Jacobians

Originated by Carl Gustav Jacob Jacobi in the mid 1800's. The use of Jacobians in multivariable calculus is essential to making a change of variables when integrating a function over its domain. So, in order to change from one coordinate system to another, you must go through a specific process to obtain the multiplier, or Jacobian.

Basic example using six steps:

Step I: Define a new coordinate system for u and v. Based on our bounds for Ω, we want:

Step II: Using u and v, isolate what x and y are equal to in terms of u and v.

Step III: Find the Jacobian for the transformation by taking the absolute value of the determinate.

Step IV: Transform the bounds of Ω to the bounds of the new region in terms of u and v, called Γ.

Step V: Transform the function in terms of u and v.

Step VI: Integrate, don't forget to multiply by the Jacobian

This also works in 3-dimensions, just with a substantial amount of extra work. The process is essentially the same, except the integral is now triple instead of double, and the determinant is a much more demanding 3x3 for the Jacobian.

As you can see, using Jacobians extremely simplifies the process of integration in some situations and using Banchoff's applets helps to demonstrate how the change in coordinate systems affects integration. For instance, staying in the first coordinate system would require three separate integrals, while the (u,v) system requires only one simple integral.