I had a question involving the notion of coordinates. Here, I will use polar coordinates on $\mathbb{R}^2$ as an example. From my admittedly lacking understanding of differential geometry, polar coordinates form a chart on the differential manifold formed by the punctured plane $\mathbb{R}^2 / \{(0,0)\}$. Now, my understanding of a chart is that it is a homeomorphism from an open subset $U$ of the manifold $M$ to an open subset of a Euclidean vector space that effectively assigns local coordinates to $U$. This is where my confusion arises.

The usual assignment of polar coordinates to $\mathbb{R}^2 / \{(0,0)\}$ is something like $(x,y) = (r\cos(\theta),r\sin(\theta))$ , but the definition of a chart from above suggests that it should instead be something like $(r,\theta) = \big(\sqrt{x^2+y^2},\arctan\big({\frac{y}{x}}\big)\big)$. And yet, we are all familiar with the fact that the former assignment is used to calculate objects like the Jacobian matrix in polar coordinates and, consequently, the Euclidean metric in polar coordinates.

My question is, why the descrepancy? I apologize if I am not explaining this well enough, I will do my best to clarify further if need be.


You are right. The usual direction for a chart $(U,\phi)$ from an atlas is: \begin{align*} \phi:U\subset M\to\mathbb{R}^n \end{align*} where $U$ is an open set of the manifold $M$ and $\phi$ a homeomorphism. This convention is used to define "abstract" manifolds.

However in some cases, like the polar case, one use a parameterization, the only difference is that the maps are in the other direction. Here instead of using $\phi$, we use $\phi^{-1}$: \begin{align*} P:\mathbb{R}^2 &\to M \\ (r,\theta)&\mapsto P(r,\theta):=\phi^{-1}(r,\theta)=(r\cos\theta,r\sin\theta) \end{align*}

This is a common pratice in Differential geometry of surfaces, because $$ \phi^{-1}((r,\theta)\in\mathbb{R}^2)=(r\cos\theta,r\sin\theta) $$ is easier to manipulate than $$\phi(m\in M)=\big(\underbrace{\sqrt{x^2+y^2}}_r,\underbrace{\arctan\big({\frac{y}{x}}}_\theta\big)\big)$$.

But theoretically, as these functions are homeomorphisms, this is equivalent.

If you have the chance to have M. Spivak book

"A Comprehensive Introduction to Differential Geometry", Vol. 1, 3rd edition,

your trouble is exactly described page 36.


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