Hi. I am taking a calculus 3 class and was sent home with this problem:
The partial differential equation representing two-dimensional heat flow on
a rectangular plate is:
DE: <du/dt> = k(<d^2u/dx^2> + <d^2u/dy^2>)
where u=u(x,y,t) is the temperature at time t at the point (x,y) in a
rectangular region in the xy-plane.
If a circular plate is employed, however, polar coordinates are utilized.
Determine the form of the above partial differential equation in polar
coordinates. Note: Use x=r cos(theta) and y=r sin(theta), thus the DE would
then yield a solution of the form: u=u(r,theta,t)
I'm really not sure how to get started on this one. It seems like I would
need an original polar equation u for the temperature in r, theta, and t so
I could attempt to take the partial derivitive of u with respect to t. I
understand what he means by x=r cos(theta) and y=r sin(theta) but where
does t come in? I'm lost...
Any help you can give would be greatly appreciated. I'm not looking for the
answer but some direction would be great.
Thanks!
The partial differential equation representing two-dimensional heat flow on
a rectangular plate is:
DE: <du/dt> = k(<d^2u/dx^2> + <d^2u/dy^2>)
where u=u(x,y,t) is the temperature at time t at the point (x,y) in a
rectangular region in the xy-plane.
If a circular plate is employed, however, polar coordinates are utilized.
Determine the form of the above partial differential equation in polar
coordinates. Note: Use x=r cos(theta) and y=r sin(theta), thus the DE would
then yield a solution of the form: u=u(r,theta,t)
I'm really not sure how to get started on this one. It seems like I would
need an original polar equation u for the temperature in r, theta, and t so
I could attempt to take the partial derivitive of u with respect to t. I
understand what he means by x=r cos(theta) and y=r sin(theta) but where
does t come in? I'm lost...
Any help you can give would be greatly appreciated. I'm not looking for the
answer but some direction would be great.
Thanks!