I am playing around, writing a program that will rotate a set of points in 3d space and I am using a bit of trigonometry for this. When I think about a 90 degree rotation visually, it seems natural that cos(pi/2) * tan(pi/2) should equal 1, and my program depends on that result. Luckily, that’s the answer that python computes. Even in an Excel spreadsheet, that’s the answer. However, analytically, the answer should be 0, since cos(pi/2) is 0. Or at least the answer should be undefined, since tan(pi/2) is infinity.
So I guess that’s the mathematical answer, that the value is undefined. But my question is, is it conventional to just agree that the answer is 1? Is that why computer programs will return a value of 1? Or is the value of 1 returned only because of technical reasons, in that cos(pi/2) is actually stored as some minuscule non-zero number and tan(pi/2) is stored as some large but finite number? Or is it both, in that the computer would return a value of 1 for unintended technical reasons, but we all agree that it’s ok, because we agree to define cos(pi/2) * tan(pi/2) as 1?
I recently got into complex analysis and I often see people use the residue theorem to solve real integrals from -∞ to ∞.
One technique that I often see people use is, let gamma 1 be the parametrization of a semicircle with radius R on the upper-half complex plane, and let gamma 2 be the line from R to -R. Combining these two curves gives us a intagral over a closed curve, which we can use the residue theorem on.
They then use the estimation theorem to prove that the integral over gamma 1 is zero, which means that the integral over gamma is equal to the original integral. My question is, what if gamma 1 is nonzero? Is there some other techniques to evaluate integrals of such kind?
On Monday I have to present an exam which will mainly be about showing and proving results, particularly showing if a set is convex, if a function is (quasi)concave/convex, proving by direct argument if a point is a miximum. We can use the textbook and our own notes so memorizing theorems won’t be a problem, so do you have any suggestions?
My only motivation for learning Abstract Algebra is to have examples to play around with for category theory.
I am thinking of self-studying the first two chapters of Aluffi’s Algebra: Chapter 0, which covers basic set theory, category theory (basics) and group theory, up to Lagrange’s theorem. I’m already familiar with abstract linear algebra from Axler.
So would linear algebra and group theory be sufficient to learn about and appreciate category theory? Or should I learn more abstract algebra? How much topology should I know?
My interest in learning category theory is in foundations of mathematics and type theory.
I’m not asking you to do my homework. This is a legitimate brain burning problem. This problem is a problem that nobody in my campus (not any math teacher) could wrap their heads around. I also asked some random people who like math and nobody could solve it. I’m asking people to prove that my solution is wrong but they can’t. I’m not asking for an alternative solution that is correct, because I acknowledge that there is a correct solution to it and thus I lost 2 points. I know the solution. I’m asking you to prove WHY my particular solution is wrong. Here’s the question: https://ibb.co/pd0bNnx
So I am in the second year of my degree in mathematical physics. Next semester I’ll be taking an introductionary course in quantum mechanics aimed at physics majors. The course won’t be “mathematician”-level rigorous but I want to understand the content on a completely rigorous and proof based level. I have a bit of time during the semester break to get some mathematics under my belt I need for this.
So far I took the following classes relevant to this:
- Two semesters of Linear Algebra and Real Analysis (both “proof based”)
- A semester of Functional Analysis where we coverd Banach and Hilbert spaces, basic operator theory and the spectral theorem for unbounded normal operators on Hilbert spaces
- A semester of Measure and integration theory where we covered basic measure theory, the Lebesgue integral and L^p spaces
- A semester of Abstract Algebra where we covered basic group, ring, and field theory (this won’t be too relevant for quantum mechanics I suppose).
What other topics of mathematics do I need to get a good understanding on introductionary quantum mechanics and it’s mathematical foundations? I guess partial differential equations are important but I don’t know how much I need to know about them. What other “big theorems” like the spectral theorem are used in quantum mechanics I need a thorough understanding of? It’d be awesome if someone could help me!