The notation does not fully elucidate the meaning of the assertion. Here is the interpretation:
The second assertion follows from:
One has by the definition and the product rule:
This is an immediate consequence of Problem 4-13 (a).
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 .
Differentiating , gives , i.e. where is the tangent vector to at .
A vector field is just a function which assigns to each an element . Given such an , define by . Then .
One has .
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 .
The first equation is just Theorem 4-7.
For the second equation, one has:
For the third assertion:
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.
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.
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.
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, .