In this video, I give examples of the types of vectors in Hilbert Space, and I introduce Dirac Notation.
Questions? Let me know in the comments!
Prereqs: What's happened in the playlist so far: https://www.youtube.com/playlist?list=PLdgVBOaXkb9Bv466YnyxslT4gIlSZdtjw
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMLXdGd2lXLXJfdDQ
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 24864
Faculty of Khan

In this video, I introduce the subject of Variational Calculus/Calculus of Variations. I describe the purpose of Variational Calculus and give some examples of problems which may be solved using techniques from Calculus of Variations.
Specifically, Calculus of Variations seeks to find a function y = f(x) which makes a functional stationary. Note: a functional is a function of functions, or a map which converts a function to a real number, kind of like how the time functional T converts a particular function/particle path to a real value of time. I would have included the Euler-Lagrange proof but it would have likely made the video over 15 minutes, which is a little too long for us.
Questions/requests? Let me know in the comments!
Prerequisites: Not many, just know Calculus 1 (obviously). For later videos, a background in ODEs will be needed, though you don't need to be an expert in them since I'll walk you through :)
Lecture Notes: https://drive.google.com/open?id=0BzC45hep01Q4M2h3MmdaMi1xTFU
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan

Views: 50174
Faculty of Khan

This video gives an introduction, complete with examples, of a Laurent series of a complex function. If you have any questions, let me know in the comments!
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMd0NmUlpiLTU3WGc
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 66243
Faculty of Khan

In this video, I introduce the Hilbert Space and describe its properties.
Questions? Let me know in the comments!
Prereqs: Previous video on vector spaces, knowledge of what real and rational numbers are.
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMVFZFaVZOWlkxX3M
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 76358
Faculty of Khan

In this video, I introduce Einstein notation (or Einstein Summation Convention), one of the most important topics in Tensor Calculus. Einstein notation is a way of expressing sums in short-form; repeated indices are used to denote the index that is summed over.
I describe the 4 major rules of Einstein notation, as well as the definitions of free and dummy indices. I also discuss some important information related to these major rules.
Questions/requests? Let me know in the comments!
Prerequisites: The videos before this one on this playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9D6zw47gsrtE5XqLeRPh27_
Lecture Notes: https://drive.google.com/open?id=1qgQvuoDU_1EScznBjWzHc_dV_GJGCsmU
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://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

Views: 15621
Faculty of Khan

My tensor series is finally here! In this video, I introduce the concept of tensors. I begin by talking about scalars, then vectors, then rank-2 tensors (whose explanation takes up the bulk of the video since these are probably the most difficult to understand out of the three).
I then move on to define tensors (without specifying their transformation properties), after which I conclude the video with a short discussion on rank-3 tensors, which may be represented by 3-D matrices/arrays.
Questions/requests? Let me know in the comments!
Pre-requisites: You basically need to know what vectors, scalars, and matrices are. Nothing much more to it. A 1st-year Physics + Linear Algebra course should be enough.
Lecture Notes: https://drive.google.com/open?id=1O5GOXA-oJsrn3j8ZHnk-CecPEA79uiJv
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special thanks to my Patrons for supporting me at the $5 level or higher:
- Jose Lockhart
- Yuan Gao
- James Mark Wilson
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques

Views: 33349
Faculty of Khan

In this video, I discuss the properties of bras, kets, and brakets.
Questions? Comment below!
Prereqs: The playlist so far: https://www.youtube.com/playlist?list=PLdgVBOaXkb9Bv466YnyxslT4gIlSZdtjw
Here's the 2nd video in the playlist: https://www.youtube.com/watch?v=7zx3MT9FgT0
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMOUpXTzVxeUExbDA
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 14837
Faculty of Khan

In this video, I briefly describe the concept of linear vector spaces. This is part of my Quantum Mechanics lectures series, and will help you understand some of the mathematics we'll be doing later on.
Questions? Let me know in the comments!
Prereqs: Basic Linear Algebra (at the first year college level) helps but is not absolutely necessary (I think).
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMYUstVmpnV3JNME0
Patreon Link: https://www.patreon.com/user?u=4354534
EDIT TO CLARIFY: Instead of "For each phi_i there is a vector 0 such that ..." it should be "There is a vector 0 such that for each phi_i ...". Similarly, for the scalar multiplication, "there is a 1 and a 0 such that for every vector psi ...". I don't want you guys to think that the '1' and the '0' vectors are different for each psi; they are the same for the entire vector space! Also, when I say that a vector space is a set of vectors psi_1, psi_2, psi_3 ..., I don't mean that it's countable. I'm just using psi_1, psi_2, and psi_3 as some example vectors.

Views: 24053
Faculty of Khan

In this video, I describe two types of Hilbert Spaces, finite-dimensional and infinite-dimensional.
Questions? Let me know in the comments!
Prereqs: The two videos in the playlist that occurred before. Playlist link: https://www.youtube.com/playlist?list=PLdgVBOaXkb9Bv466YnyxslT4gIlSZdtjw
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMdklXbnFjRGtvM2c
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 25785
Faculty of Khan

Nondimensionalization is a simple, but exceedingly useful tool that helps you reduce differential equations to their natural forms. It helps you 'get rid of' unnecessary parameters, and lets you zero in on what's important. In this video, I show the basic technique behind nondimensionalization and supplement that technique with a useful example.
Questions? Let me know in the comments.
Prereqs: Basic (1st-order) ODEs. In fact, you may not even need this since we're not solving any ODEs. We're only doing algebra to make ODEs more simple.
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMYmEtNHNrMDdsbGM
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 18344
Faculty of Khan

There's no better way to celebrate Christmas than with a 12 minute video on the Schrodinger Equation! In this lesson, I introduce Quantum Mechanics with a discussion on wavefunctions and the Schrodinger Equation (in 1-D). I show how wavefunctions can represent probability density functions (via the norm-squared), and discuss the significance of this representation.
I then introduce/revisit some basic Statistics concepts, and end the video with a proof of how the normalization of wavefunctions stays preserved with time.
Questions/requests? Ask in the comments!
Prereqs: This whole playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9Bv466YnyxslT4gIlSZdtjw
Lecture Notes: https://drive.google.com/open?id=1IA_theB0HyGh7JJMZ5nlCj8IYN0ap2ib
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan?lang=en
Special thanks to my Patrons:
- Jennifer Helfman
- Justin Hill
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
NOTE: At around 11:30-11:45, I mention how the 'boundary' integrals have to approach zero at +/- infinity. This is true for square-integrable functions that come up in Physics. However, as one of the commenters pointed out, there are exceptions (i.e. square-integrable functions that don't approach zero at infinity). These exceptions aren't found in Physics though, so we'll ignore them, but I figure they're worth mentioning as a footnote.

Views: 37231
Faculty of Khan

In this lesson, I introduce the convolution integral. I begin by providing intuition behind the convolution integral as a measure of the degree to which two functions overlap while one sweeps across the other. I demonstrate this intuition by showing that the convolution of two box functions is a triangle.
I then move on to proving the Convolution Theorem for Fourier Transforms, and discussing how it compares to the Convolution Theorem for Laplace Transforms. The proof for Fourier Transforms is relatively simple, but the proof for Laplace Transforms is a bit more difficult (if you really want to see the Laplace Transform proof, I can make another video but I've put it off for now).
Questions/requests? Let me know in the comments! Hopefully the intuition I provided was sufficiently clear.
Prereqs: Very basic knowledge of Fourier and Laplace Transforms (i.e. you just need to know what they are and what they're used for), ODEs, and integration.
Lecture Notes: https://drive.google.com/open?id=1dDWYNk5SpzhkI7ep_PS2m74El17aeMiK
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special thanks to my Patrons for supporting me at the $5 level or higher:
- Jose Lockhart
- Yuan Gao
- Justin Hill
- Marcin Maciejewski
- Jacob Soares
- Yenyo Pal
- Chi
- Lisa Bouchard

Views: 10811
Faculty of Khan

A brief introduction to Complex Functions, including basics and holomorphicity, as well as comparisons to real functions.
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMU0JHSU9qRWNJUzQ
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 48499
Faculty of Khan

In this video, I introduce operators.
Questions? Let me know in the comments!
Prereqs: The playlist so far: https://www.youtube.com/playlist?list=PLdgVBOaXkb9Bv466YnyxslT4gIlSZdtjw
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMVzVEUzRIZUhKZXM
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 13350
Faculty of Khan

This video covers the method of complex integration and proves Cauchy's Theorem when the complex function has a continuous derivative.
Some background knowledge of line integrals in vector calculus is useful to understand connections, but not necessary.
Any questions? Let me know in the comments!
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMVFd3S1NYQ0o0bEU
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 73348
Faculty of Khan

In this video, I set up and solve the brachistochrone problem, which involves determining the path of shortest travel in the presence of a downward gravitational field. This is done using the techniques of Calculus of Variations, and it will turn out that the brachistochrone can be represented by the parametric equations of a cycloid.
The Brachistochrone is a rather popular topic on Youtube, with pop-science channels like VSauce making videos about it. However, not many people actually derive the equations, so I'm hopeful that this tutorial will be a more rigorous change of pace.
Questions/requests? Let me know in the comments!
Prereqs: The first three videos of this playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9CD8igcUr9Fmn5WXLpE8ZE_
Euler-Lagrange Video: https://www.youtube.com/watch?v=sFqp2lCEvwM
Lecture Notes: https://drive.google.com/open?id=0BzC45hep01Q4TmE4T0dGNzZLMEU
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan?lang=en
Patreon Supporters:
Jennifer Helfman
Jacob Soares
Yenyo Pal

Views: 25266
Faculty of Khan

In this video, I introduce PDEs and the various ways of classifying them.
Questions? Ask in the comments below!
Prereqs: Basic ODEs, calculus (particularly knowledge of partial derivatives/what they are).
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMSFFmNHoteDdTaDg
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 33813
Faculty of Khan

In this video, I derive/prove the Euler-Lagrange Equation used to find the function y(x) which makes a functional stationary (i.e. the extremal). Euler-Lagrange comes up in a lot of places, including Mechanics and Relativity. The derivation is performed by introducing a variation in the extremal via a parameter epsilon, and setting the derivative of the functional with respect to epsilon to be zero.
My previous Variational Calculus video was very positively received, so I thought it would be appropriate to continue the series and upload the second video sooner rather than later. Also, you'll notice that the writing here is smaller, but that's because the screen I'm using now is bigger because of my new desktop.
Questions/requests? Let me know in the comments!
Prereqs: First video of my Calculus of Variations playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9CD8igcUr9Fmn5WXLpE8ZE_
Lecture Notes: https://drive.google.com/open?id=0BzC45hep01Q4MUllbWpMTndFUFk
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://www.twitter.com/FacultyOfKhan/

Views: 51007
Faculty of Khan

In this video, I solve the Legendre differential equation, using the regular series solution method.
Questions? Let me know in the comments!
Prerequisites: Series solutions intro video, Basic Calculus, Really basic ODEs (first order ODE stuff)
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMajFEOHE3VHg5N2M
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 66631
Faculty of Khan

In this video, I introduce Differential Geometry by talking about curves. Curves and surfaces are the two foundational structures for differential geometry, which is why I'm introducing this series by defining curves.
After defining level curves, parametrized curves, and tangent vectors, I solve a short example where I convert a level curve to a parametrized curve and then find its tangent vector.
Questions/requests? Let me know in the comments!
Pre-requisites: A background in Multivariable Calculus (Calculus 3) is helpful, but even if you know the material until Calculus 2, you probably still won't be lost.
Lecture Notes: https://drive.google.com/open?id=1CirfXRYfjS8eKB7TVwEWkAT-8nTzpFEQ
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special thanks to my Patrons for supporting me at the $5 level or higher:
- Jose Lockhart
- Yuan Gao
- James Mark Wilson
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques

Views: 18116
Faculty of Khan

In this video, I solve the diffusion PDE but now it has nonhomogenous but constant boundary conditions. I show that in this situation, it's possible to split the PDE problem up into two sub-problems: one which gives a steady-state solution, and another which gives a transient solution.
I show that the transient solution obeys homogenous boundary conditions, and that using the steady state solution helps to remove the non-homogeneity. Solving the transient solution is just a simple matter of separating variables, in which case these two videos should help:
https://www.youtube.com/watch?v=aq2DAkJIA2w
https://www.youtube.com/watch?v=5AkjTUD6TDw
Questions? Ask in the comments!
Prereqs: My PDE videos so far (see my playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9Ab7UM8sCfQWgdbzxkXTNVD)
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMUFZiM2kwZ3pJYjQ
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 20370
Faculty of Khan

