• Support PF! Buy your school textbooks, materials and every day products Here!

Partial derivatives extensive use

  • #1

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
 

Answers and Replies

  • #2
33,075
4,779

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.
Code:
      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.

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
 

Related Threads for: Partial derivatives extensive use

  • Last Post
Replies
2
Views
1K
Replies
5
Views
862
  • Last Post
Replies
5
Views
1K
Replies
0
Views
844
Replies
3
Views
1K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
7
Views
572
  • Last Post
Replies
1
Views
2K
Top