Calculate Surface Integral F.ndS on Sphere at Origin

[mmh37]

I am really struggling with this one:

Calculate $$\Int F.ndS$$, where

$$F = a * x^3 * i + b*y^3*j + c*z^3*k$$

where a,b and c are constants,

over the surface of a sphere of radius a, centred at the origin.

note that F and n are vectors (sorry, tried to type them in bold...but it doesn't work)

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So, this is my attempt:

convert everything in polar coordinates and integrate it

where

dS = r^2*sinx*cosz (

for only a hemisphere though...I would multiply it by 2 afterwards to make it a sphere)

the final integral is then:

$$dS = \Int {a*dS} = \Int {r^3 (a*sin^3x*sin^3z + b*sin^3x*sin^3z + c*cos^3x) * r^2*sinx*cosz}$$

And this is just a mess. What is wrong here?

 

[CarlB]

Try doing the dot product in Euclidean coordinates, but then still write the integral in polar coordiantes. The result of a dot product is a scalar, and the scalar will therefore be simpler to convert into polar coordinates than those nasty vectors.

Carl