In this video, I introduce the most basic parabolic PDE, which is the 1-D heat or diffusion equation. I show what it means physically, by discussing how it relates the concavity at a point (indicative of the average value of a function in the regions surrounding that point) to the time derivative.
In other words, the more different a parabolic PDE solution is from its surroundings, the more quickly it changes in order to better match/equilibrate with its surroundings. For example, the hotter your frying pan, the more quickly it loses heat to match the room temperature.
Questions? Ask in the comments below!
Prereqs: Basic ODEs, my ODE topics playlist, and a couple of my first two introductory PDE videos. For this lecture though, the bare minimum is a basic knowledge of PDEs (not necessarily how to solve them, just what they are) and some calculus.
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMV19SczAycXk0LWc
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 10207
Faculty of Khan

My oft-requested video has finally arrived! In this lesson, I introduce the Principle of Stationary Action to begin my newest series on Analytical Mechanics. The Principle of Stationary Action serves as an exceedingly useful tool to solve higher-level problems in Classical Mechanics as well as in other branches of Physics.
It states that the path a particle follows in space can be determined by setting the action functional stationary; essentially, we can find the particle's path by solving a Calculus of Variations problem. I show that the Principle of Stationary Action is essentially equivalent to applying Newton's 2nd Law, and afterwards, I solve a simple example problem to illustrate the Principle of Stationary Action *in action* (huehuehue).
More complex versions of the Principle of Stationary Action, in addition to more complex examples, will be solved in future videos in this playlist.
Questions/requests? Let me know in the comments!
Pre-reqs: My Calculus of Variations playlist, until the 7th video - https://www.youtube.com/playlist?list=PLdgVBOaXkb9CD8igcUr9Fmn5WXLpE8ZE_
Lecture Notes: https://drive.google.com/open?id=1EmIVdtHEhbocjoRDeSxCmhnZqxGwK9J8
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special thanks to my Patrons for supporting me at the $5 level or higher:
- Jose Lockhart
- Yuan Gao
- James Mark Wilson
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques

Views: 5639
Faculty of Khan

In this video, I introduce the Frobenius Method to solving ODEs and do a short example.
Questions? Ask them below!
Prerequisites: Regular series solutions of ODEs (basically those two series videos I made).
Also, one of the comments pointed out that I made a mistake at 8:45: when you plug r1=1 into the coefficient of the a1 term, you get 3! For r2=0.5 you get 1. Still, that doesn't change the fact that plugging these values of r into the other terms will give us non-zero coefficients for the a1, a2, a3, etc.
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMZnFhX3AxYXhjZ3c
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 93988
Faculty of Khan

After 1 month of self-imposed exile to Med School Island (my apologies for the long wait btw), I'm finally back with a video! This time, I've made a tutorial on deriving the Beltrami Identity for Calculus of Variations. The Beltrami Identity is an equation which arises from the Euler-Lagrange equation when the integrand of the functional has no explicit or direct dependence on the independent variable x.
This was originally supposed to be part of my Brachistochrone problem video, but I decided to make it an independent topic to avoid confusion.
Questions/requests? Let me know in the comments!
Euler-Lagrange Derivation: https://www.youtube.com/watch?v=sFqp2lCEvwM
Prerequisites: The first two videos of this playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9CD8igcUr9Fmn5WXLpE8ZE_
Lecture Notes: https://drive.google.com/open?id=0BzC45hep01Q4a1EzWFYwWVhtQjQ
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan?lang=en

Views: 8140
Faculty of Khan

In this video, I state and derive the Cauchy Integral Formula. In addition, I derive a variation of the formula for the nth derivative of a complex function.
If you have any questions, ask in the comments!
Link to the stuff I wrote/Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMVWZSUEU1b1VEY0k
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 51011
Faculty of Khan

In this video, I will prove the Residue Theorem, using results that were shown in the last video. Note that the theorem proved here applies to contour integrals around simple, closed curves.
Any questions? Leave them in the comments below!
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMd0NmUlpiLTU3WGc
Patreon Link: https://www.patreon.com/user?u=4354534
CORRECTION: Just a note that the sum at 0:30 and 6:15 should start at j = 1. Annotations are gone now so my best bet is to update the description with corrections. Regardless, this is a fairly minor detail and shouldn't affect things too much.

