Jacobians and Surface integrals

IniquiTrance

Why is it that when we evaluate a surface integral of:

f(x, y ,z) over dS, where

x = x(u, v)
y = y(u, v)
z = z(u, v)

dS is equal to ||r_u X r_v|| dA

Why don't we use the jacobian here when we change coordinate systems?

HallsofIvy

Because you are NOT "changing coordinate systems"- not in the sense of replacing one 3 dimensional coordinate system with another or replacing one 2 dimensional coordinate system with another. The Jacobian is the determinant of an n by n matrix and so requires that you have the same dimension on both sides. That is not the situation when you have a two dimensional surface in a three dimensional space.

IniquiTrance

What would be a case then where the jacobian matrix would be used in evaluating a surface integral?

Thanks for the response.

IniquiTrance

Would the Jacobian be used if:

x = x(u, v, w)
y = y(u, v, w)
z = z(u, v, w)

?

HallsofIvy

Yes, but of course that's not a "surface integral"- that's changing from one three-dimensional coordinate system to another. You might, after forming the integral over a surface, decide that the integral would be simpler if you chose different coordinates, that is a different parameterization, for the surface. Then you would use the Jacobian to change from one two-dimensional coordinate system to another.