Exercises: Chapter 3, Section 6

1. Use Theorem 3-14 to prove Theorem 3-13 without the assumption that .

Let Then is open and Theorem 3-13 applies with in place of the in its statement. Let be a partition of unity subordinate to an admissible cover of . Then is a partion of unity subordinate to the cover . Now is absolutely convergent, and so also converges since the terms are identical. So, . By Theorem 3-14, we know that . Combining results, we get Theorem 3-13.

2. If and , prove that in some open set containing we can write , where is of the form , and is a linear transformation. Show that we can write if and only if is a diagonal matrix.

We use the same idea as in the proof of Theorem 3-13. Let be a point where . Let , and . Then . Define for , . Then . So we can define on successively smaller open neighborhoods of , inverses of and . One then can verify that . Combining results gives

and so .

Now, if is a diagonal matrix, then replace with . for and . Then the have the same form as the and .

On the other hand, the converse is false. For example, consider the function . Since is linear, ; so is not a diagonal matrix.

3. Define by .
1. Show that is 1-1, compute , and show that for all . Show that is the set of Problem 2-23.

Since , to show that the function is 1-1, it suffices to show that and imply . Suppose . Then implies that (or ). If , it follows that . But then and has the same value, contrary to hypothesis. So, is 1-1.

One has

So, for all in the domain of .

Suppose , i.e. and . If , then implies and so . But then contrary to hypothesis. On the other hand, if , then let and let be the angle between the positive -axis and the ray from (0,0) through . Then .

2. If , show that , where

(Here denotes the inverse of the function .) Find P'(x,y). The function is called the polar coordinate system on .

The formulas for and follow from the last paragraph of the solution of part (a). One has . This is trivial from the formulas except in case . Clearly, . Further, L'H@ocirc;pital's Rule allows one to calculate when by checking separately for the limit from the left and the limit from the right. For example, .

3. Let be the region between the circles of radii and and the half-lines through 0 which make angles of and with the -axis. If is integrable and , show that

If , show that

Assume that and . Apply Theorem 3-13 to the map by . One has and . So the first identity holds. The second identity is a special case of the first.

4. If , show that

and

For the first assertion, apply part (c) with . Then . Applying (c) gives .

The second assertion follows from Fubini's Theorem.

5. Prove that

and conclude that

One has and the integrands are everywhere positive. So

Since part (d) implies that , the squeeze principle implies that also.

But using part (d) again, we get also exists and is (since the square root function is continuous).