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## Homework Statement

Using Cylindrical coordinates, find the total flux through the surface of the ellipsoid defined by x

^{2}+ y

^{2}+ ¼z

^{2}= 1 due to an electric field

**E**= x

**x**+ y

**y**+ z

**z**(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:

s

^{2}cos

^{2}φ + s

^{2}sin

^{2}φ + ¼z

^{2}= 1

solving for s looks like s = √(1-¼z

^{2})

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 = ∫∫ √(1-¼z

^{2}) dφdz = 2π

^{2}

How to use this to find Flux, I am unsure of.

I know to convert

**E**to cylindrical so

**E**= scosφ

**x**+ ssinφ

**y**+ z

**z**, but don't know what to do about the unit vectors.

Is the flux just ∫

**E⋅**

*d*

**a**? and if so, how do I take the dot product and what do I do about the unit vectors?