Finding flux through ellipsoid in Cylindrical Coordinates

[BrianA.]

Homework Statement

Using Cylindrical coordinates, find the total flux through the surface of the ellipsoid defined by x² + y² + ¼z² = 1 due to an electric field E = xx + yy + zz (bold denoting vectors | x,y,z being the unit vectors)

Calculate ∇⋅E and then confirm the Gauss's Law

Homework Equations

Cylindrical Coordinates being used: (s,φ,z)
Conversion to Cylindrical Coordinates:
x = scosφ
y = ssinφ
z= z

Surface Element of a Cylinder:
da = sdφdz

The Attempt at a Solution

I converted the ellipsoids equation into cylindrical, so it looks like:
s²cos²φ + s²sin²φ + ¼z² = 1

solving for s looks like s = √(1-¼z²)

I solved for both the volume and surface area of the ellipse through integration. Surface Area was as Follows:
∫ (from -2 to 2) ∫ (from 0 to 2π) sdφdz = ∫∫ √(1-¼z²) dφdz = 2π²

How to use this to find Flux, I am unsure of.
I know to convert E to cylindrical so E = scosφx + ssinφy + zz, but don't know what to do about the unit vectors.

Is the flux just ∫E⋅da ? and if so, how do I take the dot product and what do I do about the unit vectors?

 

[Chestermiller]

Are you allowed to use the divergence theorem?

Chet

 

[BrianA.]

I think that's expected.
The two are related right?

 

[Chestermiller]

I think that's expected.
The two are related right?

Yes. Please state the divergence theorem in terms of E. In your problem, the divergence of E is a constant. What is that constant?

Chet