Khan 2020-01-11 | This video is a continuation of my series on Differential Geometry, and is a discussion about closed and periodic curves.

In this video, I begin by discussing the difference between open and closed curves. I then move on to define important terms, such as a periodic curve, a closed curve, and the period of the curve. I then prove an important theorem that if a smooth regular curve is periodic, then its unit-speed reparameterization (i.e. reparameterization with respect to arc length per the previous video) is also periodic.

Questions/requests? Let me know in the comments!

Pre-reqs: the previous videos in my playlist youtube.com/playlist?list=PLdgVBOaXkb9DJjk8V0-RkXTnD4ZXUOFsc
Lecture Notes: drive.google.com/open?id=1JZjgA997IVsqrTmYK4R4hjAhQl1UJ6KN
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

#DifferentialGeometry #ArcLength #PeriodicCurve

Arc Length as a Parameter | Differential Geometry 3

Faculty of Khan 2020-01-10 | After a long hiatus from Differential Geometry, I return to this highly-requested series as promised! The New Year is upon us and there is no better way to celebrate than with a discussion on reparametrizing with respect to arc length.

In this video, I begin by discussing how reparametrizing with respect to arc length is a valid choice, and why doing this procedure is important in the context of Differential Geometry (re: unit-speed curves). I then prove two theorems related to using arc length as the parameter and the resultant unit speed properties of the curve.

Questions/requests? Let me know in the comments!

Pre-reqs: the previous videos in my playlist youtube.com/playlist?list=PLdgVBOaXkb9DJjk8V0-RkXTnD4ZXUOFsc
Lecture Notes: drive.google.com/open?id=1WWXM084fohiIqbeCwinT4bwBVTMSF9hl
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

#DifferentialGeometry #ArcLength #UnitSpeedCurve

Supercritical and Subcritical Pitchfork Bifurcations | Nonlinear Dynamics and Chaos

Faculty of Khan 2019-10-13 | Get your pitchforks out everyone, because this video is about pitchfork bifurcations, and is another continuation to the Bifurcations videos in my Nonlinear Dynamics series!

There are two types of pitchfork bifurcations: supercritical and subcritical. Supercritical bifurcations are bifurcations in which varying a parameter changes the stability of one fixed point from stable to unstable, and causes two stable fixed points to appear. A subcritical bifurcation is the opposite, in that an unstable fixed point becomes stable, while two new unstable fixed points appear after crossing the bifurcation point.

Questions/requests? Let me know in the comments!

Pre-reqs: The videos before this one on this playlist: youtube.com/playlist?list=PLdgVBOaXkb9C8iPDD5xW0jT-c3dtP4TR5

Lecture Notes: drive.google.com/open?id=1xus7TiyNmpvgn7m_olEQhMcs9GzWT-H1
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- Cesar Garza
- Daigo Saito
- Alvin Barnabas
- Aldo Mendes Martins
- Yenyo Pal
- Lisa Bouchard
- Richard Mcnair
- Patapom
- Bernardo Marques
- Jacob Soares
- Otar Kemularia
- Eugene Bulkin

#NonlinearDynamics #Bifurcations #FacultyofKhan

Transcritical Bifurcations | Nonlinear Dynamics and Chaos

Faculty of Khan 2019-09-29 | This video is about transcritical bifurcations, and is a continuation to the Bifurcations videos in my Nonlinear Dynamics series.

Transcritical bifurcations are bifurcations in which varying a parameter causes two fixed points to fuse and become a half-stable fixed point, after which they switch stability. I also solve an example of a transcritical bifurcation, which ultimately reduces to its normal form.

Questions/requests? Let me know in the comments!

