Jacobians and Surface integrals

In summary, when evaluating a surface integral of a function over a two-dimensional surface in a three-dimensional space, the Jacobian is not used since it requires the same dimension on both sides. However, it may be used when changing from one three-dimensional coordinate system to another.
  • #1
IniquiTrance
190
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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?
 
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  • #2
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
What would be a case then where the jacobian matrix would be used in evaluating a surface integral?

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

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

?
 
  • #5
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.
 

1. What is a Jacobian and how is it related to surface integrals?

A Jacobian is a mathematical concept used to describe the relationship between two coordinate systems. In the context of surface integrals, the Jacobian represents the change in variables from one coordinate system to another. It is used to convert surface integrals from one coordinate system to another, making calculations easier.

2. How do you calculate the Jacobian of a surface integral?

The Jacobian of a surface integral can be calculated by taking the partial derivatives of the transformation equations between the two coordinate systems and arranging them in a specific way. This resulting matrix is then used to convert the integral to the desired coordinate system.

3. What is the significance of the Jacobian in multivariable calculus?

The Jacobian is an important concept in multivariable calculus because it allows for the conversion of integrals from one coordinate system to another. This is especially useful in solving problems involving multiple variables, as it simplifies calculations and allows for more efficient solutions.

4. Can the Jacobian be negative?

Yes, the Jacobian can be negative. This occurs when the transformation between coordinate systems results in a reversal of orientation. In surface integrals, this means that the direction of the normal vector may be flipped, resulting in a negative Jacobian value.

5. How is the Jacobian used in real-world applications?

The Jacobian is used in many real-world applications, including physics, engineering, and economics. It is particularly useful in problems involving fluid dynamics, heat transfer, and optimization. In these fields, the Jacobian allows for the conversion of integrals to different coordinate systems, making calculations more efficient and accurate.

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