MATH 200 WEEK 10 WEDNESDAY THE JACOBIAN CHANGE

MATH 200 WEEK 10 - WEDNESDAY THE JACOBIAN & CHANGE OF

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 uvplane. 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

MATH 200 ▸ Let’s label the corners of S as follows ▸ A(u 0, v 0) ▸ B(u 0 + Δu, v 0): B is a a little to the right of A ▸ C(u 0, v 0 + Δv): C is a little higher that A ▸ D(u 0 + Δu, v 0 + Δv): D is a little higher and to the right of A

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(u 0, v 0), y(u 0, v 0)) ▸ B’(x(u 0 + Δu, v 0), y(u 0 + Δu, v 0)) ▸ C’(x(u 0, v 0 + Δv), y(u 0, v 0 + Δv)) ▸ D’(x(u 0 + Δu, v 0 + Δv), y(u 0 + Δu, v 0 + 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 ZCOMPONENT BECAUSE CROSS PRODUCTS ARE ONLY DEFINED FOR

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 APPROXIMATE BECAUSE WE’RE MISSING THE L

MATH 200 ▸ Applying this idea to both vectors, we get

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

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 xyplane

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 TRANSFOR M THIS REGION INTO AN UPRIGHT RECTANGL E

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

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

MATH 200 ▸ For the bounds, we have ▸ Finally, our integral becomes

MATH 200

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

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

MATH 200 ▸ For the Jacobian, we need the partials of x=v/u and y=v ▸ xu = -v/u 2 ▸ xv = 1/u ▸ yu = 0 ▸ Lastly, we need to convert the integrand (the function we’re integrating) ▸ yv = 1 ▸ 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, xy 2 = 1, and xy 2 = 2. Evaluate the following integral. R

MATH 200 ▸ Let u=xy and v=xy 2 ▸ 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/y 2 ▸ Setting those equal: u/y = v/y 2 ▸ Solve for y: y = v/u ▸ Plug into the u-equation: u=x(v/u) ▸ x = u 2/v

MATH 200 ▸ Jacobian: ▸ Setup: