Jacobians and Surface integrals

  • #1
190
0

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

  • #2
HallsofIvy
Science Advisor
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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.
 
  • #3
190
0
What would be a case then where the jacobian matrix would be used in evaluating a surface integral?

Thanks for the response.
 
  • #4
190
0
Would the Jacobian be used if:

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

?
 
  • #5
HallsofIvy
Science Advisor
Homework Helper
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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.
 

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