Expressing a surface in cartesian coordinates from spherical

[acedeno]

Homework Statement

The following equation describes a surface in spherical coordinates. θ =pi/4
Write the equation in the cartesian coordinates?

that is, (r,θ,Ø) to (x,y,z)

Homework Equations

x=rsinθcosØ
y=rsinθsinØ
z=rcosθ

r=sqrt(x^2+y^2+z^2)
θ=cos^-1(z/r)
Ø=tan^-1(y/x)

The Attempt at a Solution

I'm pretty stumped. The only start I can get is that if θ=pi/4, this means that r and phi have to work around this. Allowing r to be from 0 to infinity. I'm not really sure what happens to phi. I'm thinking that it can be anywhere from 0 to 2pi.

 

[WJSwanson]

If your surface is simply defined as

[itex]\theta = \pi/4[/itex]

then all you need to do is solve your conversion factor from theta of

[itex]\theta = cos^{-1}(\frac{z}{r}) = cos^{-1}(\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}})[/itex]

Since your value of theta is a constant, you just have

[itex]\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}} = cos(\frac{\pi}{4})[/itex]

Although you may want to make sure you have your notation correct. Generally I've seen

[itex]z = r cos\phi, x = r cos\theta sin\phi, y = r sin\theta sin\phi[/itex]

and if you did in fact get your notation mixed up (which it's possible that you did or didn't, but it's worth checking) that changes the nature of your calculations.