The Jacobian & Change of variables Math 200 Week 10 - Wednesday The Jacobian & Change of variables

Math 200 Goals Be able to convert integrals in rectangular coordinates to integrals in alternate coordinate systems

Math 200 Definition Transformation: A transformation, T, from the uv-plane to the xy-plane is a function that maps (u,v) points to (x,y) points. x = x(u,v); y = y(u,v)

Math 200 Example Consider the region in the rθ-plane bounded between r=1, r=2, θ=0, and θ=π/2 The transformation T:x=rcosθ, y=rsinθ maps the region in the rθ-plane into this region in the xy-plane Transformations need to be one-to-one and must have continuous partial derivatives

Math 200 Jacobian The area of a cross section in the xy-plane may not be exactly the same as the area of a cross section in the uv- plane. We want to determine the relationship; that is, we want to determine the scaling factor that is needed so that the areas are equal.

Math 200 Image of s under t Suppose that we start with a tiny rectangle as a cross section in the uv-plane with dimensions u and v. The image will be roughly a parallelogram (as long as our partition is small enough). T: x=x(u,v), y=y(u,v) x and y are functions of u and v

Let’s label the corners of S as follows A(u0,v0) Math 200 Let’s label the corners of S as follows A(u0,v0) B(u0 + Δu,v0): B is a a little to the right of A C(u0,v0 + Δv): C is a little higher that A D(u0 + Δu,v0 + Δv): D is a little higher and to the right of A

What happens to these points under the transformation T? Math 200 What happens to these points under the transformation T? Well, we have transformation functions x(u,v) and y(u,v), so we can plug the coordinates of the points A, B, C, and D to get the corresponding points in the xy-plane For the image of a point P under T, we’ll write T(P) = P’, so we get: A’(x(u0,v0),y(u0,v0)) B’(x(u0 + Δu,v0), y(u0 + Δu,v0)) C’(x(u0,v0 + Δv), y(u0,v0 + Δv)) D’(x(u0 + Δu,v0 + Δv), y(u0 + Δu,v0 + Δv))

Math 200 We want to know how the area of R compares to the area of S. R is a parallelogram, so its area is the cross-product of the vectors a and b: We’ve added the z-component because cross products are only defined for vectors in 3-space

It’s approximate because we’re missing the limit Math 200 To simplify these vectors down to something more manageable, we look to the limit definition of the partial derivative: From calc 1 Notice that the first component of the vector a looks like the numerator of the limit definition of partial derivative of x with respect to u It’s approximate because we’re missing the limit

Applying this idea to both vectors, we get Math 200 Applying this idea to both vectors, we get

Now we can take the cross product Math 200 Now we can take the cross product For the area of R, we get

Conclusions We’ve shown that the area of R is The area of S is Math 200 Conclusions We’ve shown that the area of R is The area of S is So the scaling factor we were looking for is |xuyv - xvyu| We can write this as the determinate of a special matrix called the Jacobian Matrix:

Math 200 Example Compute the Jacobian (i.e., the determinate of the Jacobian Matrix) for the transformation from the rθ-plane to the xy- plane

Math 200

Math 200 Theorem Given a transformation T from the uv-plane to the xy-plane, if f is continuous on R and the Jacobian is nonzero, we have For example, when converting from rectangular to polar, we have

Math 200 Example Consider the following double integral: R It would be nice If we could transform this region into an upright rectangle

Let’s take the transformation T to be Math 200 Let’s take the transformation T to be To convert our xy-integral to a uv-integral, we want the Jacobian We could solve for u and v in our transformations OR we could use the convenient fact that

First, we need the partial derivatives: Math 200 First, we need the partial derivatives: Plug it all in to the Jacobian Matrix Take the reciprocal to get

Finally, our integral becomes Math 200 For the bounds, we have Finally, our integral becomes

Math 200

Example 2 Use the transformation x = v/u, y=v to evaluate the integral Math 200 Example 2 Use the transformation x = v/u, y=v to evaluate the integral Let’s first convert the region from the xy-plane to the uv-plane The region extends from the line y=0 to the line y=x From x=0 to x=1

So the region looks the same in the uv-plane Math 200 Apply the conversion formulas x=v/u and y=v y=0 becomes v=0 y=x becomes v=v/u u=1 x=0 becomes v/u=0 v=0 x=1 becomes v/u=1 v=u In summary, the region extends from the line v=0 to the line v=u Bounded by u=1 So the region looks the same in the uv-plane

For the Jacobian, we need the partials of x=v/u and y=v Math 200 For the Jacobian, we need the partials of x=v/u and y=v xu = -v/u2 xv = 1/u yu = 0 yv = 1 Lastly, we need to convert the integrand (the function we’re integrating) Now we have all the pieces we need to set up the integral

Math 200 At this point we need to do integration by parts twice to finish the problem

Math 200

Math 200 Example 3 Let R be the region enclosed by xy = 1, xy = 2, xy2 = 1, and xy2 = 2. Evaluate the following integral. R

This gives us a nice rectangular region in the uv-plane Math 200 Let u=xy and v=xy2 This gives us a nice rectangular region in the uv-plane u=1 to u=2 and v=1 to v=2 To find the Jacobian we’ll need to solve for x and y in terms of u and v Solve each equation for x to get x = u/y and x = v/y2 Setting those equal: u/y = v/y2 Solve for y: y = v/u Plug into the u-equation: u=x(v/u) x = u2/v

Math 200 Jacobian: Setup: