spherical to cartesian?

by yoamocuy
Tags: cartesian, conversion, spherical
 P: 40 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?
 Emeritus Sci Advisor HW Helper Thanks PF Gold P: 11,521 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.
P: 40
 Quote by vela 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.
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.

Emeritus