# Partial derivatives extensive use

1. Nov 2, 2012

### shivam jain

1. The problem statement, all variables and given/known data

let u be a function of x and y.using x=rcosθ y=rsinθ,transform the following expressions in the terms of partial derivatives with respect to polar coordinates:(d^u/dx^2(double derivative of u with respect to x)+d^2u/dy^2(double derivative of u with respect to y)
2. Relevant equations
chain rule in partial derivatives

3. The attempt at a solution
first i differentiated u with respect to theta by using chain rule and then with respect to r also by using chain rule.first has no r term wheras 2nd has r terms so no way the terms can cancel also.please tell me how to proceed or method i should use to solve this problem

2. Nov 3, 2012

### Staff: Mentor

Are these the partials you need to find?

$$\frac{\partial^2 u}{\partial r^2}~ \text{and}~\frac{\partial^2 u}{\partial^2 \theta}$$

If so, what did you get for these first partials?
$$\frac{\partial u}{\partial r}$$
$$\frac{\partial u}{\partial \theta}$$

For problems like this I find it helpful to draw a diagram of the relationships between all the variables.
Code (Text):

x ------ θ
/
u
\  y -------r

Although I can't show them, there are also lines between x and r and between y and θ.

The idea is that there are two ways to get from u to θ (through x and y), and there are two ways to get from u to r (also through x and y). This helps to get across the idea that each partial involves the sum of two terms.

Using subscripts to indicate partials, and letting u = f(x, y), we have
uθ = fx * xθ + fy * yθ, and
ur = fx * xr + fy * yr

To get the second partials (we don't call them double partials), you need to differentiate uθ with respect to θ, and differentiate ur with respect to r.