How do you evaluate the vector calculus question over an ellipsoid?

[BrandonATC]

The question is

Let r=r(x,y,z) where it is the distance from a point O. Evaluate

\oint\nabla(1/r)*ndS

(where * is the dot product)

over the ellipsoid

x^2/4 +y^2/9+z^2/25=1

I thought the answer was 0 since the ellipsoid is a simply connected region in R^3 and the div(grad(1/r))= 0 since 1/r is harmonic.

But some people in the class said the answer was -4pi

Thanks

 

[tiny-tim]

Welcome to PF!

Hi BrandonATC! Welcome to PF!

(have a pi: π and a del: ∇ and try using the X² icon just above the Reply box )

… I thought the answer was 0 since the ellipsoid is a simply connected region in R^3 and the div(grad(1/r))= 0 since 1/r is harmonic.

No, it's not harmonic at the origin.

 

[BrandonATC]

thank you, just wondering how do you show the answer is -4pi?