(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A field is given in spherical coordinates as F=[cos(θ)/r^{2}]∙a_{r}+[sin(θ)/r]∙a_{θ}. Express F in terms of x, y, z, a_{x}, a_{y}, a_{z}

2. Relevant equations

a_{r}∙a_{x}=sin(θ)cos(∅)

a_{r}∙a_{y}=sin(θ)sin(∅)

a_{r}∙a_{z}=cos(θ)

a_{θ}∙a_{x}=cos(θ)cos(∅)

a_{θ}∙a_{y}=cos(θ)sin(∅)

a_{θ}∙a_{z}=-sin(θ)

x=r*sin(θ)*cos(∅)

y=r*sin(θ)*sin(∅)

z=r*cos(θ)

r=√(x^{2}+y^{2}+z^{2})

cos(θ)=z/r

∅=tan^{-1}(y/x)

3. The attempt at a solution

cos(θ)/r^{2}*[sin(θ)cos(∅)a_{x}+sin(θ)sin(∅)a_{y}+cos(θ)a_{z}]+sin(θ)/r*[cos(θ)cos(∅)a_{x}+cos(θ)cos(∅)a_{y}-sin(θ)a_{z}]

z/r^{3}*[sin(θ)cos(∅)a_{x}+sin(θ)sin(∅)a_{y}+cos(θ)a_{z}]+sin(θ)/r*[cos(θ)cos(∅)a_{x}+cos(θ)cos(∅)a_{y}-sin(θ)a_{z}]

(z*r)/r^{4}*[sin(θ)cos(∅)a_{x}+sin(θ)sin(∅)a_{y}+cos(θ)a_{z}]+sin(θ)/r*[cos(θ)cos(∅)a_{x}+cos(θ)cos(∅)a_{y}-sin(θ)a_{z}]

z/r^{4}*[xa_{x}+ya_{y}+za_{z}]+sin(θ)/r*[cos(θ)cos(∅)a_{x}+cos(θ)cos(∅)a_{y}-sin(θ)a_{z}]

z/(x^{2}+y^{2}+z^{2})^{3}*[xa_{x}+ya_{y}+za_{z}]+sin(θ)/r*[cos(θ)cos(∅)a_{x}+cos(θ)cos(∅)a_{y}-sin(θ)a_{z}]

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?

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# Spherical to cartesian?

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