The Laplace Equation in Polar Coordinates

In summary, the equation above is equal to: \frac{\partial^2f}{\partial r^2}+\frac{1}{r^2} \frac{\partial ^2f}{\partial \theta^2} + \frac{1}{r} \frac{\partial f}{\partial r}= 0
  • #1
thejinx0r
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Homework Statement



[tex]\frac{\partial^2f}{\partial x^2}+\frac{\partial ^2f}{\partial y^2}= 0[/tex]


Homework Equations



Show that the equation above is equal to:
[tex]\frac{\partial^2f}{\partial r^2}+\frac{1}{r^2} \frac{\partial ^2f}{\partial \theta^2} + \frac{1}{ r} \frac{\partial f}{\partial r}= 0[/tex]

The Attempt at a Solution



So, let [tex]f=(r,\theta)[/tex]
[tex] r = \sqrt{x^2+y^2}[/tex]
[tex]\theta = tan^{-1}(y/x) [/tex]
then by the chain rule, partial f partial r is:

http://www.texify.com/img/%5CLARGE%5C%21%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20r%7D%20%5Cfrac%7Bx%7D%7B%20%5Csqrt%7Bx%5E2%20%2B%20y%5E2%7D%20%7D%20%2B%20%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20r%7D%20%5Cfrac%7By%7D%7B%20%5Csqrt%7Bx%5E2%20%2B%20y%5E2%7D%20%7D.gif

And I have something similar for partial f partial theta,
but I'm not sure if what I'm doing is right...

Because, I applied the chain rule to get the top part, (which I do not know how to write a matrix in latex :S).

But I've never applied the chain rule twice to get a second order differential.
So I'm stuck there. I tried to replace df/dr with d^f/dr^2 and similarly with the theta and multiplying it by the gaussian matrix (dr/dx dr/dy ; dt/dx dt/dy) [dt = d theta].

But then I realized that I would not get the 3rd term because of the way I just changed df/dr with d^f/dr^2 which is probably wrong on my part ...
 
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  • #2
One of your LaTeX images contained an error and didn't load properly. Was it supposed to be:

[tex]\frac{\partial f}{\partial r}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial r}[/tex]

If so, then just use this again:

[tex]\frac{\partial ^2 f}{\partial r^2}= \frac{\partial }{\partial r} \left( \frac{\partial f}{\partial r} \right)=\frac{\partial}{\partial x} \left( \frac{\partial f}{\partial r} \right) \cdot \frac{\partial x}{\partial r}+\frac{\partial }{\partial y} \left( \frac{\partial f}{\partial r} \right) \cdot \frac{\partial y}{\partial r}[/tex]

Substitute [itex]\frac{\partial f}{\partial r}[/itex] into this, and compute [itex]\frac{\partial x}{\partial r}[/itex] and [itex]\frac{\partial y}{\partial r}[/itex].
 
  • #3
So I fixed up the first post.

But it does help me do the second derivative and shows that I sort of did write it out wrong

:)
 

FAQ: The Laplace Equation in Polar Coordinates

1. What is the Laplace equation in polar coordinates?

The Laplace equation in polar coordinates is a mathematical equation that describes the distribution of potential and electric fields in a two-dimensional, circularly symmetric system. It is used to solve problems in electrostatics and fluid dynamics.

2. How is the Laplace equation derived in polar coordinates?

The Laplace equation in polar coordinates is derived from the Laplace equation in Cartesian coordinates by using the chain rule to convert the derivatives from Cartesian coordinates to polar coordinates. This results in the Laplace equation being expressed in terms of the radial and angular coordinates.

3. What is the significance of the Laplace equation in polar coordinates?

The Laplace equation in polar coordinates is significant because it allows for the solution of problems that have circular symmetry, which cannot be solved using the Laplace equation in Cartesian coordinates. It is also used in many real-world applications, such as in the analysis of electric fields and fluid flow.

4. How is the Laplace equation solved in polar coordinates?

The Laplace equation in polar coordinates is solved using separation of variables, where the solution is expressed as a product of two functions, one depending on the radial coordinate and the other on the angular coordinate. These functions are then solved separately using techniques such as Fourier series or Bessel functions, and the final solution is obtained by combining them.

5. What are some examples of problems that can be solved using the Laplace equation in polar coordinates?

The Laplace equation in polar coordinates can be used to solve problems such as the electric potential and field around a charged conducting sphere or the velocity field in a circularly symmetric flow. It is also applied in the study of heat conduction in circularly symmetric systems and the behavior of waves in circular domains.

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