Videos uploaded by user “Faculty of Khan” for the 2018

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: 19428
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: 45734
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: 14944
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: 7228
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: 23192
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: 21784
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: 14027
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: 4797
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: 5853
Faculty of Khan

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: https://drive.google.com/open?id=1uMzWmrJXWyeUXUTwBDnmyCwmO6ENRdEW
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
- 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'.

Views: 3127
Faculty of Khan

In this video, I continue my lessons on Einstein notation (or Einstein Summation Convention), by explaining how parentheses work in Einstein Notation. This is followed by an explanation of some Einstein Notation identities, non-identities, and the Kronecker Delta symbol.
This should wrap up the videos on Einstein notation, because in the next video on Tensor Calculus, I'm going to go more in-depth into actual Tensor Algebra!
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=1hRjB60YhSdw7FB8m4208hIs6-uTV_WWj
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: 12992
Faculty of Khan

In this video, I introduce the concept of Laplace Transforms to PDEs. A Laplace Transform is a special integral transform, and when it's applied to a differential equation, it effectively integrates out one of the independent variables to make the differential equation a simpler equation. Once we solve this simpler equation, we can take the inverse Laplace Transform (with the help of tables) and obtain the solution to the original differential equation.
After introducing Laplace Transforms, I apply the method of Laplace Transforms to a simple example involving the heat equation on a semi-infinite domain. After some computation, we end up with a complimentary error function as our solution.
I'm also pleased to announce that after several infuriating months of trying to find a way to display the cursor on my recording, I have finally achieved success. The cursor can be seen as the yellow dot, and I hope that it will make my videos easier to follow. Please be sure to congratulate me on this achievement by writing 'thank mr cursor' in the comments section.
Prerequisites: Basic knowledge of Laplace Transforms from ODEs (though I've tried to give a sufficiently thorough review without getting too thorough) and the first 3 videos of this playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9Ab7UM8sCfQWgdbzxkXTNVD
Lecture Notes: https://drive.google.com/open?id=14uoU3rUmARL7HVTyw9FQBC_pFPNse_eH
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: 11898
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: 5211
Faculty of Khan

In this video, I define the expectation value of position for a wavefunction psi and use that to derive the expectation value of momentum as well as the expressions for the position and momentum operator. I then show that in general, any Classical Mechanics quantity can be determined from the wavefunction using a combination of the position and/or momentum operators.
There's quite a bit of math involved, so if you have any questions, let me know in the comments! Stay tuned for the next video (coming soon!) in which I will use the expressions for the momentum and position operators to derive the Heisenberg Uncertainty Principle.
Prerequisites: All the previous videos in both of these playlists (Playlist 1: https://www.youtube.com/playlist?list=PLdgVBOaXkb9AtG88OsK_c8FDEBDLCC6_9, Playlist 2: https://www.youtube.com/playlist?list=PLdgVBOaXkb9Bv466YnyxslT4gIlSZdtjw)
2nd Postulate Video: https://youtu.be/-h2lU2FXy48?t=9m30s
Previous Video: https://www.youtube.com/watch?v=kUm4q0UIpio
Lecture Notes: https://drive.google.com/open?id=1ppTeR55YdIXox2ncf0obr8Na8vaFEIbk
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: 7764
Faculty of Khan

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: https://www.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: https://drive.google.com/open?id=1Jb0EBzEXPh18mpHaEEKg16nOUXB5HmyG
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: 1962
Faculty of Khan

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: https://www.youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB
ML Inequality Video: https://www.youtube.com/watch?v=sEyVa_W1Syo
Lecture Notes: https://drive.google.com/open?id=1ZQCg9XhPXkqIvjl5H3UIPKkPMaK5GARH
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
- Yuan Gao
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau

Views: 6057
Faculty of Khan

In this video on Analytical Mechanics, I discuss extensions to the Lagrange equations when working with systems of multiple particles or when constraints are involved. I also discuss constraints in classical mechanics, including holonomic (scleronomic + rheonomic) and non-holonomic constraints.
I finish the video with a strategy on how to solve problems involving multiple particles and constraints using the Lagrange equations, which is a nice segue to my next video, where I solve an actual Action Problem.
Questions/requests? Let me know in the comments!
Prereqs: The videos before this one on my playlist - https://www.youtube.com/playlist?list=PLdgVBOaXkb9DSSqQZWfBrZy_rOljWmA3j
Lecture Notes: https://drive.google.com/open?id=12DXpoikDTXqKvZZDejS-WvZE2BiayERS
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: 2487
Faculty of Khan

