Problem 1

Write out the chain rule for each of the following functions.

(a). Where

(b). Where

(c). Where

Problem 2

Using the chain rule evaluate and given that the function , with , , and .

Problem 3

Consider the functions , , and , where , , and . Assuming that and determine the gradient of the function at .

Problem 4

Let be an arbitrary differentiable function defined on the entire real line. Show that the function defined on the entire plane as satisfies the partial differential equation