Chapter 12.5, Problem 55E

Walking on a surface Consider the following surfaces specified in the form z = f(x, y) and the oriented curve C in the xy-plane.

1. a. In each case, find z’ (t).
2. b. Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing).

55.     $z=\sqrt{1-x^2-y^2,}C:x=e^{-t},y=e^{-t}t\ge \frac{1}{2}\ln 2$