Local Coordinates and the Local Immersion Theorem Explained


The way to think about coordinate systems is totally different than we might have treated them in a freshman calculus course or even a course on multivariable calculus course. In this article, I’ll attempt to try and explain how you should think about this topic.

The best way to start is to try and forget everything you know about coordinates from before and dive straight into the definition.

Definition

Let \(X\) be a manifold (\(C^{\infty}\)). Then a chart or a local coordinate system is a mapping \(\phi: U \to V\) where \(\phi\) is a diffeomorphism mapping an open subset of \(X\) to an open subset of a euclidean space.

Yes, a coordinate system is simply just a mapping, a function from a piece of a manifold to a piece of a euclidean space. Furthermore, we write \(\phi\) as \((x_1, x_2, …, x_k)\) (with \(k\) being the dimension of \(U\)). In this case, each individual \(x_i\) is called a local coordinate function.

How does this fit in with my previous understanding of coordinates?

You may have thought coordinates as something which was implicitly connected to the space you were working in (e.g. \(\mathbb{R}, \mathbb{R}^2, …\)) in the way that giving a coordinate was the same as pointing out a unique place on the space. This is in fact, still correct. In this case, let \(X = \mathbb{R}^n\) and if our neighbourhood \(U \subset X\) is of dimension \(k\), then we let it be the identity mapping.