David Robertson ---------------------------------------------------------------------------------- djr0045

Week 9 ---------------------------------------------------------------------------------------------- Problem 5.5.7

DEFINITION:

Define coordinates q and p in a region of the (y,z) plane by y = q²sinh(p) and z = q²cosh(p).

Find formulas for

by implicit differentiation. (Answers will be in terms of q and p.)

COMMENT:

Remember that cosh²(x) - sinh²(x) = 1.

SOLUTION:

This problem is solved by first setting up the Jacobian matrix as follows.

Now, the inverse of the Jacobian matrix, J^-1, needs to be computed. In this calculation, the following formula will be used.

Now, for the computation of J^-1.

Thus, comparing matrices, we know that the answer the problem is stated by the four expressions