Views: 41311
Faculty of Khan

In this video, I describe 3 techniques behind finding residues of a complex function: 1) Using the Laurent series, 2) A residue-finding approach for simple poles, and 3) A residue-finding approach for non-simple poles.
I also prove/verify these techniques, which are ultimately going to be used to calculate complex integrals (and even real integrals) when applying the Residue Theorem.
Questions/suggestions? Let me know in the comments! Also, yes, I spelled 'technique' wrong at 8:50. Pls forgive my transgression.
Prereqs: The playlist so far (the first 7 videos, especially the Laurent series and residue theorem one): https://www.youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMNHN4YjNRZ3NfNzA
Support my Patreon: https://www.patreon.com/user?u=4354534

Views: 42752
Faculty of Khan

In this video, I describe the application of Green's Functions to solving PDE problems, particularly for the Poisson Equation (i.e. A nonhomogeneous Laplace Equation). I begin by deriving the 2 Green Identities, after which I use those identities to come up with an equation for the solution to the Poisson Equation.
Most of the derivations I've done in this video apply to a 3-dimensional case, but as I explain at the end, you could just as easily apply these methods to a 2-D situation as well (in fact, it's slightly easier than with 3-D)!
Questions/requests? Let me know in the comments!
Prerequisites: The first two videos of this playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9Ab7UM8sCfQWgdbzxkXTNVD and this video on Green's Functions for ODEs are essential: https://www.youtube.com/watch?v=Jws70qd-XRw - the rest of the videos in the PDE playlist are helpful but optional.
Lecture Notes: https://drive.google.com/open?id=1uP8WlVed4T6zVcm8y-3WwSQn2GPY_7nz
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special thanks to my Patrons:
- Jennifer Helfman
- Justin Hill
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard

Views: 18364
Faculty of Khan

In this video, I convert the Navier-Stokes equation for an incompressible, Newtonian fluid to a dimensionless form. The Reynold's number comes up as one of the parameter groups that results from the nondimensionalization.
Any questions/comments? Leave them below and I'll get back to you!
Prereqs: Previous nondimensionalization video. Even though Navier-Stokes is mentioned, you don't need to know much Fluid Mechanics. It helps to know about Reynold's number, but the lesson is basically a giant exercise in algebra so not knowing some of the dimensionless parameters won't screw you over. Some really basic knowledge of partial derivatives also helps.
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMbnVyQmFRREpISEE
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 13159
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: https://www.youtube.com/watch?v=sSj7z-pz-yY
Prereqs: The videos before this one on the playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB
Lecture Notes: https://drive.google.com/open?id=0BzC45hep01Q4SWFiaW8wMzFqekk
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan/

Views: 16644
Faculty of Khan

In this video, I continue my introduction to tensors by talking about the transformation property that defines tensors, that tensors are invariant under a change of coordinate system.
After describing this transformation property using 3 examples of tensors, I then talk about the intuition behind this property. I finish the video by elaborating on the differences between matrices and tensors.
Questions/requests? Let me know in the comments!
Prerequisites: Previous video(s) on Tensors: https://www.youtube.com/playlist?list=PLdgVBOaXkb9D6zw47gsrtE5XqLeRPh27_
Lecture Notes: https://drive.google.com/open?id=121Ov4gWoZ3le2P0v2OAS0X9S3vEz8zIb
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special thanks to my Patrons for supporting me at the $5 level or higher:
- Jose Lockhart
- Yuan Gao
- James Mark Wilson
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan

Views: 16509
Faculty of Khan

In this video, I introduce the concept of separation of variables and use it to solve an initial-boundary value problem consisting of the 1-D heat equation and a couple of homogenous Dirichlet boundary conditions.
Questions? Ask in the comments!
Prereqs: My ODEs stuff, and the PDEs stuff I've covered up till now.
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMY19ZT0xnbU9meGM
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 41957
Faculty of Khan

