# Spherical to cartesian?

1. Sep 4, 2011

### yoamocuy

1. The problem statement, all variables and given/known data
A field is given in spherical coordinates as F=[cos(θ)/r2]∙ar+[sin(θ)/r]∙aθ. Express F in terms of x, y, z, ax, ay, az

2. Relevant equations

ar∙ax=sin(θ)cos(∅)
ar∙ay=sin(θ)sin(∅)
ar∙az=cos(θ)
aθ∙ax=cos(θ)cos(∅)
aθ∙ay=cos(θ)sin(∅)
aθ∙az=-sin(θ)
x=r*sin(θ)*cos(∅)
y=r*sin(θ)*sin(∅)
z=r*cos(θ)
r=√(x2+y2+z2 )
cos(θ)=z/r
∅=tan-1(y/x)

3. The attempt at a solution
cos(θ)/r2*[sin(θ)cos(∅)ax+sin(θ)sin(∅)ay+cos(θ)az]+sin(θ)/r*[cos(θ)cos(∅)ax+cos(θ)cos(∅)ay-sin(θ)az]

z/r3*[sin(θ)cos(∅)ax+sin(θ)sin(∅)ay+cos(θ)az]+sin(θ)/r*[cos(θ)cos(∅)ax+cos(θ)cos(∅)ay-sin(θ)az]

(z*r)/r4*[sin(θ)cos(∅)ax+sin(θ)sin(∅)ay+cos(θ)az]+sin(θ)/r*[cos(θ)cos(∅)ax+cos(θ)cos(∅)ay-sin(θ)az]

z/r4*[xax+yay+zaz]+sin(θ)/r*[cos(θ)cos(∅)ax+cos(θ)cos(∅)ay-sin(θ)az]

z/(x2+y2+z2)3*[xax+yay+zaz]+sin(θ)/r*[cos(θ)cos(∅)ax+cos(θ)cos(∅)ay-sin(θ)az]

That's about as far as I've gotten. I'm not even sure if what I've done so far is on the right track or not :/ I'm not sure what to do with the 2nd half of this equation?

Last edited: Sep 4, 2011
2. Sep 4, 2011

### vela

Staff Emeritus
Use the fact that $\vec{F} = (\vec{F}\cdot\hat{a}_x)\hat{a}_x + (\vec{F}\cdot\hat{a}_y)\hat{a}_y + (\vec{F}\cdot\hat{a}_z)\hat{a}_z$.

Calculate $\vec{F}\cdot \hat{a}_x$ using the various dot products you listed above. Then convert from the spherical variables to the Cartesian variables.

3. Sep 4, 2011

### yoamocuy

Isn't that what I did above? I didn't originally show the ax, ay, az in my work but I just added them in there for clarity.

4. Sep 4, 2011

### vela

Staff Emeritus
Oh, OK. I didn't see the unit vectors in your original attempt, so I figured you were doing it all wrong and didn't bother to look too closely.

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