Pre-reqs: The videos before this one on this playlist: youtube.com/playlist?list=PLdgVBOaXkb9C8iPDD5xW0jT-c3dtP4TR5
Lecture Notes: drive.google.com/open?id=1jwIMJVksbeX5xplKyfhKJbDRTj-zO1HP
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- Cesar Garza
- Daigo Saito
- Alvin Barnabas
- Aldo Mendes Martins
- Yenyo Pal
- Lisa Bouchard
- Richard Mcnair
- Patapom
- Bernardo Marques
- Jacob Soares
- Otar Kemularia

Rouches Theorem and the Fundamental Theorem of Algebra

Faculty of Khan 2019-09-07 | In this video, I prove Rouche's Theorem, which relates the zeros of a function f(z) and the sum of two functions f(z) and g(z), assuming that certain conditions are satisfied.

Following the proof, I apply Rouche's Theorem to an important example involving the proof of the Fundamental Theorem of Algebra. I'm sure that the pure math lovers who watch my videos will be excited to see this proof!

Questions/requests? Let me know in the comments!

Pre-reqs: Every video before this one in my Complex Variables playlist: youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB

Lecture Notes: drive.google.com/open?id=1Fm0RdZj5DAX460v8RvBrY0ilhYjBwhxg
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- Cesar Garza
- Alvin Barnabas
- Aldo Mendes Martins
- Yenyo Pal
- Lisa Bouchard
- Richard Mcnair
- Patapom
- Bernardo Marques
- Jacob Soares
- Otar Kemularia

The Argument Principle | Complex Variables

Faculty of Khan 2019-08-23 | I'm finally back from my self-imposed exile, and there's no better way to celebrate my return than with a new video on the Argument Principle!

In this lesson, I derive the Argument Principle in complex variables, which relates the winding number of a complex function with respect to a traversal over a contour C to the number of zeros and poles of that function inside C.

This video will be particularly useful when it comes to proving Rouche's Theorem, which will be the next video coming up. Questions/requests? Let me know in the comments!

Pre-reqs: Every video before this one in my Complex Variables playlist: youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB

Lecture Notes: drive.google.com/open?id=1W8E1xLxwECVX1VJjKNkCF5992W0bmRCs
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- Cesar Garza
- Alvin Barnabas
- Aldo Mendes Martins
- Yenyo Pal
- Lisa Bouchard
- Richard Mcnair
- Guillaume Chereau
- Patapom
- Bernardo Marques
- Jacob Soares
- Otar Kemularia

The Kramers-Kronig Relations | Complex Variables

Faculty of Khan 2019-04-10 | In this video, I begin my new series on Advanced Topics in Complex Variables with a lesson on deriving the Kramers-Kronig relations. The Kramers-Kronig relations allow the determination of the real part of a complex function from the imaginary part and vice-versa.

For this reason, they come up in many places in Physics that make use of complex variables. I begin this derivation by discussing the Cauchy Principal Value, setting up the assumptions, and performing an integration over a slightly modified semicircular contour. Then, I use the definition of the Principal Value, the polar representation of complex numbers, Jordan's Lemma, and Cauchy's Theorem to make some simplifications, after which I complete the derivation.

You'll notice that a lot of this proof takes elements from multiple videos in my Complex Variables playlist, so I highly recommend you watch those so that you can best engage with the content presented in this video!

Questions/Requests? Let me know in the comments!

Pre-reqs: My entire Complex Variables playlist: youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB
Improper Integral Video: youtube.com/watch?v=a2ocry6LBe0

Lecture Notes: drive.google.com/open?id=1dxk08qUIfItegpJwG2FXww8fF0Ozj7yt
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- Anonymous
- Cesar Garza
- Odissei
- Alvin Barnabas
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau
- Patapom
- Vitor Ciaramella
- McKay Oyler
- Dieter Walter Reule

Tensor Operations: Contractions, Inner Products, Outer Products

Faculty of Khan 2019-03-27 | In this video, I continue the discussion on tensor operations by defining the contraction, inner product, and outer product. I provide some short examples of each of these operations, which will hopefully solidify your understanding of how these operations work.

Questions/requests? Let me know in the comments!

