How to calculate a Vector Field in Spherical Coordinates I am having trouble with the following problem. I keep on getting a long unmanagable result - so any suggestion as to where I've gone wrong/how to do this would be a lifesaver! Please?
Consider a Vector Field in $\mathbb R^3 $   Given By F($\bar{x}$)=$\bar{\varepsilon}  \times \bar{x}$
Where $\bar{\varepsilon}$ is a fixed non-zero vector and $\bar{x}$ is some variable vector
Compute this Vector Field in Spherical coordinates.
I have assumed that they want me to express this field using spherical coordinates bases $\bar{e  }_{p}$ $\bar{e  }_{\phi}$ $\bar{e  }_{\theta}$
$\bar{e  }_{p}$ =$\cos \theta \sin \phi $ $\bar{i}$ + $\sin \theta \sin \phi $$\bar{j}$ +  $\cos \phi$$\bar{k}$
$\bar{e  }_{\phi}$ = $\cos \theta$ $\cos \phi $$\bar{i}$ + $\cos \phi $$ \sin \theta $$\bar{j}$ - $ \sin \phi $$\bar{k}$
$\bar{e  }_{\theta}$ = $ \sin \theta $$\bar{i}$  +$\cos \theta $$\bar{j}$
Let $\bar{x}$ = x$\bar{i}$+y$\bar{j}$ + z$\bar{k}$ 
In Spherical Coordinates x=p $\cos \theta \sin \phi $
y=p $\sin \theta \sin \phi $
 z=p $\cos \phi$
$\bar{x}$=p $\cos \theta \sin \phi $$\bar{i}$ += p $\sin \theta \sin \phi $$\bar{j}$ +p $\cos \phi$$\bar{k}$
Then  $\bar{x}$= p$\bar{e  }_{p}$ 
I then calculated the Fixed Vector in relation to the spherical coordinate base
$\bar{\varepsilon}$ = a$\bar{i}$+b$\bar{j}$ + c$\bar{k}$   (Vector in relation to Cartesian Base) where a,b,c, are constants 
$\bar{\varepsilon}$ = a($\cos \theta \sin \phi $$\bar{e  }_{p}$ + $\cos \phi \cos\theta $$\bar{e  }_{\phi}$ - $\sin \theta $$\bar{e  }_{\theta}$) + b($\sin \theta \sin \phi $$\bar{e  }_{p}$ + $\cos \phi \sin \theta $$\bar{e  }_{\phi}$ + $\sin \theta $$\bar{e  }_{\theta}$) + c($\cos\phi $$\bar{e  }_{p}$ - $\sin \phi $$\bar{e  }_{\phi}$ (used inverse relation between bases)
$\bar{\varepsilon}$ = (a$\cos \theta \sin \phi $ + b$\sin \theta \sin \phi $+ c($\cos\phi $)$\bar{e  }_{p}$ + (a$\cos \phi \cos\theta $+b$\cos \phi \sin \theta $ - c$\sin \phi $)$\bar{e  }_{\phi}$ +(b$\sin \theta $-a$\sin \theta $)$\bar{e  }_{\theta}$
I then took the cross-product of these two vectors.(I know how to do the cross-product so that's not an issue - however I am not sure that what I have done here is correct and I am extremely uncomfortable with the result. Where have I gone wrong? Any help, suggestion or comment would very very gratefully recieved
 A: I'll take a different approach here. Suppose $\vec{\varepsilon}$ is fixed vector, or constant vector field with respect to the cartesian frame. Then as we write $\vec{\varepsilon}$ in the spherical frame the coefficients manifest a point-dependence. In particular, there exist functions $\varepsilon_{\rho},\varepsilon_{\phi},\varepsilon_{\theta}$ such that 
$$\vec{\varepsilon} = \varepsilon_{\rho}\widehat{e}_{\rho}+\varepsilon_{\phi}\widehat{e}_{\phi}+\varepsilon_{\theta}\widehat{e}_{\theta} $$
we can calculate these coefficient functions by $\varepsilon_{\rho}=\vec{\varepsilon} \cdot \widehat{e}_{\rho}$, $\varepsilon_{\phi}=\vec{\varepsilon} \cdot \widehat{e}_{\phi}$ and $\varepsilon_{\theta} =\vec{\varepsilon} \cdot \widehat{e}_{\theta}$. This follows from the orthonormality of the spherical frame. (you can check, and I think you already realize $\widehat{e}_{\rho} \cdot \widehat{e}_{\rho}=1, \widehat{e}_{\rho} \cdot \widehat{e}_{\phi}=0$ etc...). However, you should also realize that $\{ \widehat{e}_{\rho}, \widehat{e}_{\phi}, \widehat{e}_{\theta} \}$ forms a right-handed-triple in the sense that the cross-products of  $\widehat{e}_{\rho}, \widehat{e}_{\phi}, \widehat{e}_{\theta}$ share the same patterns as that of the standard cartesian frame:
$$ \widehat{e}_{\rho} \times \widehat{e}_{\phi} =  \widehat{e}_{\theta}, \ \ 
 \widehat{e}_{\phi} \times \widehat{e}_{\theta} =  \widehat{e}_{\rho}, \ \
\widehat{e}_{\theta} \times \widehat{e}_{\rho} =  \widehat{e}_{\phi} $$
Now, as you point out, $\vec{x} = \rho \widehat{e}_{\rho}$ thus,
$$ \vec{F} = \vec{\varepsilon} \times \vec{x}  = (\varepsilon_{\rho}\widehat{e}_{\rho}+\varepsilon_{\phi}\widehat{e}_{\phi}+\varepsilon_{\theta}\widehat{e}_{\theta} ) \times \rho \widehat{e}_{\rho} = -\rho \varepsilon_{\phi}\widehat{e}_{\theta} 
+ \rho \varepsilon_{\theta}\widehat{e}_{\phi}$$
now, we just need to calculate those dot-products and we're done. 
That said, I'd rather use some notation like $\vec{A}$ instead of $\vec{ \varepsilon}$ since $e$ and $\varepsilon$ look so similar. In $\vec{A}$-notation,
$$ \vec{F} = \vec{A} \times \vec{x}  = (A_{\rho}\widehat{e}_{\rho}+A_{\phi}\widehat{e}_{\phi}+A_{\theta}\widehat{e}_{\theta} ) \times \rho \widehat{e}_{\rho} = -\rho A_{\phi}\widehat{e}_{\theta} 
+ \rho A_{\theta}\widehat{e}_{\phi}$$
