# Jacobians and Surface integrals

## Main Question or Discussion Point

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 ||ru X rv|| dA

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

## Answers and Replies

HallsofIvy
Homework Helper
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.

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

Thanks for the response.

Would the Jacobian be used if:

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

?

HallsofIvy