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Mastering Change of Variables in Multiple Dimensions

Learn how to simplify integrals by changing variables in 2D and beyond, including Jacobian determinants and practical examples.

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Mastering Change of Variables in Multiple Dimensions

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  1. Change of Variables In 2-D,  We changed the variables to make the integral easier.

  2. For double integrals, we change the variables to u and v.  Just like in 2-D where dx = ?? du, we need to have dy dx = ?? du dv.  This extra stuff is called the Jacobian

  3. Def. If x = g(u,v) and y = h(u,v), then the Jacobian of x and y with respect to u and v is (absolute value of the determinant)

  4. Ex. Find the Jacobian for the change of variables x = u cos v and y = u sin v.

  5. This means that where R is in the xy-plane and S is in the rθ-plane.

  6. The point is to make a simpler, more easily integrated region.

  7. Ex. Use the change of variables x = 2u + v, y = u +2v to evaluate the integral where R is the triangular region with vertices (0,0), (2,1), and (1,2).

  8. Ex. Let R be the region bounded by x – 2y = 0, x – 2y = -4, x + y = 1, and x + y = 4. Find a change of variables, sketch the image in the uv-plane, calculate the Jacobian, and evaluate .

  9. Def. If x = g(u,v,w), y = h(u,v,w), and z = j(u,v,w), then the Jacobian of x, y, and z with respect to u, v and w is

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