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

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

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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)

4. 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

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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.

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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

7. 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

8. 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))

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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

10. 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

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

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

13. 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:

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

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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

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

18. 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

19. 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

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

21. Math 200

22. 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

23. 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

24. 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

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

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

28. 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

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Jacobian:
Setup: