Course Introduction
Welcome to the Advanced Math For Physics -course!
First of all, I want to congratulate you for taking action towards learning new things. That being said, to learn new things, you also have to put in the work to really understand them.
First things first, it’s important to know what exactly the goal of this course is.
To put it simply, this course is aimed to help you build an understanding of the most important areas of mathematics, such as vector calculus and calculus of variations.
However, learning just math is a bit boring without knowing how to apply those concepts (to me, at least) and that’s why this course will have a heavy emphasis on physics also.
With each lesson, you will learn a new mathematical concept (or several of them) as well as step-by-step examples of exactly how this concept can be applied to various areas of physics.
This will help you understand both how to calculate stuff, but also to apply the calculations to something practical, such as electromagnetism.
With each lesson, you will be given a problem set that can be found in the Workbook (note that all of the problem sets have not been created yet).
You’ll also find step-by-step solutions to each problem set there, but I highly recommend actually trying to do the practice problems on your own first. You really won’t be able to learn a new concept effectively if you don’t practice using that concept.
My personal recommendation is that you go through a single lesson at once, really try to internalize everything and then do the problem set associated with that lesson. After that, you can move on to the next lesson.
As new materials get added to the course, I will be updating this page regularly for instructions.
For now, however, I’d recommend you to just go ahead and move on to the first lesson named Coordinates, Vectors & Basis Vectors.
Report Mistakes/Errors In The Course Material
In case you encounter any errors, could you please fill them out in the form below? This helps me keep track of them in order to fix them later and only takes you a couple seconds!
The tangent of theta is given by the opposite over the adjacent side of a right triangle. However, in your section on vector components you state the tangent of theta is vx/vy, which would be adjacent over opposite.
Yes, you are completely correct. That seems to be a typo or mistake on my part, but it’s been updated now. Thank you for pointing it out and apologies for any confusion this might have caused!
There appears to be an error in your geometric definition of the cross product. If the vector u has components (0,1,0) and the vector v components (1,0,0) I get u x v has components (0,0,-1) = – z^. The algebraic definition seems to presume a right-handed coordinate system while the picture has a left-hand coordinate system.
Actually, the geometric and the algebraic definitions given technically differed by a minus sign, but they were both consistent with each other. This is because in the geometric definition, we had v x u, but in the algebraic definition, we instead had u x v. As you might know, the cross product is anticommutative, so the two definitions did give the same results. I’ve updated that lesson now to make it more clear.
On a positive note, I really like your integration of the physics with the math. Your style is also very easy to follow. However, I do find some things that might confuse people. For example, in the introductory section on the integral you have a misleading graph. dx1, dx2, dx3 should be the vertical changes in position and should not lie as represented along the curve. The horizontal changes should be shown as dt. Then it makes sense that the dx1 + dx2 + dx3 is the change in position.
Actually, yes, you are correct. I should not have put the curve in a t,x-coordinate system and labeled the same curve as x. What I was going after in that diagram was x being the distance (arc length, essentially) along the curve itself. I’ve made an update to that diagram now.
Thanks for pointing it out! The course is still a work-in-progress and there might be things that are unclear… your feedback is appreciated!
Hi
It may be that I am not skilled enough to navegate the course properly but I am quite unable to find any of the 11 worksheets and any of the 44 practice problems.
Please let me know how to find these.
Hi! The worksheets are available from the Gumroad course dashboard. You can access it from the link in the confirmation email you should have gotten when purchasing this course. Let me know if you’re unable to find this in your email inbox.
As a recent buyer of this course (today !) I want to have all the files as a PDF to be able to study offline whenever I want.
I already managed to download the cheatheet and the problems.
But where are the Part 1, Part 2 and Part 3 PDFs to be downloaded ?
Also where can I find the 44 practice problems ?
I’d really appreciate your reply.
Best regards
Victor Santos
victormsantos@netcabo.pt
Portugal
Hi, unfortunately the lessons themselves are only available online. This is an older course, so they haven’t been made into PDF’s, but in the Lagrangian mechanics course, everything is in PDF form.
For the practice problems, they should be available in the Gumroad dashboard. You were able to access this since you already downloaded the cheat sheet?
In the answer to problem 2.3 (c) you show the derivative of the velocity having zeros at t=0 and t= 2pi, which is true, but should there not also be a zero at t= pi?
There should, yeah. I forgot to mention that one for some reason. But the maximum is still at t=0, so the answer is the same regardless.
Good morning:
I would like to ask you about the publication of the next lessons. I have studied all the “Advanced Math For Physics” that have been published so far. I have the intention to study the “General Relativity Bundle” too, but I would like to cover the Mathematics of General Relativity part first. I would like to know if the publication of Lesson 3 (Covectors, Dual Vectors & One-Forms) and the followings are intended to be published soon or if I should switch to the General Relativity Bundle in the meantime. Thank you
Best Regards
J.H.
Hi! I’m currently in the middle of publishing a couple books and finishing work on my other course on Lagrangian mechanics and field theory. My intention is to get back to work on the Math For GR -course after these have been finished. I cannot give you an exact date yet unfortunately, but it likely won’t be before the end of the year. My recommendation is that you begin with the GR Bundle, as there are many articles in there that don’t require too many mathematical prerequisites. Besides, it might even be a good thing to familiarize yourself with general relativity before even doing the Mathematics For General Relativity -course. The best article to begin with in the bundle would probably be “General Relativity For Dummies: An Intuitive Introduction”.
Ville – two comments about “The Dot Product” section:
1. The number “5” is missing from the expression “vx= and ux = 2”
2. In the diagram for the component of v along u, shouldn’t the angle between u and the dotted red line be 90 degrees?
Yes, you are completely correct on both, those seem to be my mistakes. I’m actually in the middle of converting these lessons into PDF’s (Part 1 should be done in the next few days), so I’ll have these fixed. Thank you!
2 more quick comments:
1) In problem 2.1, the function has an exp(x^2 y^2) but in the solution, it just has exp(y^2)
2) In problem 4 (Brachistocrone) the velocity is given as sqrt(2gy). This doesn’t make sense to me, as it seems that the velocity should increase as y decreases. Am I missing something?
Thanks!
Thank you for pointing those out!
1) For this problem, I remember changing the problem from having the function exp(x^2y^2) to just exp(y^2), but looks like I forgot to update it from the problem description. So, the solutions should be correct.
2) Looks like you are correct on this, the velocity should be sqrt(2E/m-2gy). The correct expression and solution can actually be found in the “Examples of Variational Calculus” -lesson in Part 3.
I’ll have to fix these, but thank you again for letting me know! I try to be more careful with checking everything thoroughly with the newer lessons I’m creating, but on those older lessons, I seem to have made surprisingly many little mistakes… My apologies for the trouble.
For Problem 4.4, I’m getting that the B field is in the -y direction (not +y as in the solutions manual). Also, the sketch of the wave in the solution’s manual has a left-handed coordinate system.
I think you’re correct. Looks like I forgot a minus signs coming from the chain rule there. Could you please report the mistake in this form:
https://forms.gle/mKMopLbwgpmeYkE56
This just helps me keep track of all the errors, so it’s easier for me to fix them later. Thanks!
The end of the “Stokes and Divergence Theorem” section has no summary – just FYI. BTW, you have a very nice physical description of the fundamental theorem of calculus in that lesson – definitely thought provoking! Looking forward to going through the Mathematics of Relativity lessons.
Thank you, I’m glad you found it helpful! And yeah, I’m aware of the missing lesson summary – not sure where it has gone.