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What goes wrong with this argument?

Suppose I have a curve $\gamma: [a, b] \rightarrow R^n$. We express this in polar coordinates as $\gamma(t) = (r(t), \theta(t))$. The derivative of $\gamma$ is just taken componentwise, so $\gamma'(t) = (r'(t), \theta'(t))$. Now the length of $\gamma$ is given by $\int_a^b \| \gamma'(t) \| \; dt = \int \|(r'(t), \theta'(t))\| \; dt$. In polar coordinates, the norm of $(r'(t), \theta'(t))$ is simply $r'(t)$. It follows that the length is just $r(b) - r(a)$, but this is clearly wrong. Where are the mistakes?

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2 Answers

The differential of arc length is $\sqrt{(dr)^2+(r d\theta)^2}$ in polar coordinates.

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When you have $\gamma(t) = (x(t),y(t))$, it is then right to say $\gamma'(t) = (x'(t),y'(t))$. However, when you are in other coordinate systems, you need to be careful. For instance, if you move to the $r-\theta$ coordinate system, and parameterize $\gamma(t)$ by $(r(t),\theta(t))$, it is incorrect to say that the curve $\gamma'(t)$ is $(r'(t),\theta'(t))$.

To make it clear, let $\gamma_1(t) = (x(t),y(t))$ and $\gamma_2(t) = (r(t),\theta(t))$ be the parameterizations of the same curve in different coordinate systems. $\gamma_1$ is in the $X-Y$ coordinate system, while $\gamma_2$ is the parameterization of the same curve in $r-\theta$ coordinate system.

Note that $x(t) = r(t) \cos(\theta(t))$, $y(t) = r(t) \sin(\theta(t))$, $r(t) = \sqrt{x(t)^2 + y(t)^2}$, $\tan(\theta(t)) = \frac{y(t)}{x(t)}$.

$\gamma_1'(t) = (x'(t),y'(t)) = (r' \cos(\theta) - r \sin(\theta) \theta',r' \sin(\theta) + r \cos(\theta) \theta')$.

If you express this in $r-\theta$ coordinate system, $\gamma_2'(t) = (\sqrt{r'^2 + (r \theta')^2},\tan^{-1}(\frac{r' \sin(\theta) + r \cos(\theta) \theta'}{r' \cos(\theta) - r \sin(\theta) \theta'})) = (r_1(t),\theta_1(t))$.

So the length of the curve is $\int {||\gamma_1'(t)|| dt}=\int {\sqrt{x'(t)^2 + y'(t)^2} dt}=\int {\sqrt{r'^2 + (r \theta')^2} dt}=\int {r_1(t) dt}=\int {||\gamma_2'(t)|| dt}$.

Note

$r(t) = ||\gamma_1(t)||$.

Note that $r'(t) = ||\gamma_1(t)||' \neq ||\gamma_1'(t)||$.

You need to apply chain rule to get the relation between $r'(t)$ and $\gamma_1'(t)$.

$r'(t) = \frac{\partial r}{\partial x} x'(t) + \frac{\partial r}{\partial y} y'(t) = \frac{x}{\sqrt{x^2+y^2}} x'(t) + \frac{y}{\sqrt{x^2+y^2}} y'(t)$ $r'(t) = \cos(\theta(t)) x'(t) + \sin(\theta(t)) y'(t) = (\cos(\theta),\sin(\theta)) \cdot \gamma_1'(t)$

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