Hey lytwynk.
Hint: Use the chain rule to get u_xx in terms of (p,theta,phi). (du/dx = du/dp*dp/dx + du/dtheta*dtheta/dx + du/dphi*dphi/dx).
Hello,
We were given the question below as an exercise but I am a bit confused and am in need of help.
Spherical coordinates (p, theta, phi) are defined by
x=pcos(theta)sin(phi)
y=psin(theta)sin(phi)
z=pcos(theta)
Compute Laplace's equation u_{xx} + u_{yy} + u_{zz} = 0 in spherical coordinates.
I would really appreciate any help.
I tried this question for about an hour already and haven't really gotten anywhere. I understand the chain rule but I don't really understand how to apply it to this question. If you have any other tips or a bit more of an explanation I would really appreciate it.
I gave you a hint with getting Ux in terms p, theta, and phi. To get Uxx you differentiate again. You do the same thing to get Uyy and Uzz.
Again here is the hint:
(du/dx = Ux = du/dp*dp/dx + du/dtheta*dtheta/dx + du/dphi*dphi/dx).
Now for Uxx: Let V = Ux
Uxx = dV/dp*dp/dx + dV/dtheta*dtheta/dx + dV/dphi*dphi/dx
I understand what you are saying to do but I don't know how... The functions in this question are U(x,y,z), x(p, theta, phi), y(p, theta, phi) and z(p, theta, phi).
I am confused here: du/dx = Ux = du/dp*dp/dx + du/dtheta*dtheta/dx + du/dphi*dphi/dx
du/dp: p is in x,y and z. Do I then have to differentiate those wrt p, then just multiply all the values by dp/dx, because you can't really differentiate and get a value for dp/dx...?
I am just getting pretty confused with this problem.
You can differentiate and get dp/dx by writing p as a function of the other variables.
The concept I have used for the relationship is here:
Total derivative - Wikipedia, the free encyclopedia
I'm sorry I really can't understand what you are trying to say or show me. I have rearranged the problem a bit.
It says there are spherical coordinates implying an equation f(x,y,z) = x^{2 }+ y^{2} + z^{2 }- p^{2}, which for this case I will call U, given x, y and z which are all functions of p, theta, phi.
so now for dU/dx = df/dx(dx/dp + dx/d(theta) + dx/d(phi))... or dU/dx = 2x(dx/dp + dx/d(theta)+ dx/d(phi))... is this correct?
Now if that is right from here I don't really understand how to get d^{2}u/dx^{2}.
Once I figure this out I will be able to do the same thing for d^{2}u/dy^{2} and d^{2}u/dz^{2}, then I can finish the problem.
I'll give you a bit more of a hint. Consider your equations:
x=pcos(theta)sin(phi)
y=psin(theta)sin(phi)
z=pcos(theta)
dx/dp = cos(theta)sin(phi) and as long dx/dp != 0 then dp/dx = 1/(dx/dp).
Alternatively you can re-arrange the equation to get p in terms of (x,y,z) by calculating the Jacobian and inverting it or by finding a clever relation. Anyway lets use the derivative expression that dx/dy = 1/(dy/dx) where dy/dx != 0.
So I posted the formula to get Ux.
Note that dU/dp is not known so you just leave this as is (and also for dU/dphi and dU/dtheta).
You should be able to finish this.
Ok. So I understand dx/dp = cos(theta)sin(phi) and as long dx/dp != 0 then dp/dx = 1/(dx/dp). Now How is dU/dp unknown because since U is a function of x,y,z and x,y,z are know can it not also be differentiated? And then like do I just continue the same technique to get Uxx Uyy and Uzz?
You just leave the dU/dother terms as they are: they are just place-holders.
Think of them as variables where a particular function will give a realization just as x is a variable and x = 2 is a realization.