Calculating the partial derivative in polar coordinates

  • #1
james weaver
27
4
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.
 

Answers and Replies

  • #2
PeroK
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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/
 
  • #3
PhDeezNutz
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I’m also lost on the ##\frac{\partial z}{\partial f}## expression when ##z## is literally defined as ##z = f##.
 
  • #4
Orodruin
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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
 

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