In this video, I solve the catenary problem. A catenary is a curve that describes the shape of a string hanging under gravity, fixed on both of its ends. Here, I determine the equation of the catenary for a uniform string with both ends fixed at the same height using the techniques of Variational Calculus (with constraints).
I begin by deriving the two integrals necessary to solve the constrained variation problem. The first integral is the constraint integral, according to which the length of the string is fixed. The second integral is formed by evaluating the total gravitational potential energy of the entire string (the functional to minimize).
After setting up the integrals, I create a composite functional K which includes my potential energy + (Lagrange multiplier)*(constraint integral). Applying the Euler-Lagrange equation/Beltrami Identity to K and solving the resulting differential equation gives me the equation for my catenary. Finally, I compute the 3 unknown constants in the catenary equation using the constraint and the two boundary conditions, and I show that the equation of a catenary takes the form of a hyperbolic cosine.
Questions/requests? Let me know in the comments!
Pre-reqs: This playlist (especially the video on constraints + multiple dependent variables): https://www.youtube.com/playlist?list=PLdgVBOaXkb9CD8igcUr9Fmn5WXLpE8ZE_
Lecture Notes: https://drive.google.com/open?id=12r-pbUpd6ffyBgrJ7TwOm3yxkIo9CEbt
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: 11913
Faculty of Khan

In this video, I begin by deriving the Euler-Lagrange Equation for multiple dependent variables. I show that in order to make a functional involving multiple y's stationary, it is necessary to solve an Euler-Lagrange equation for each of those y's. This is going to be useful when we work in 2-D or 3-D coordinate systems to solve Action Problems in Classical Mechanics.
In the second part of the video, I show how to approach variational problems when there are one or more constraints involved. The technique described comes from Lagrange multipliers and is a relatively simple one. This will also come in handy for my classical mechanics videos where there are constraints imposed on the particle's motion.
Questions/requests? Let me know in the comments!
Prereqs: Just these two videos (you could probably watch the rest of the playlist too, which is what I would recommend):
1. https://www.youtube.com/watch?v=6HeQc7CSkZs&index=1&list=PLdgVBOaXkb9CD8igcUr9Fmn5WXLpE8ZE_&t=0s
2. https://www.youtube.com/watch?v=sFqp2lCEvwM&list=PLdgVBOaXkb9CD8igcUr9Fmn5WXLpE8ZE_&index=2
Lecture Notes: https://drive.google.com/open?id=1i4vmv1ElkHX9jXaWDcKhHhI9ppGu8NBP
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: 7442
Faculty of Khan

In this video, I solve an example problem in Quantum Mechanics which involves normalizing a wavefunction, finding expectation values, and then using those expectation values to verify that the Heisenberg Uncertainty Principle holds true.
This problem should solidify what we've learned so far in Quantum Mechanics.
Questions/requests? Let me know in the comments!
Video on the expectation value of Q(x,p): https://www.youtube.com/watch?v=Egu4i8umpoM
Prerequisites: The videos before this one in this playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9AtG88OsK_c8FDEBDLCC6_9
Lecture Notes: https://drive.google.com/open?id=1HMjtG2bxRvnx2vudUwRuyt0rQmgN_rC0
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: 3440
Faculty of Khan

After a long hiatus from this Nonlinear Dynamics, I have finally returned with a 4th video! In this lesson, I begin with proving that oscillations for a 1-D autonomous dynamical system of the form dx/dt = f(x) are IMPOSSIBLE.
Then, I introduce the concept of potentials of differential equations. Potentials are another useful way of examining a differential equation and are convenient quantities because they behave in a manner similar to potential energies in Physics (i.e. physical systems prefer a state of lower potential energy, similar to how dynamical systems here tend to lower their potential over time).
After introducing potentials, I finish off by solving a quick example problem which shows how potentials can be utilized to ascertain the behaviour of a nonlinear ODE.
Questions/comments? Let me know below!
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=1Uyr2UB5hugKQaBi7JAlyFYvGj_KIww2O
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
- Chi
- Lisa Bouchard
- Bernardo Marques