Pre-reqs: The previous videos in the playlist - youtube.com/playlist?list=PLdgVBOaXkb9D6zw47gsrtE5XqLeRPh27_
Lecture Notes: drive.google.com/open?id=1PUSPfoI7g8lcU0-Gzcxi9n9pBdMXjjKY
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- Anonymous
- Cesar Garza
- Odissei
- Alvin Barnabas
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau
- Patapom
- Vitor Ciaramella
- McKay Oyler
- Dieter Walter Reule

EDIT: At 8:35, when I write the components of a, I meant to use superscripts instead of subscripts! a is a contravariant tensor, so superscripts are the way to go!

Tensor Transformation Laws: Contravariant, Covariant, and Mixed Tensors

Faculty of Khan 2019-03-26 | In this video, I shift the discussion to tensors of rank 2 by defining contravariant, covariant, and mixed tensors of rank 2 via their transformation laws. After laying down these laws (get it?), I clearly describe the notation for a tensor of a particular rank. Specifically, I define a (p,q) tensor as a tensor with a contravariant rank of p (i.e. p indices in the superscript) and covariant rank of q (i.e. q indices in the subscript).

Then, I define the transformation law for a general (p,q) tensor. I finish the video by defining a couple of simple tensor operations: summation, scalar multiplication, and the extension of the two: linear combination.

Questions/requests? Let me know in the comments!

Pre-reqs: The previous videos in the playlist - youtube.com/playlist?list=PLdgVBOaXkb9D6zw47gsrtE5XqLeRPh27_
Lecture Notes: drive.google.com/open?id=1EkKWJPxplILIxH4pgHKOVLyk3yMAVIrp
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- Anonymous
- Cesar Garza
- Odissei
- Alvin Barnabas
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau
- Patapom
- Vitor Ciaramella
- McKay Oyler
- Dieter Walter Reule

Contravariant and Covariant Vectors | 2/2

Faculty of Khan 2019-03-24 | In this video, I define contravariant and covariant vector components. Here, we define contravariant and covariant vector components according to their transformation definitions. I also define invariants, which are quantities with intrinsic significance.

I describe two examples of contravariant and covariant tensors. For contravariant tensors, I use the tangent vector of a parametric curve. For a covariant tensor, I use the example of a gradient vector of a scalar field.

Questions/requests? Let me know in the comments!

Pre-reqs: The previous videos in the playlist - youtube.com/playlist?list=PLdgVBOaXkb9D6zw47gsrtE5XqLeRPh27_
Lecture Notes: drive.google.com/open?id=1E2I-LpQcKV-ERMUXqYMOtIH2dZs5EM5E
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- Anonymous
- Cesar Garza
- Odissei
- Alvin Barnabas
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau
- Patapom
- Vitor Ciaramella
- McKay Oyler
- Dieter Walter Reule

Contravariant and Covariant Vectors | 1/2

Faculty of Khan 2019-03-24 | In this video, I describe the meaning of contravariant and covariant vector components. As mentioned in a previous video, tensors are invariant under coordinate transformations. However, tensor components transform according to specific rules; they may either transform in a contravariant manner (i.e. the opposite manner as the basis vectors) or in a covariant manner (i.e. the same manner as the basis vectors).

I discuss contravariant and covariant components and give some intuition of what they mean by using two examples. The next video will cover a more rigorous definition of contravariant and covariant vector components.

Questions/requests? Let me know in the comments!

Pre-reqs: The previous videos in the playlist - youtube.com/playlist?list=PLdgVBOaXkb9D6zw47gsrtE5XqLeRPh27_
2nd video on Tensors: youtube.com/watch?v=j6DazQDbEhQ&list=PLdgVBOaXkb9D6zw47gsrtE5XqLeRPh27_&index=3&t=0s
Lecture Notes: drive.google.com/open?id=1CVYgE5iEGAeuSe16L1x0ULYiL0GVTgJn
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- Anonymous
- Cesar Garza
- Odissei
- Alvin Barnabas
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau
- Patapom
- Vitor Ciaramella
- McKay Oyler
- Dieter Walter Reule

Coordinate Transformations and Curvilinear Coordinates | Tensor Calculus

Faculty of Khan 2019-03-23 | In this video, I go over concepts related to coordinate transformations and curvilinear coordinates. I begin with a discussion on coordinate transformations, after which I move on to curvilinear coordinates.

