2020年12月21日 · The way they did geometry from the ancient Greeks to Descartes. Coordinates and their maps are the foundation of standard differential geometry. The theory is coordinate …

2009年10月20日 · The property you are interested in is known as being normal.For affine varieties, the definition of normal is just that the coordinate ring is integrally closed, and the …

Differential geometry doesn't see rational points. Neither does algebraic geometry over $\mathbb C$. EDIT Considering the comments of the OP, I think I should be more explicit in my answer.

2024年9月11日 · What is true is that there always exists one such coordinate system where the expansion holds. These are called "Bochner coordinates" or "K-coordinates". You can just …

2016年10月22日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for …

1.The Euler-Lagrange equations are coordinate independent, but if the Lagrangian is a bit more sophisticated it is difficult to come-up with an invariant form. One convenient way to do this is …

On an oriented Riemannian 2-manifold, every harmonic function u has a harmonic conjugate function v, defined by the relation dv = * du, where * denotes the Hodge star operator. Near a …

2017年11月16日 · $\begingroup$ @BigbearZzz, I have yet to find a rigorous and readable introduction to differential (and Riemannian) geometry. I advise against Chavel's cited book, …

Note also that you can push forward the standard coordinate fields in the plane by an arbitrary diffeomorphism and obtain random-looking vector fields that Lie-commute by construction. …

I'm trying to motivate a bit of algebraic geometry in an abstract algebra course (while simultaneously trying to learn a bit of algebraic geometry), and I thought that it might be nice to …