Centroid of a Hemisphere in Spherical Coordinates

[whereisnomar]

Homework Statement

Find the centroid of the hemisphere, z=sqrt(a-x^2-y^2).

Homework Equations

z(bar)=(1/m)* Surface integral(z dS)
dS= magnitude of the magnitude of normal vector * dA

The Attempt at a Solution

I use the gradient to the hemisphere to get the magnitude of the normal vector... which comes out to be 2a. But the book says to use the cross product of the partial derivatives of a parametrization of the function as a parametric surface. And I get the wrong answer when I use the gradient.

Why can't you use the gradient to get the normal needed for a surface integral?

 

[Dick]

If you say f(x,y)=sqrt(a^2-x^2-y^2) then the position vector on the surface is r=[x,y,f(x,y)]. And if I take the function to find the gradient of as z-f(x,y), then I get that the gradient vector happens to be the same as the cross product r_x x r_y. But neither one of them has norm 2a. Can you show what you did? In general taking a gradient will give you a normal, but it won't necessarily give you the same normal in terms of magnitude as r_x x r_y. The latter is the correct one to use.

 

[HallsofIvy]

I would be inclined to write the hemisphere in spherical coordinates with $\rho= a$.