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.
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.
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.
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 .
(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, .
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.
For the first assertion, apply part (c) with . Then . Applying (c) gives .
The second assertion follows from Fubini's Theorem.
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).