# Polar coordinate and chart on manifold.

Let $M$ a manifold of dimension $2$.

Q1) I know that we can have charts $U$ that gives coordinates $(x_1,x_2)$ and charts $V$ that gives coordinates $(y_1,y_1)$. Now are polar coordinates is given by a new charts $W$ or from each charts $U$ and $V$ I can get polar coordinate defined as $$(x_1,x_2)=(r\cos\theta ,r\sin\theta )$$ and $$(y_1,y_2)=(\rho\cos\varphi,\rho\sin\varphi) \ \ ?$$

Q2) For a function $f:M\to \mathbb R^2$ for example. Do we have that in $U$ that $\nabla f=(\frac{\partial f}{\partial x-1},\frac{\partial f}{\partial x_2})$ and in $V$ that $\nabla f=(\frac{\partial f}{\partial y_1},\frac{\partial f}{\partial y_2})$ ? Or the gradient is defined in a specific charts only ?

The answer to Q1) for $U$ depends on the image of the chart in the $x_1,x_2$ plane. Let me use the notation $f : U \to \mathbb R^2$ for the coordinate map, where $f(p)=(x_1(p),x_2(p))$ for each $p \in U$. Also, for the image of $U$ under this coordinate map, let me use the notation $$\text{image}(f) = \{(x_1(p),x_2(p)) \mid p \in U\} \subset \mathbb R^2$$ The problem is that polar coordinates are not defined for all subsets of the plane. For instance, polar coordinates are not defined for any subset that contains the origin $(0,0)$. Also, polar coordinates are not defined for any subset that contains a circle around the origin, because the angle coordinate $\theta$ is not well-defined (going all the way around the circle increases $\theta$ by $2\pi$). However, polar coordinates are defined for any simply connected subset that does not contain the origin. So, what you can conclude is that if $\text{image}(f)$ is simply connected and does not contain the origin, or even if $\text{image}(f)$ is contained in a simply connected set that does not contain the origin, then yes, polar coordinates can be defined on $U$. For a very specific example, if $U$ is homeomorphic to the open 2-disc then $\text{image}(f)$ is also homeomorphic to the open 2-disc and is therefore simply connected; hence, polar coordinates are defined as long as $\text{image}(f)$ does not contain the origin.