Views: 1742
Faculty of Khan

In this video, I introduce the subject of Heat Transfer. 'Heat Transfer' is a bit of redundant term; as I mention in the video, 'heat' (by definition) already refers to energy in transfer. However, engineers don't care about this and since Heat Transfer is an engineering subject, we're going to go with the engineering convention (mentioned in the video) when defining 'heat' and 'heat transfer'.
After explaining the semantics, I describe the 3 laws of Thermodynamics to establish a foundation for the study of heat transfer. I then briefly outline the 3 methods of heat transfer and two basic principles underlying these methods (i.e. heat is only transferred spontaneously from hot to cold and heat transfer only occurs when there is a temperature difference - pretty intuitive stuff).
Questions/comments? Let me know below!
Prerequisites: Basic Physics, even 1st year undergrad Physics might be overkill; you basically need to know what energy and temperature are (I trust that your high school covered those things; if not, then some serious changes need to be made to the public education system). Knowing the laws of thermodynamics helps but I explain them here so it's not *that* necessary.
Lecture Notes: https://drive.google.com/open?id=1xUHnDqhRgLs87uoNZ9shhsO4Wtkgt0te
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: 1332
Faculty of Khan

In this video, I use the skills we've developed in the previous videos on Calculus of Variations/Analytical Mechanics to solve an Action Problem involving a block sliding down a moving inclined plane.
I begin by formulating the problem, determining the kinetic and potential energies, and then solving the Lagrange equations for the 2 'particle' (i.e. block + wedge) system.
Questions/requests? Let me know in the comments!
Prerequisites: The videos before this one on my playlist - https://www.youtube.com/playlist?list=PLdgVBOaXkb9DSSqQZWfBrZy_rOljWmA3j
Lecture Notes: https://drive.google.com/open?id=1b6NXapv6kIHnpGcJDHChnBevp3aElQb1
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: 1749
Faculty of Khan

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: https://www.youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB
Improper Integral Video: https://www.youtube.com/watch?v=a2ocry6LBe0
Lecture Notes: https://drive.google.com/open?id=18w4I-qVTCKbkWqh_qk4H2KwrFRvIFwm1
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

Views: 3714
Faculty of Khan

In this video, I derive and discuss the Heisenberg Uncertainty Principle, perhaps one of the most famous relationships in Quantum Mechanics. I start by using the Generalized Uncertainty Principle (derived here: https://www.youtube.com/watch?v=qgkWUuqoSVM) to determine the commutator of the position and momentum operators whose expressions I found in the previous video (link: https://www.youtube.com/watch?v=Egu4i8umpoM).
Once I derive the Heisenberg Uncertainty Principle, I devote much of the video to explaining what it means - *that it's a statement on the fundamental nature of quantum mechanical particles which arises from certain mathematical principles (e.g. Fourier Transforms) describing those particles*. In the video, I state that the Heisenberg Uncertainty Principle 'is a consequence of mathematics', but keep in mind that this mathematics is used to model quantum mechanical particles. Thus, in the context of Quantum Mechanics, the Heisenberg Uncertainty Principle is a physical fact that can be derived mathematically (that's what I mean when I say 'consequence of mathematics').
Towards the end of the video, I emphasize that the Uncertainty Principle is NOT a statement about observer-induced limitations on measurements; that's the Observer Effect. The Heisenberg Uncertainty Principle is rather difficult to wrap one's head around, at least at first, so I encourage you to ask questions and comment down below!
Prerequisites: All the previous videos in this playlist (Playlist 1: https://www.youtube.com/playlist?list=PLdgVBOaXkb9AtG88OsK_c8FDEBDLCC6_9), and all the videos in this playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9Bv466YnyxslT4gIlSZdtjw.
Lecture Notes: https://drive.google.com/open?id=1S7BUUttlepXFHBaLAmryWqzXy8JhXd5k
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
EDIT: The deBroglie formula at 6:40 should have hbar instead of h.

Views: 3966
Faculty of Khan

In this video, I celebrate the Faculty of Khan's 10,000 subscriber milestone with a Q and A video! Be sure to let me know in the comments if you'd like to see a Medical Analysis video (as extra content in addition to my regular Math/Physics videos)!
Questions/requests? Let me know in the comments!
Pre-requisites: None
Lecture Notes: None.
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: 1032
Faculty of Khan

In this video, I discuss how to determine the set of points which satisfies a system of two linear inequalities. I begin by deriving the solution set for a system of linear inequalities where the corresponding lines intersect, then I go on to derive the solution set for the system of inequalities with parallel lines.
I'd like to thank James Mark Wilson, one of my patrons, for providing the content and inspiration for this video! Also, as promised, here is the example problem and solution: https://drive.google.com/open?id=1ET6F5SunBbYZYWHiOk5lfS5HDvK1rU5Y
Questions/requests? Let me know in the comments!
Prerequisites: Basic Grade 8/9 algebra (i.e. knowledge of lines, slopes, linear systems of 2 equations).
Lecture Notes: https://drive.google.com/open?id=13xM6SUU6fAnei_BILxoAmyskR54o8VSe
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

Views: 1475
Faculty of Khan

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