# Differentiating a polar function

• kgal
In summary, when changing to polar coordinates, the partial derivatives of ∂z/∂r and ∂z/∂θ can be calculated using the chain rule for functions of two variables. Specifically, ∂z/∂r = (∂z/∂x)(∂x/∂r) + (∂z/∂y)(∂y/∂r) and ∂z/∂θ = (∂z/∂x)(∂x/∂θ) + (∂z/∂y)(∂y/∂θ).
kgal

## Homework Statement

let z=f(x,y) be a differentiable function. If we change to polar coordinates, we make the substitution x=rcos(θ), y=rsin(θ), x^2+y^2=r^2 and tan(θ) = y/x.
a. Find expressions ∂z/∂r and ∂z/∂θ involving ∂z/∂x and ∂z/∂y.
b. Show that (∂z/∂x)^2 + (∂z/∂y)^2 = (∂z/∂r)^2 + (1/r^2)(∂z/∂θ)^2.

## The Attempt at a Solution

a. i understand that f(x,y) in polar is f(r,θ) but don't understand how to calculate the partial derivatives of ∂z/∂x and ∂z/∂y because there is not know function for z...

Do you know the chain rule for functions of two variables?

in the specific case of this problem they come out like this:
∂z/∂r = (∂z/∂x)(∂x/∂r) + (∂z/∂y)(∂y/∂r)
∂z/∂θ = (∂z/∂x)(∂x/∂θ ) + (∂z/∂y)(∂y/∂θ)

right?

## What is a polar function?

A polar function is a mathematical function that describes a relationship between a point and a fixed point, called the pole, in a polar coordinate system. It is expressed in terms of the distance from the pole (r) and the angle from a fixed reference direction (θ).

## How do you differentiate a polar function?

To differentiate a polar function, you can use the polar form of the derivative formula, which is given by dr/dθ = (dr/dx)(dx/dθ) + (dr/dy)(dy/dθ). You can find the derivatives of r with respect to x and y by using the chain rule and then plug them into the formula.

## What is the difference between finding the derivative of a polar function and a Cartesian function?

The main difference between finding the derivative of a polar function and a Cartesian function is that in polar coordinates, the independent variable is the angle (θ) instead of the traditional x-coordinate used in Cartesian coordinates. This means that when finding the derivative, you must consider the effect of both the change in radius (r) and the change in angle (θ).

## What are the common types of polar functions?

The most commonly used types of polar functions include circles, cardioids, roses, and limaçons. These functions are all defined by a specific relationship between the distance from the pole (r) and the angle from the reference direction (θ).

## Why is differentiating polar functions important?

Differentiating polar functions is important because it allows us to find the instantaneous rate of change of a polar function at any given point. This is useful in many fields of science and engineering, such as physics, astronomy, and mechanics.

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