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Background

Suppose $M$ is a smooth $n$-manifold and $(U,\varphi=(x^i))$ a smooth chart on $M$. For each point $p\in U$ we know that $(\frac{\partial}{\partial x^i}|_p)_{i=1}^n$ is a basis for the tangent space $T_pM$. If $(\tilde U,\tilde\varphi=(\tilde x^i))$ is another smooth chart on $M$ with $U \cap \tilde U\ne \emptyset$, then for each point $p\in U \cap \tilde U$ we have the following formula for the basis - change:

$$(\frac{\partial}{\partial x^1}|_p,\dots,\frac{\partial}{\partial x^n}|_p)=(\frac{\partial}{\partial \tilde x^1}|_p,\dots,\frac{\partial}{\partial \tilde x^n}|_p) \cdot \operatorname{Jac}(\tilde\varphi\circ\varphi^{-1})_{\varphi(p)} \quad[1]$$

where $\operatorname{Jac}(\tilde\varphi\circ\varphi^{-1})_{\varphi(p)}$ is the Jacobian matrix of $(\tilde\varphi\circ\varphi^{-1})$ in $\varphi(p)$.

Fact

Suppose we have the following vector field on $\mathbb{R}^2$: $X=2x \frac{\partial}{\partial x}$. Let $f:\mathbb{R}^2 \to \mathbb{R}, (x,y)\mapsto x^2$.

We want to compute the coordinate expression for $X$ in polar coordinates (on some open subset on which they are defined) and show that it is not equal to $\frac{\partial f}{\partial r}\frac{\partial }{\partial r}+ \frac{\partial f}{\partial \theta}\frac{\partial }{\partial \theta}$.

My understanding and execution

Let $M=\{(x,y)\in \mathbb{R}^2 : x>0\}$, and

$N=(0,+\infty)\times (-\frac{\pi}{2},\frac{\pi}{2})$.

Let $F=M\to N, (x,y)\mapsto (\sqrt {x^2+y^2},\arctan\frac{y}{x})$.

Then $F$ is a diffeomorphism with inverse $F^{-1}:N \to M, (r,\theta)\mapsto (r\cos\theta, r\sin\theta)$.

Issue 1: Polar coordinates are, by definition, the component functions of the map $F$, right? So we have $F=(r, \theta)$.

Great Issue 2: I have an ambiguity on how to interpret $\frac{\partial}{\partial r}$ and $\frac{\partial}{\partial \theta}$.

(a) Should I interpret $(r,\theta)$ also as standard coordinates on $N$ and thus $\frac{\partial}{\partial r}$ and $\frac{\partial}{\partial \theta}$ are simply the ordinary partial derivatives on $N$? (Just like $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ on $M$?)

(b) I know that if $M$ is a smooth $n$-manifold and $(U,\varphi=(x^i))$ is a smooth chart on $M$, then for each point $p\in U$ we have that $\frac{\partial}{\partial x^i}|_p=d\varphi^{-1}_{\phi(p)}(\partial_i|_{\phi(p)})$ where $\partial_i$ is the standard $i$-th partial derivative on $\mathbb{R}^n$. So thinking of my $F$ as a smooth chart, I should interpret $\frac{\partial}{\partial r}|_{(x,y)}=dF^{-1}_{F(x,y)}(\frac{\partial}{\partial x}|_{F(x,y)})$ for each $(x,y)\in M$.

Which one between (a) and (b) is the correct interpretation?

Issue 3 The vector field I'm looking for, say $Y$, should be the pushforward of $X$ by $F$?

Issue 4 Applying formula [1] I have

$$(\frac{\partial}{\partial x}|_{(x,y)},\frac{\partial}{\partial y}|_{(x,y)})=(\frac{\partial}{\partial r}|_{(x,y)},\frac{\partial}{\partial \theta}|_{(x,y)}) \cdot \operatorname{Jac}(F)_{(x,y)}$$ which i can rewrite

$$(\frac{\partial}{\partial x}|_{(r\cos\theta,r\sin\theta)},\frac{\partial}{\partial y}|_{(r\cos\theta,r\sin\theta)})=(\frac{\partial}{\partial r}|_{(r\cos\theta,r\sin\theta)},\frac{\partial}{\partial \theta}|_{(r\cos\theta,r\sin\theta)}) \cdot \operatorname{Jac}(F)_{(r\cos\theta,r\sin\theta)},$$ from which I have

$$\frac{\partial}{\partial x}|_{(r\cos\theta,r\sin\theta)}=\cos\theta \frac{\partial}{\partial r}|_{(r\cos\theta,r\sin\theta)}-\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}|_{(r\cos\theta,r\sin\theta)};$$ finally we have $$X=2x\frac{\partial}{\partial x}=2r\cos\theta\frac{\partial}{\partial x}|_{(r\cos\theta,r\sin\theta)}=2r\cos^2\theta \frac{\partial}{\partial r}|_{(r\cos\theta,r\sin\theta)}-2\sin\theta\cos\theta\frac{\partial}{\partial \theta}|_{(r\cos\theta,r\sin\theta)}.$$ I think there is something wrong with this and whit using formula [1] in this context. Instead, using the formula for computing the pushforward of $X$ by $F$, I got $$(F_*X)_{(r\,\theta)}=2r\cos^2\theta \frac{\partial}{\partial r}|_{(r,\theta)}-2\sin\theta\cos\theta\frac{\partial}{\partial \theta}|_{(r,\theta)}$$ which sounds correct.

So is there some "abuse of notation" which is confusing me? Am I right in saying that strictly speaking I cannot use formula [1]?

Please excuse me for the very long post, I have done the possible to make it clear.

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