I give 3 important examples of curvilinear coordinates: polar, cylindrical, and spherical. Afterwards, I finish the video by defining the Jacobian matrix and its determinant, the Jacobian.

Questions/requests? Let me know in the comments!

Pre-reqs: The previous videos in the playlist - youtube.com/playlist?list=PLdgVBOaXkb9D6zw47gsrtE5XqLeRPh27_
Lecture Notes: drive.google.com/open?id=1rrdneshnzStDXi62CTEURivd9uMYwM-M
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- Anonymous
- Cesar Garza
- Odissei
- Alvin Barnabas
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau
- Patapom
- Vitor Ciaramella
- McKay Oyler
- Dieter Walter Reule

The Vertebrobasilar Circulation | Neurology

Faculty of Khan 2019-02-11 | Don't worry guys, I haven't crossed over to the Medicine side; this is just a video for a class assignment. It's also a pilot for an idea I have brewing in my head for my channel, but that idea's only for much later in the future. I'll resume with my Math and Physics stuff right after this (*cough* Time Dilation *cough*).

This video covers the basics of the vertebrobasilar circulation, which supplies the brainstem, spinal cord, cerebellum, and the posterior part of the cerebrum. If you've been staring at your Anatomy book for the past 5 hours trying to make sense of things, then hopefully this video can explain the same concepts in just over 7 minutes. Questions/requests? Let me know in the comments!

Pre-reqs: Basic knowledge of vertebra anatomy, meninges, and basic knowledge of the major heart blood vessels. However, I've explained the relevant portions just to refresh your memory.
Lecture Notes: drive.google.com/open?id=1JvqdHDMG1mzQ5MW-PFv91MAabbJ4I-HA
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons:
- James Mark Wilson
- Cesar Garza
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau
- Patapom
- Elm Mara
- Vitor Ciaramella
- McKay Oyler
- Dieter Walter Reule

Arc Length and Reparameterization | Differential Geometry 2

Faculty of Khan 2019-02-09 | In this video, I continue my series on Differential Geometry with a discussion on arc length and reparametrization. I begin the video by talking about arc length, and by deriving the arc length formula for a parametrized curve. Then, I discuss the ideas of speed and unit-speed curves, which are quite important in Differential Geometry.

I then move on to a discussion on reparametrization. I define reparametrization, give a short example to provide necessary context, and then finish the video by defining regular and singular points on a curve.

Questions/requests? Let me know in the comments!

Pre-reqs: The previous videos in the playlist - youtube.com/playlist?list=PLdgVBOaXkb9DJjk8V0-RkXTnD4ZXUOFsc
Lecture Notes: drive.google.com/open?id=19bbVU9HczuCaFJ64ceQBncwoYqrfsh5H
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- James Mark Wilson
- Cesar Garza
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau
- Patapom
- Elm Mara
- Vitor Ciaramella
- McKay Oyler
- Dieter Walter Reule

Basic Linear Algebra Concepts for Tensors

Faculty of Khan 2019-02-09 | In this short video, I go over some very basic concepts in linear algebra that will be relevant to tensors later on in the series. These include Linear Transformations and General Transformations. I finish the video with the Chain Rule for Partial Derivatives (which will also come in handy in the future) expressed in terms of Einstein Notation.

My lecture series on Tensors assumed knowledge of Linear Algebra, so this should be a fairly elementary review for most people. Nonetheless, in my effort to be as self-contained as possible, I made this 'detour' video just in case people needed it. In any case, we have the foundation necessary to tackle the actual Mathematics of Tensors, starting from the next video in the series.

