# Jacobian of polar coordinates

## Homework Statement

Change of coordinates from rectangular (x,y) to polar (r,θ). Not sure what's wrong with my working..

## Answers and Replies

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pasmith
Homework Helper
Your error is that you have differentiated $y = x \tan\theta$ incorrectly. You should get
$$\frac{\partial y}{\partial \theta} = x\sec^2\theta + \frac{\partial x}{\partial \theta}\tan\theta$$
because $x$ is also a function of $\theta$.

Also, to work out $\partial x/\partial r$, you would need to differentiate
$$r^2 = x^2 + y^2$$
with respect to $r$ with $\theta$ held constant, which again gives
$$2r = 2x\frac{\partial x}{\partial r} + 2y\frac{\partial y}{\partial r}$$
because y is also a function of $r$.

If you're trying to find derivatives with respect to r and $\theta$, it's best to start from $x = r\cos\theta$, $y = r\sin\theta$.

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