# Exercises: Chapter 2, Section 4

1. Find expressions for the partial derivatives of the following functions:
1. , , .

2. Let . For , the limit

if it exists, is denoted and called the directional derivative of at , in the direction .

1. Show that .

This is obvious from the definitions.

2. Show that .

3. If is differentiable at , show that and therefore .

One has

which shows the result whenever . The case when is trivially true.

The last assertion follows from the additivity of the function .

3. Let be defined as in Problem 2-4. Show that exists for all , but if , then is not true for all and all .

With the notation of Problem 2-4, part (a) of that problem says that exists for all . Now suppose . Then , But .

4. Let be defined as in Problem 1-26. Show that exists for all , although is not even continuous at (0,0).

By Problem 1-26 (a), for all .

1. Let be defined by

Show that is differentiable at 0 but is not continuous at 0.

Clearly, is differentiable at . At , one has since .

For , one has . The first term has limit 0 as approaches 0. But the second term takes on all values between -1 and 1 in every open neighborhood of . So, does not even exist.

2. Let be defined by

Show that is differentiable at (0,0) but that is not continuous at .

The derivative at (0, 0) is the zero linear transformation because , just as in part (a).

However, for where is as in part (a). It follows from the differentiability of , that and are defined for . (The argument given above also shows that they are defined and 0 at .) Further the partials are equal to up to a sign, and so they cannot be continuous at 0.

5. Show that the continuity of at may be eliminated from the hypothesis of Theorem 2-8.

Proceed as in the proof of Theorem 2-8 for all . In the case, it suffices to note that follows from the definition of . This is all that is needed in the rest of the proof.

6. A function is homogeneous of degree if for all and . If is also differentiable, show that

Applying Theorem 2-9 to gives . On the other hand, and so . Substituting in these two formulas show the result.

7. If is differentiable and , prove that there exist such that

Following the hint, let . Then . On the other hand, Theorem 2-9 gives . So, we have the result with .