Calculating the partial derivative in polar coordinates

In summary, the problem involves finding the partial derivatives of z with respect to r and theta when z is a function of x and y, which are themselves functions of polar coordinates. The chain rule needs to be applied in order to find these derivatives.
  • #1
james weaver
28
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.
 
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  • #2
james weaver said:
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
I’m also lost on the ##\frac{\partial z}{\partial f}## expression when ##z## is literally defined as ##z = f##.
 
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  • #4
PhDeezNutz said:
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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Related to Calculating the partial derivative in polar coordinates

What is a partial derivative in polar coordinates?

A partial derivative in polar coordinates is a mathematical concept used to calculate the rate of change of a function with respect to one variable while holding all other variables constant. It is represented by the symbol ∂ and is often used in multivariable calculus.

How do you calculate the partial derivative in polar coordinates?

The partial derivative in polar coordinates can be calculated by taking the derivative of the function with respect to the desired variable and then substituting in the corresponding polar coordinate expressions for that variable. This can be done using the chain rule and the product rule, if necessary.

What is the difference between a partial derivative and a total derivative in polar coordinates?

A partial derivative in polar coordinates only considers the rate of change of a function with respect to one variable, while holding all other variables constant. A total derivative, on the other hand, takes into account the changes in all variables simultaneously. In polar coordinates, the total derivative is often represented by the symbol d.

Why are polar coordinates useful in calculating partial derivatives?

Polar coordinates are useful in calculating partial derivatives because they provide a more intuitive understanding of how the variables in a function are related. This can make it easier to visualize and solve problems involving multiple variables and their rates of change.

Can the partial derivative in polar coordinates be applied to any function?

Yes, the partial derivative in polar coordinates can be applied to any function that is expressed in terms of polar coordinates. This includes functions that are not explicitly written in polar form, as they can be converted using the appropriate trigonometric identities.

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