On being able to play with a topic

I’ve always felt like I don’t have a decent grasp on a topic until I can “play” with it. For example, I have lots of fun asking questions in elementary real analysis, and to some extent topology and linear algebra, which usually leads to nice problems and more fun questions.

But with certain topics, like complex analysis, abstract algebra, and basically anything at the grad level, I can’t play with them. I don’t know how to come up with interesting questions to ask, and I can’t create any problems. I can understand definitions on a surface level, and solve textbook exercises somewhat fluently, but I still don’t feel like I have enough intuitive understanding to play with the topic.

So, some questions:

1) Usually the best way to really get to know a subject is to play with it, but how do you get good enough to play when you can’t play yet?

2) Does anyone else share this view on playing? Are there also topics for you that you feel comfortable playing with, and some where you just feel stuck?

submitted by /u/WaltWhit3
[link] [comments]

Why should we pursue “symmetry” in Gauss Quadrature ?

In the context of the Gauss quadrature of order 2 for triangle, determining the weights and quadrature points involves solving a system of quadratic equations.

In this document (http://math2.uncc.edu/~shaodeng/TEACHING/math5172/Lectures/Lect_15.PDF), section C.1.b it is mentioned that solving such system with 1 quadrature point is not possible, which is obviously explained by the over-determination of the system. However, the reason why N_g = 2 is not chosen is beyond me. It is mentioned that N_g = 3 is chosen for symmetry reasons. It is not clear to me WHY symmetry is a required property in this case.

In https://math.stackexchange.com/questions/2200409/why-is-symmetry-in-gaussian-quadrature-over-a-triangle-a-beneficial-feature, it is mentioned that symmetry might mean invariance to some transformations. However, if the function to approximate is of degree 2 or less, the quadrature is said to be exact. Therefore I don’t really understand why one would require invariance to rotation.

Is this “symmetry” property only required when one wants to approximate a polynomial of degree greater than the Gauss quadrature ?

submitted by /u/henker92
[link] [comments]

Writing My First Real Analysis Exam Tomorrow

The class is going pretty well for me but I’m still pretty nervous. For reference, we cover everything from elementary set theory to point set topology, sequences, limits of functions, continuity (uniform and lipschitz) and basic compactness (Heine-Borel specifically) [roughly the first 5 chapters of Bartle ] . For those of you further along, anything you wish you knew when you were in my position? Did one problem kill you on your first final? Any sage worthy advice?

submitted by /u/sidmad
[link] [comments]