Start watching the video. Pause after a few minutes to look sth up on wikipedia regarding the topic on hand. Spend 3 hrs researching and trying out stuff. Find the open browser tab with the video. Watch video til the end and realize, that you just learned, what he explained in the following minutes. You, sir, are an absolutely amazing teacher.
I quite often split multi-integral calculations in this way (over radius and sphere). It is common that the integral over the sphere needs to be split again into an integral over the inclination (or latitude) and a sphere of lower dimension.
Thank you for sharing this high class mathematics that I have never seen it. It would be great motivation why I have to study harder. You are such a great teacher. Thank you!!!
This is a really cool way to do polar integrals, wish I'd been taught this. I had a question for you, by the way, regarding how Multivariable Calculus is traditionally taught. I'm prepared to go into a Mathematics PhD in order to be a professor, and I find it a little weird how Multivariable calculus is taught. Some schools split multivariable calculus into two sections (the first doing partial derivatives and multiple integrations, and the second doing vector calculus with line, surface integrals and divergence/stokes theorem.) The end of multivariable calculus seems like a PERFECT lead-in to Differential Geometry for differential forms, hodge operators, generalized stokes' theorem, etc. However, those topics aren't taught typically in any undergrad courses, and is only touched on in one graduate-level course at my university. Is there any reason this connection isn't more smooth? I've only seen one multivariable/vector calculus book that goes into differential geometry/forms. Do you see a reason why this isn't included after the discussion of divergence/stokes' theorem? For schools that put multivariable calculus in one section, I can see a time constraint not allowing this, but if you are doing two sections of multivariable/vector calculus, I don't see a reason why this wouldn't happen. It seems natural to include it. Thanks again.
Firstly, I really think the first step would be to move from traditional vector calculus to geometric calculus, which does cover the generalized Stokes theorem and Cauchy's theorem in a single stroke (heh). So there is time to cover some introductory differential geometry in such a course.
It's like when we integrate along a fixed value of x or y (a strip of area in the input space) and then integrate with respect to the strip of area over the entire region over which we are integrating when we do double integrals in Cartesian coordinates.
Hey, Dr Peyam! Awesome video as always. May I ask which drawing software you are using? I will be working as a tutor this semester and would very much like to use this program you are using!
Thanks for explaining so nicely! I have a question: @8:20...I thought we had wanted to integrate f over D. But, the level surfaces are of another function H...Where does H come from...How does it relate to f and D?
So, you say we take the Hausdorff measure of the hypersurface. Wikipedia indicates that in order to define a Hausdorff measure, one needs a notion of distance, so I am wondering which distance you are using here? The two natural ones seem to be the metric induced by the restriction of the metric of R^n on the level sets, or the infemum of the length of curves contained in the hypersurface that connects the two points? Or do these both give rise to the same (Hausdorff) measure?
@@drpeyam Oh that is weird, because the wikipedia article says that the normal measure on the sphere is the one induced the smaller angle between the two points (which should also be the smallest distance of a path connecting the points on the sphere when the radius is 1) en.wikipedia.org/wiki/Spherical_measure
I want to be like Dylan, Dr. Peyam etc. Solving Complex Mathematics looks so cool! I want to do research in Quantum Mathematics. Such that: I think Complex numbers r the subset of Quantum numbers. Simultaneously quantum numbers are the subset of the complex numbers. (I m not taking about binary digits) -A Brand New Extended Number System 🙏🏻
I read in a book about advanced Calculus something called "A comparison criterion for convergence" and it's the following; Let {𝑎ₙ} converge to 𝑎. Then {𝑏ₙ} converges to 𝑏 if and only if there is a non negative number 𝐶 and an index 𝑁₁ such that: | 𝑏ₙ - 𝑏 | ≤ 𝐶 | 𝑎ₙ - 𝑎 |. Im having a lot of trouble understanding what this even means, and apparently its used for the rest of the book, can anyone explain it for me? Thanks! ❤️
My limited analysis knowledge wasn't enough to understand this :(. What exactly is the surface measure on a sphere? Is it just the jacobian wrapped up to look pretty?
Eeemmm, so what? It's exactly the same formula, and ofc everyone who takes a serious calculus class is taught to do this. It's not in any general way better than the "usual" way to integrate a function (as I said, it is the same formula, actually) , only when f(x) is spherically/radially symmetric has it an advantage over the "standard" formula. In fact, in all other cases you still need to use the general dphi-dtheta integration first to determine the measure on a spherical/radial layer.
