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?
Use the fact that [itex]\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[/itex]. Calculate [itex]\vec{F}\cdot \hat{a}_x[/itex] 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 a_{x}, a_{y}, a_{z} in my work but I just added them in there for clarity.
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.