Next semester (fall 2021) I am planning on taking a grad-student level differential topology course but I have never studied differential geometry which is a pre-requisite for the course. My plan is to self-study in the summer and this semester so that I do not have to waste a semester taking a differential geometry course which will ruin my ...

This course is an introduction to differential geometry. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature. The text used for this course is: Kuhnel, Wolfgang. Differential Geometry: Curves – Surfaces – Manifolds. Student mathematical library, vol. 16.

This is why differential geometry in Euclidean space is so much easier-the space comes equipped with very natural charts(i.e. Cartesian,plane and cylindrical polar coordinates,spherical coordinates). You're in luck since differential geometry of all the mathematical disciplines, has the largest number of clear textbooks for self learning.

2012年8月29日 · I'm interested in computations with vector-valued differential forms on Riemannien manifolds. Wedge product, exterior derivative, likely Hodge dual, tensor product, contractions of all kinds. Basically just rewriting and then possibly casting equations into …

2015年3月19日 · Also, there is the mathematical practicality that you want every point of the manifold to be inside and not on the edge of some set, particularly if you're doing calculus or differential geometry. You want a one-size-fits all, seamless, definition: you don't want to have to be dealing with onesided limits at edges of a partition.

I've come across the picture you're looking for in physics textbooks before (say, in classical mechanics). A bit of googling and I found this one for you!

2016年7月3日 · $\begingroup$ I know little about robotics, but in certain cases people do model the motions of robotics in the language of differential geometry. It makes sense because physical constraints (e.g., the length of an arm is fixed) can result in a background geometry different from Euclidean spaces.

2019年10月13日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

"Differential 2-form" roughly means "integrand of surface integrals" and so forth. The point of the construction mentioned is that you can represent a differential $1$-form as a cotangent vector field, which gives you a way to work with a differential $1$-form in terms of its values at individual points. $\endgroup$ –

The differential forms approach is indeed very powerful. What Hestenes points out in his From Clifford Algebra to Geometric Calculus is that to give a complete treatment of differential geometry of manifolds, you need various structures. In the book, you will find an alternative.