How can I calculate the gradient in a new coordinate system?

Question

Let $f: \mathbb{R}^{2} \to \mathbb{R}$, where $f(x,y) = x^{2}+y^{3}$. Calculate $\nabla{f}$ in the new coordinate system $(r,\theta)$, where $$ x = r\cos{\theta}$$ $$ y = r\sin{\theta}$$

I know that $\nabla{f}$ is a covariant vector. How can I answer this in terms of $r$ and $\theta$

One option is to take $\nabla f(x,y)$ then convert to polar. — Andrew Tawfeek, Sep 11 '17 at 15:02
Related: https://math.stackexchange.com/questions/586848/how-to-obtain-the-gradient-in-polar-coordinates — Hans Lundmark, Sep 11 '17 at 15:10

score 0 · Accepted Answer · answered Sep 11 '17 at 15:19

0

Notice that $\nabla f = (2x,3y^2)$, and by applying the transformation

$$\begin{cases} x = r \cos \theta \\ y = r \sin \theta \end{cases}$$

we obtain:

$$\nabla f(r, \theta) = (2r\cos \theta, 3r^2\sin^2 \theta) = 2r \cos \theta \hat{i} + 3r^2 \sin^2 \theta \hat{j}$$

answered Sep 11 '17 at 15:19

Thank you! I'm sorry if this was too simple of a question, I didn't know that was what was required. Can you explain how $\nabla f$ being a covariant vector is relevant? – user8358234 Sep 11 '17 at 19:34

How can I calculate the gradient in a new coordinate system?

1 Answers1