Partial derivatives extensive use

[shivam jain]

Homework Statement

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)

Homework Equations

chain rule in partial derivatives

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

 

[Mark44]

Homework Statement

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)

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.

      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
u_θ = f_x * x_θ + f_y * y_θ, and
u_r = f_x * x_r + f_y * y_r

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

Homework Equations

chain rule in partial derivatives

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