Comment by GaussBonnet | Hacker News Reader

Comment by GaussBonnet | Hacker News Reader

Parent Full thread

The point of differential forms is that they give a way to express geometric theorems in a coordinate free way. Coordinates are seen as obscuring the pure geometric content of theorems. They are sometimes necessary artifacts of doing concrete calculations, but the idea is that geometry shouldn't depend on a choice of coordinates.

The important ideas can be found in pages 9-10 in this link:

https://www.math.ucla.edu/~tao/preprints/forms.pdf

Note halfway down page 9 we get a really clean equation for how to change variables (change coordinates) in an abstract way.

Also note the simple form that the general n-dimensional stokes theorem takes in terms of differential forms at the top of this page:

https://en.wikipedia.org/wiki/Stokes%27_theorem

That they allow the expression of substantial theorems in concise form is a clue that they are the "right" way to do differential geometry.

> [...] but the idea is that geometry shouldn't depend on a choice of coordinates

Indeed, in "Tensor Geometry" (Dodson & Poston), they note:

> Most modern "differential geometry" texts use a coordinate-free notation almost throughout. This is excellent for a coherent understanding, but leaves the physics student quite unequipped for the physical literature, or for the specific physical computations in which coordinates are unavoidable. Even when the relation to classical notation is explained, as in the magnificent [Spivak], pseudo-Riemannian geometry is barely touched on. This is crippling to the physicist, for whom spacetime is the most important example, and perverse even for the geometer. Indefinite metrics arise as easily within pure mathematics (for instance in Lie group theory) as in applications, and the mathematician should know the differences between such geometries and the positive definite type. In this book therefore we treat both cases equally, and describe both relativity theory and (in Ch. IX, §6) an important "abstract" pseudo Riemannian space, SL(2;R).

Reply on news.ycombinator.com