Problem 5. Divergence and volume:
a) Let S be a closed smooth surface, and let E denotes the region inside S. Show the relation:
Vol(E)=(1)/(3)∯_(S)(x,y,z)*hat(n)dS,
where Vol(E) denote the volume of E.
b) Use the above result and find the volume of the ellipsoid:
(x^(2))/(a^(2))+(y^(2))/(b^(2))+(z^(2))/(c^(2))=1
Note that the natural parametrization of an ellipsoid is: x=asin(phi )cos( heta ),y=bsin(phi )sin( heta )
and z=ccos(phi ).
Problem 5. Divergence and volume
a) Let S be a closed smooth surface, and let E denotes the region inside S. Show the relation:
Vol(E)=
(x,y,z).nds
where Vol(E) denote the volume of E
b) Use the above result and find the volume of the ellipsoid:
x2 y2 z2
Note that the natural parametrization of an ellipsoid is: x=sin()cos(), y=bsin()sin( and z=ccos().