1. The problem statement, all variables and given/known data 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 2. Relevant 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 3. 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?