Questions/requests? Let me know in the comments!

Pre-reqs: The previous videos in the playlist - youtube.com/playlist?list=PLdgVBOaXkb9D6zw47gsrtE5XqLeRPh27_
Lecture Notes: drive.google.com/open?id=1ZWnssKy7wghon0UHOzzNOtsncYRj4KsR
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- James Mark Wilson
- Cesar Garza
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau
- Patapom
- Elm Mara
- Vitor Ciaramella
- McKay Oyler
- Dieter Walter Reule

Introducing Special Relativity and the Relativity of Simultaneity

Faculty of Khan 2019-01-08 | In this video, I begin a new series on Special Relativity and discuss the Relativity of Simultaneity. Special Relativity is based on two postulates: 1) That the Laws of Physics hold in all inertial reference frames, and 2) That the speed of light is constant in all inertial reference frames.

I discuss these postulates, as well as their implications. Then, I define events and give an especially pertinent (and spicy) example with the Youtube Rewind video of 2018, after which I finish the video with an explanation of the relativity of simultaneity. This concept will be re-visited in future videos when I go over spacetime diagrams, but this should be a good introduction.

This series should be a good companion to my concurrent Tensor Calculus and Differential Geometry series (I'm going to upload the next videos on these next!) - which should serve as a nice prelude to General Relativity. Questions/requests? Let me know in the comments!

Pre-reqs: Just know what coordinate systems are and know that speed = distance/time. Knowledge of grade 12 Classical Mechanics helps but isn't absolutely essential.
Lecture Notes: drive.google.com/open?id=1gXPrsZGL92TKVMHKRvZA66gScT6XL74X
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- James Mark Wilson
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau
- Patapom
- Elm Mara
- Vitor Ciaramella
- McKay Oyler

Laplaces Method and the Stirling Approximation

Faculty of Khan 2018-12-31 | In this video, I begin with a discussion on Laplace's Method. Laplace's Method is a technique used to approximate the integral of the exponential of a large number times a function with a unique global maximum.

Here, I describe the assumptions underlying the application of Laplace's Method. This method also comes in handy when deriving the Stirling formula using the definition of the Gamma Function, which is what we do in the second part of this video.

Questions/requests? Let me know in the comments!

Gamma Function Video: youtube.com/watch?v=PwCl7vVyXwY
Pre-reqs: My Gamma Function Video and knowledge of 1st-year undergraduate Calculus (especially knowledge of integrating the exponential of -x^2 from -infinity to infinity).
Lecture Notes: drive.google.com/open?id=1Jb0EBzEXPh18mpHaEEKg16nOUXB5HmyG
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- James Mark Wilson
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau
- Patapom
- Elm Mara
- Vitor Ciaramella
- Cooper Wang

The Geodesic Problem on a Sphere | Calculus of Variations

Faculty of Khan 2018-12-03 | In this video, I set up and solve the Geodesic Problem on a Sphere. I begin by setting up the problem and using the Euler-Lagrange Equation to determine the equation of the geodesic on a sphere. Then, I take a quick detour and explain the concept of a great circle, which is formed by the intersection of a plane passing through the center of a sphere and the sphere's surface.

I finish the video by showing that the geodesic on a sphere carries the exact same form as the great circle. This leads to the conclusion that the curve representing the shortest distance between two points on a sphere is an arc on the great circle connecting those two points.

Questions/requests? Let me know in the comments!

Pre-reqs: The videos before this one in this playlist: youtube.com/playlist?list=PLdgVBOaXkb9CD8igcUr9Fmn5WXLpE8ZE_
Previous Geodesic Video: youtube.com/watch?v=f8ACx2iN6fk&list=PLdgVBOaXkb9CD8igcUr9Fmn5WXLpE8ZE_&index=4&t=0s
Lecture Notes: drive.google.com/open?id=1PNGrNpjTjG-dsuCMJaTC5uGPKgcWEC3E
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- James Mark Wilson
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau
- Patapom
- Elm Mara
- Vitor Ciaramella
- Cooper Wang

How to Integrate Fourier Integrals | Complex Variables

Faculty of Khan 2018-11-05 | In this video, I demonstrate the technique of performing improper Fourier integrals using the methods of Complex Variables. The technique involves setting up a semicircular contour, and using a combination of the Residue Theorem and Jordan's Lemma to eventually compute the integral.

