Gradient and Divergence in spherical coordinates

[physicss]

Homework Statement

Hello, are my answers correct?

I had to calculate the gradient of f=z and f=xy in spherical coordinates.
My solution for f= z is: cos(θ) er+ (-sin(θ)) e(θ)

f=xy

rsin^2(θ)sin(2φ) er+ rcos(θ)cos(φ)sin(θ)sin(φ) e(θ)+ rcos(2φ)sin(θ) e(φ)

I also had to calculate the divergence of the following vectorfield (Image)

My result is: (scalarfield)

Sin^2(θ)cos^2(φ) er + rcos(2θ)cos^2(φ) e(θ)-rsin(θ)cos(2φ) e(φ)


Thanks in advance

Relevant Equations

er, e(θ) and e(φ) are the spherical base vectors

Vectorfield for the divergence

 

[haruspex]

rsin^2(θ)sin(2φ) er+ rcos(θ)cos(φ)sin(θ)sin(φ) e(θ)+ rcos(2φ)sin(θ) e(φ)

I get a slightly different $\vec e_\theta$.
Please learn to use LaTeX.

 

[haruspex]

My result is: (scalarfield)

Sin^2(θ)cos^2(φ) er + rcos(2θ)cos^2(φ) e(θ)-rsin(θ)cos(2φ) e(φ)

If it is a scalar, how come it has $\vec e_r$ etc?
I also note a dimensional inconsistency.

 

[lvirany]

You can get the gradient in spherical (or any other) coordinates by direct substitution, using basic linear algebra. From there you can get div, curl, and the Laplacian just by vector operations: