# Exercises: Chapter 4, Section 2

1. If and , show that and .

The notation does not fully elucidate the meaning of the assertion. Here is the interpretation:

The second assertion follows from:

2. If , show that .

One has by the definition and the product rule:

1. Let be a differentiable curve in , that is, a differentiable function . Define the tangent vector of at as . If , show that the tangent vector to at is .

This is an immediate consequence of Problem 4-13 (a).

2. Let and define by . Show that the end point of the tangent vector of at lies on the tangent line to the graph of at .

The tangent vector of at is . The end point of the tangent vector of at is which is certainly on the tangent line to the graph of at .

3. Let be a curve such that for all . Show that and the tangent vector to at are perpendicular.

Differentiating , gives , i.e. where is the tangent vector to at .

4. If , define a vector field by
1. Show that every vector field on is of the form for some .

A vector field is just a function which assigns to each an element . Given such an , define by . Then .

2. Show that .

One has .

5. If , define a vector field by

For obvious reasons we also write . If , prove that and conclude that is the direction in which is changing fastest at .

By Problem 2-29,

The direction in which is changing fastest is the direction given by a unit vector such thatt is largest possible. Since where , this is clearly when , i.e. in the direction of .

6. If is a vector field on , define the forms

1. Prove that

The first equation is just Theorem 4-7.

For the second equation, one has:

For the third assertion:

2. Use (a) to prove that

One has by part (a) and Theorem 4-10 (3); so .

Also, by part (a) and Theorem 4-10 (3); so the second assertion is also true.

3. If is a vector field on a star-shaped open set and , show that for some function . Similarly, if , show that for some vector field on .

By part (a), if , then . By the Theorem 4-11, is exact, i.e. . So .

Similarly, if , then and so is closed. By Theorem 4-11, it must then be exact, i.e. for some . So as desired.

7. Let be a differentiable function with a differentiable inverse . If every closed form on is exact, show that the same is true of .

Suppose that the form on is closed, i.e. . Then and so there is a form on such that . But then and so is also exact, as desired.

8. Prove that on the set where is defined, we have

Except when , the assertion is immediate from the definition of in Problem 2-41. In case , one has trivially because is constant when and (or ). Further, L'H^{o}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, .