I begin by explaining the 5-step process in solving problems involving a Fourier integral (i.e. when you're integrating f(x)*cos(ax) or f(x)*sin(ax)) with infinite limits. After explaining the process, I apply it to a simple example problem.

Questions/Requests? Let me know in the comments!

Pre-reqs: The videos before this one on this playlist: youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB
Improper Integral Video: youtube.com/watch?v=a2ocry6LBe0

Lecture Notes: drive.google.com/open?id=18w4I-qVTCKbkWqh_qk4H2KwrFRvIFwm1
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- James Mark Wilson
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau
- Patapom
- Elm Mara

The Stirling Approximation: a 5-minute Derivation!

Faculty of Khan 2018-10-29 | In this quick video, I use the definition of integration/Riemann sums to derive the Stirling Approximation or the Stirling Formula, which is a way to approximate the factorial of a large number.

There are multiple ways to derive the Stirling Formula; I've just shown one of the simple ones here. The drawback is that this is a less powerful derivation; perhaps this could be a hint for a more rigorous proof in a later video???

Special thanks to James Mark Wilson for suggesting this video!

Questions/Requests? Let me know in the comments!

Pre-reqs: Basic Calculus (i.e. you should know what integration means).

Lecture Notes: drive.google.com/open?id=1uMzWmrJXWyeUXUTwBDnmyCwmO6ENRdEW
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- James Mark Wilson
- Yuan Gao
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau
- Patapom

ERRATA: At 4:25, evaluating the integral should actually give you NlnN - N + 1, but since N is large, I automatically neglected the '1'.

Jordans Lemma Proof | Complex Variables

Faculty of Khan 2018-09-23 | In this video, I prove Jordan's Lemma, which is one of the key concepts in Complex Variables, especially when it comes to evaluating improper integrals of polynomial expressions which also have either sine or cosine multiplying them.

I begin by proving Jordan's Inequality, which then leads nicely into the proof of Jordan's Lemma. Note here that Jordan is pronounced in the French manner (i.e. as 'Jordon').

Questions/Requests? Let me know in the comments!

Pre-reqs: The videos before this one on this playlist: youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB
ML Inequality Video: youtube.com/watch?v=sEyVa_W1Syo

Lecture Notes: drive.google.com/open?id=1ZQCg9XhPXkqIvjl5H3UIPKkPMaK5GARH
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- James Mark Wilson
- Yuan Gao
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau

Introducing Bifurcations: The Saddle Node Bifurcation

Faculty of Khan 2018-08-31 | Welcome to a new section of Nonlinear Dynamics: Bifurcations! Bifurcations are points where a dynamical system (e.g. differential equation) undergoes a significant change in its dynamical behaviour when a certain parameter in the differential equation crosses a critical value.

In this video, I explain saddle node bifurcations. These are bifurcations in which varying a parameter causes the appearance of a half-stable fixed point, followed by two fixed points from nothing. I discuss bifurcation diagrams, bifurcation points, and describe the concept of normal forms.

Questions/requests? Let me know in the comments!

Pre-reqs: The videos before this one on this playlist: youtube.com/playlist?list=PLdgVBOaXkb9C8iPDD5xW0jT-c3dtP4TR5

Lecture Notes: drive.google.com/open?id=1mt_5XJqUB6wtST-J0KBlRhSJY5v7lM7q
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan

Special thanks to my Patrons for supporting me at the $5 level or higher:
- Jose Lockhart
- James Mark Wilson
- Yuan Gao
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair