Calculating the partial derivative in polar coordinates

[james weaver]

Homework Statement

show the relationship between rectangular and polar partial derivatives

Relevant Equations

symbolic

Hello, I am trying to solve the following problem:

If $z=f(x,y)$, where $x=rcos\theta$ and $y=rsin\theta$, find $\frac {\partial z} {\partial r}$ and $\frac {\partial z} {\partial \theta}$ and show that $\left( \frac {\partial z} {\partial x}\right){^2}+\left( \frac {\partial z} {\partial y}\right){^2}=\left( \frac {\partial z} {\partial r}\right){^2}+\frac 1 {r^2}\left( \frac {\partial z} {\partial \theta}\right){^2}$

I am a little lost on how, I know I need to use the chain rule. This is what i have so far:

$\frac {\partial z} {\partial r}=\frac {\partial z} {\partial f}\frac {\partial f} {\partial r}$
and
$\frac {\partial z} {\partial \theta}=\frac {\partial z} {\partial f}\frac {\partial f} {\partial \theta}$

So i can find them symbolically, but not sure how to explicitly. If anyone has a good video they can shoot my way I would appreciate that as well. Thanks.

 

[PeroK]

I am a little lost on how, I know I need to use the chain rule. This is what i have so far:

$\frac {\partial z} {\partial r}=\frac {\partial z} {\partial f}\frac {\partial f} {\partial r}$
and
$\frac {\partial z} {\partial \theta}=\frac {\partial z} {\partial f}\frac {\partial f} {\partial \theta}$

These are not correct. Try this to get you started:

https://tutorial.math.lamar.edu/classes/calciii/chainrule.aspx

Or, you could look at my Insight:

https://www.physicsforums.com/insights/demystifying-chain-rule-calculus/

 

[PhDeezNutz]

I’m also lost on the $\frac{\partial z}{\partial f}$ expression when $z$ is literally defined as $z = f$.

 

[Orodruin]

I’m also lost on the $\frac{\partial z}{\partial f}$ expression when $z$ is literally defined as $z = f$.

Presumably it is therefore equal to one, which essentially results in a rather trivial statement.

OP: You want to apply the chain rule to the case when you express x and y as functions of the polar coordinates, ie,
$$
\frac{\partial}{\partial r}f(x(r,\theta),y(r,\theta))=\ldots
$$
etc