I'm trying to prove that some mapping of a manifold is also a manifold using the constant rank theorem but I don't really understand the meaning of rank in this context.
As I understand it, the prescription is as follows:
Let M be a manifold with local coordinates $(x^1 \dots x^m)$
Let $\theta:M\rightarrow N$ be an immersion:
$\theta(x^1 \dots x^m) = (y^1 \dots y^n)$
Then N is a manifold if $d\theta = \frac{\partial\theta}{\partial x^1} dx^1 \dots \frac{\partial\theta}{\partial x^n} dx^n $
has constant rank.
This has something to do with ensuring a constant number of elements of the $\frac{\partial\theta}{\partial x^i}$'s are linearly independent I think? and usually has something to do with making sure they don't evaluate to 0's in the domain.
Essentially, if someone could tell me how to calculate rank in this situation, identify cases where it changes so we don't have a manifold and/or correct anything else I've misunderstood, that would be amazing. Examples would also be incredible as I can't find many online but no worries if not.
Thanks!