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Videos uploaded by user “Faculty of Khan”
Introduction to Dirac Notation
 
03:25
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
Introduction to Calculus of Variations
 
06:41
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
Laurent Series of Complex Functions
 
07:19
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
An Introduction to Hilbert Spaces
 
05:17
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
Einstein Summation Convention: an Introduction
 
09:00
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
Introduction to Tensors
 
11:15
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
Dirac Notation: Properties and Neat Rules
 
03:35
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
Mathematical Basis of Quantum Mechanics: Introduction to Vector Spaces
 
03:25
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
Introduction to Hilbert Spaces: Important Examples
 
03:35
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
Introduction to Nondimensionalization
 
07:25
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
Introduction to Quantum Mechanics: Schrodinger Equation
 
12:38
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
Introducing Convolutions: Intuition + Convolution Theorem
 
11:08
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
Introduction to Complex Functions
 
09:50
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
Introduction to Operators in Quantum Mechanics
 
03:35
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
Complex Integrals and Cauchy's Integral Theorem.
 
07:57
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
The Brachistochrone Problem and Solution | Calculus of Variations
 
12:14
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
Introduction to Partial Differential Equations: Definitions/Terminology
 
09:07
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
Derivation of the Euler-Lagrange Equation | Calculus of Variations
 
07:51
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
Solving ODEs by Series Solutions: Legendre's ODE
 
11:06
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
Introduction to Differential Geometry: Curves
 
10:25
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
Solving the 1-D Heat/Diffusion PDE: Nonhomogenous Boundary Conditions
 
07:26
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
Introducing Parabolic PDEs (1-D Heat/Diffusion Eqn): Intuition and Maximum Principle
 
07:09
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
The Principle of Stationary Action
 
13:37
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
Introduction to the Frobenius Method
 
11:26
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
Beltrami Identity Derivation | Calculus of Variations
 
04:21
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
Cauchy's Integral Formula and Proof
 
08:50
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
Residue Theorem and Proof
 
07:03
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
How to find the Residues of a Complex Function
 
14:14
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
Introducing Green's Functions for Partial Differential Equations (PDEs)
 
11:35
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
Nondimensionalizing the Navier-Stokes Equation
 
08:08
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
Computing Improper Integrals using the Residue Theorem | Cauchy Principal Value
 
13:41
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
Introduction to Tensors: Transformation Rules
 
07:53
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
Solving the 1-D Heat/Diffusion PDE by Separation of Variables (Part 1/2)
 
11:09
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
The Geodesic Problem on a Plane | Calculus of Variations
 
07:10
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
The Cauchy-Riemann Relations and some Theorems
 
08:46
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
Sturm-Liouville Theorem and Proof
 
09:19
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
D'Alembert Solution to the Wave Equation
 
09:05
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
Complex Integration: The ML Inequality Proof and Example
 
10:51
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
Bessel Function of the 2nd Kind | 2nd solution of Bessel's Equation
 
05:51
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
Using Green's Functions to Solve Nonhomogeneous ODEs
 
09:39
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
Introducing Bifurcations: The Saddle Node Bifurcation
 
13:34
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
Solving the 1-D Heat/Diffusion PDE: Nonhomogenous PDE and Eigenfunction Expansions
 
08:45
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
Solving ODEs by the Power Series Solution Method
 
11:06
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
Introduction to Nonlinear Dynamics
 
09:56
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
Commutators and Eigenvalues/Eigenvectors of Operators
 
10:48
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
Quantum Mechanics: Examples of Operators | Hermitian, Unitary etc.
 
07:43
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
The Geodesic Problem on a Sphere | Calculus of Variations
 
10:01
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
Channel Introduction
 
05:19
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
Computing Definite Integrals using the Residue Theorem
 
12:13
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
Real Analysis Introduction: Sets and Set Operations
 
08:56
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