Math 242, Spring 2019, Assignment 12

"When I think of Euclid even now, I have to wipe my sweaty brow."

- C. M. Bellman

Read:

1. Section 15.8.
2. Section 15.9.

Carefully define the following terms, then give one example and one non-example of each:

1. $\frac{\partial(x,y)}{\partial(u,v)}$ (the Jacobian of the two-dimensional coordinate system $(u,v)$).
2. $\frac{\partial(x,y,z)}{\partial(u,v,w)}$ (the Jacobian of the three-dimensional coordinate system $(u,v,w)$).
3. $\rho, \theta,$ and $\phi$ (the spherical coordinates of a point in three-dimensional space).

Carefully state the following theorems (you do not need to prove them):

1. Formula for the reduction of double integrals to iterated integrals in an arbitrary two-dimensional coordinate system (i.e. $dA=\dots$).
2. Formula for the reduction of double integrals to iterated integrals in an arbitrary three-dimensional coordinate system (i.e. $dV=\dots$).
3. Formulas to convert between rectangular and spherical coordinates.
4. Formula for the reduction of double integrals to iterated integrals spherical coordinates (i.e. $dV=\dots$).

Solve the following problems:

1. Section 15.8, problems 1, 3, 5, 7, 9, 11, 15, 17, 19, 20, 21, 23, 25, and 41.
2. Section 15.9, problems 1, 3, 11, 13, 15, 17, 21(a-b), and 23.

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