In this short (hehe) video, I set up and solve the Geodesic Problem on a Plane. A geodesic is a special curve that represents the shortest distance between two points on a particular surface. Here, because the geodesic problem is being solved on a plane, I show that the geodesic on a plane is a straight line. To do this, I set up a length functional, and then using the Euler-Lagrange equation, I solve for the equation of the geodesic path. The computation here isn't that difficult, but I said I'd cover geodesics when I started this series so I figured I'd do this for the sake of completeness.
Now, I didn't mention this in the video because I thought it was a bit tangential to the topic, but geodesics become quite important in General Relativity. General Relativity essentially has two important (sets of) equations. The first are the Einstein Field Equations, which describe how spacetime (which can be thought of as a 4-D surface) curves under the influence of mass and energy. The second set are the geodesic equations, which describe how light and matter travel in this 'curved' spacetime. In other words, light travels along geodesics in spacetime (that's why light 'bends' in gravity).
The geodesic equation in General Relativity can, in fact, be derived using Euler-Lagrange. In this case, the dS isn't just sqrt(dx^2 + dy^2) but is a lot more complicated, and involves the metric tensor. I'll cover this in more depth once I start General Relativity, but I figured I'd briefly discuss an application for the interested folks out there.
ERRATA: At 3:29, I meant to say y' instead of y.
Pre-reqs: The first two videos of this playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9CD8igcUr9Fmn5WXLpE8ZE_
Euler-Lagrange Video: https://www.youtube.com/watch?v=sFqp2lCEvwM
Lecture Notes: https://drive.google.com/open?id=1bjirQAVyzE39-3GHnnVLTVSuOATD-1o5
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special thanks to my Patrons:
- Tom
- Jennifer Helfman
- Justin Hill
- Jacob Soares
- Yenyo Pal
- Chi
- Lisa Bouchard

Views: 11661
Faculty of Khan

This second video goes over the Cauchy-Riemann relations. If you have any questions/comments, let me know below!
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMU2hWWEpBVW40TTQ
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 25367
Faculty of Khan

In this video, I prove the Sturm-Liouville Theorem and explain the ideas of eigenvalues and eigenfunctions. It's a particularly useful video that's going to be referenced when we begin solving PDEs, so pay attention (particularly to the statement, not so much the proof)!
Questions? Ask me in the comments below!
Prereqs: Basic ODEs, Calculus. Linear Algebra (particularly eigenvalues and eigenvectors) helps because there's one occasion when I reference them. It's not mandatory though.
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMZEJnZlR1d3g4eFk
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 38420
Faculty of Khan

In this video, we derive the D'Alembert Solution to the wave equation. We use the general solution found in the last couple of videos to solve a Wave PDE problem in an infinite domain with two initial conditions (initial displacement and initial velocity). The resulting solution is the D'Alembert Solution/D'Alembert Formula.
We also show earlier on in the video that the Wave Equation consists of the sum of a forward travelling wave (f(x+ct)) and a backward travelling wave (g(x-ct)), because if we're at the same relative location on the wave (e.g. x+ct/x-ct is a constant), then that relative location has to have a decreasing x (backward travelling) for f(x+ct) and an increasing x (forward travelling) for g(x-ct). I hope my explanation here wasn't too confusing because I feel like that was one of the trickier parts of the video. If you have any questions, let me know!
Questions/requests? Ask me in the comments!
Prereqs: This playlist, especially videos 9-12: https://www.youtube.com/playlist?list=PLdgVBOaXkb9Ab7UM8sCfQWgdbzxkXTNVD
Lecture Notes: https://drive.google.com/open?id=0BzC45hep01Q4MXVvZWpQVzhYSzQ
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan/
Special thanks to my Patrons:
- Jennifer Helfman
- Jacob Soares

Views: 13243
Faculty of Khan

This video proves the ML Inequality (aka Estimation Lemma) for complex integrals and does a short example involving it. The ML Inequality is quite useful because it helps establish upper bounds on your complex/contour integrals. It was a request that someone made yesterday, so the fact that I managed to complete the request in one day is probably quite fortunate lol.
Questions/Requests/Feedback? Let me know in the comments! Also, I got a new microphone, which is why my voice sounds much better now.
Prereqs: The 1st 3 videos of the complex variables playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB
Patreon Link: https://www.patreon.com/user?u=4354534
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMQXNJbUtxdzlidjQ

Views: 16997
Faculty of Khan

In this video, I briefly describe how to obtain the Bessel function of the second kind, which is also the 2nd solution to Bessel's equation when the order of the ODE is an integer. People requested this video in the past so I hope this video is sufficient!
Deriving Y_n is fairly simple: take a particular linear combination of J_p and J_-p, and then let p approach n. I don't go through the entire calculation, because it's tedious and doesn't really teach much, but I refer you to this document (by Jim Lambers at Berkeley) if you're interested in the details:
https://math.berkeley.edu/~linlin/121B/LambersBessel.pdf
Questions/requests? Let me know in the comments!
Prereqs: The first 8 videos of this playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9ATVsK2Q84ghjBgIk5faHNc
Lecture Notes: https://drive.google.com/open?id=0BzC45hep01Q4UlV4STFrN2FkS28
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan

Views: 15550
Faculty of Khan

In this video, I describe how to use Green's functions (i.e. responses to single impulse inputs to an ODE) to solve a non-homogeneous (Sturm-Liouville) ODE subject to ANY arbitrary input f(x). All this involves is integrating the Green's function with the input f(x) over the domain. The input, by the way, is the function that makes the ODE non-homogeneous.
This approach is made possible because of the orthogonality relation of the Sturm-Liouville Theorem and the fact that the eigenfunctions of a REGULAR Sturm-Liouville problem form a complete set.
By the way, at 2:05, I say that we can make the eigenvalue 'any real number we want'. I should clarify that and say that we can make the 'n' in lambda_n 'whatever integer we want'. Lambda is still restricted by the boundary conditions. Nonetheless, the point that a smaller set of functions (eigenfunctions of a REGULAR Sturm-Liouville Problem) can be used to describe a larger set of functions (all nice/smooth functions) still holds.
Questions/requests? Let me know in the comments! The oft-requested PDE version of my Green's Function video is coming out soon, so stay tuned!
Prereqs: Know the Sturm-Liouville Theorem. Also, this playlist will help: https://www.youtube.com/playlist?list=PLdgVBOaXkb9ATVsK2Q84ghjBgIk5faHNc
Sturm-Liouville Video: https://www.youtube.com/watch?v=_F0ck1JncLE
Lecture Notes: https://drive.google.com/open?id=0BzC45hep01Q4cW5WU1ZkeHBqTnc
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan/

Views: 25800
Faculty of Khan

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: https://www.youtube.com/playlist?list=PLdgVBOaXkb9C8iPDD5xW0jT-c3dtP4TR5
Lecture Notes: https://drive.google.com/open?id=1mt_5XJqUB6wtST-J0KBlRhSJY5v7lM7q
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://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

Views: 3283
Faculty of Khan

In this video, I give a brief outline of the eigenfunction expansion method and how it is applied when solving a PDE that is nonhomogenous (i.e. contains a source term).
Questions? Ask me in the comments!
Prereqs: The previous videos in this PDE playlist, especially the two lectures on separation of variables. Knowledge of the Sturm-Liouville Theorem is also helpful.
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMM1I2VnVzN0VtblE
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 15786
Faculty of Khan

This video covers the basics of the series solution method of solving ODEs. There's an example there to help solidify the concepts taught.
If you have any questions, let me know in the comments!
EDIT: Also, at 9:56, I forgot the powers of x, which are 2k and 2k+1 on the series.
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMdnhvTmlnbk1hZjg
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 23251
Faculty of Khan

Greetings, Youtube!
This is the first video in my series on Nonlinear Dynamics. Comment below if you have any questions, and if you like the video, let me know.
Also, if you have any more requests for what subjects you want to see, tell me in the comments section!
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 9930
Faculty of Khan

In this video, I introduce the concept of commutators and eigenvalues/eigenvectors in Quantum Mechanics. After stating some properties (my apologies for inundating you with a bunch of statements), I then move on to discussing the 2nd Postulate of Quantum Mechanics.
This should all be a nice build-up to the video where I finally prove the Generalized Uncertainty Principle!
Questions? Ask in the comments!
Prereqs: The videos in my Quantum Mechanics playlist before this one - https://www.youtube.com/playlist?list=PLdgVBOaXkb9Bv466YnyxslT4gIlSZdtjw
Lecture Notes: https://drive.google.com/open?id=1CKKULBhbk0c_0g9cDwBB6V3_tjAFLNAi
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan?lang=en
Special Thanks to my Patrons!
- Jennifer Helfman
- Justin Hill
- Jacob Soares
- Yenyo Pal

Views: 11213
Faculty of Khan

In this video, I describe 4 types of important operators in Quantum Mechanics, which include the Inverse, Hermitian, Unitary, and Projection Operators. I also give examples of the latter three and describe the properties of all these operators.
Hermitian Operators in particular are important, for they correspond to physically measurable variables in Quantum Mechanics (we'll learn more about those later in the series).
Questions? Ask in the comments!
Previous Video: https://www.youtube.com/watch?v=KvS7Z0rEutE
Prereqs: First 6 videos of my Quantum Mech playlist (these videos represent just the introductory math necessary for QM): https://www.youtube.com/playlist?list=PLdgVBOaXkb9Bv466YnyxslT4gIlSZdtjw
Lecture Notes: https://drive.google.com/open?id=0B_urJu4cgDhMT0FBTGstUzhGaG8
Patreon: https://www.patreon.com/user?u=4354534

Views: 15468
Faculty of Khan

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: https://www.youtube.com/playlist?list=PLdgVBOaXkb9CD8igcUr9Fmn5WXLpE8ZE_
Previous Geodesic Video: https://www.youtube.com/watch?v=f8ACx2iN6fk&list=PLdgVBOaXkb9CD8igcUr9Fmn5WXLpE8ZE_&index=4&t=0s
Lecture Notes: https://drive.google.com/open?id=1PNGrNpjTjG-dsuCMJaTC5uGPKgcWEC3E
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://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

Views: 3376
Faculty of Khan

This video isn't going to involve any tutorial lessons (unfortunately, though I'm planning to put up a lesson hopefully in a week or less). However, it should serve as a good introduction to my channel, my motivations, and how you can use the content of my channel to its fullest potential.
Questions? Feedback? Let me know in the comments below!
Prereqs: None
Lecture Notes Link: https://drive.google.com/open?id=0B_urJu4cgDhMbl9SVDNLa0psUGc
Support my Patreon: https://www.patreon.com/user?u=4354534

Views: 15314
Faculty of Khan

In this video, I show how to evaluate definite integrals involving sines and cosines by taking advantage of the polar representation of complex numbers and then applying the Residue Theorem. I look at one simple example and one complicated example of an integration which takes advantage of the Residue Theorem.
In many ways, the Residue Theorem makes life a lot more simple by allowing the evaluation of integrals that would otherwise be difficult to compute using the techniques you learned from Calculus 2. I also hope that this video satisfies some of the requests I've been getting on this topic.
Prereqs: Everything before this video in this playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB
Questions/requests? Let me know in the comments!
Lecture Notes: https://drive.google.com/open?id=0BzC45hep01Q4d2dTVjE1S2ktZnM
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan

Views: 16016
Faculty of Khan

Keepin' it real with my introduction to REAL Analysis! I talk about sets, set notation, and set operations. The next video will introduce functions, one of the fundamental concepts in Analysis.
Questions/requests? Let me know in the comments!
Pre-reqs: Knowledge of Calculus I and II is helpful (especially for later videos) but not necessary for this one. I feel like even an 8th grader could understand this lesson, so there are hardly any major pre-requisites.
Lecture Notes: https://drive.google.com/open?id=1iY5w1xqk3YPgOP28iZlrXJL9GvKKfN2q
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special thanks to my Patrons for supporting me at the $5 level or higher:
- Jose Lockhart
- James Mark Wilson
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
EDIT: At 6:50, I mention that the empty set is denoted by phi: the symbol for the empty set is actually based on a Norwegian letter, not 'phi'.

Views: 3739
Faculty of Khan