6.2 When is a Vector Field v the Gradient of a Scalar Field f in some Region R?

A vector field that is the gradient of a potential in R is said to be conservative in R. v is also said to be derivable from a potential, and f is often called a potential function for v. An important fact about derivatives is that in computing mixed second derivatives, such as , the order of differentiation is irrelevant. Thus and are the same thing.

Proof

It follows from this fact that if v is the gradient of something then we must have

, and similarly,

These are restatements of the irrelevance of order in computing mixed second derivatives.The statement that v is the gradient of a function f in region R is equivalent to the statement that these combinations of derivatives are zero in R, if R is simply connected. (A region is connected if there is a path from any point in it to any other, which lies entirely in R. It is simply connected if every simple closed path in it can be shrunk down to a point somehow without encountering any point not in R.)

Proof

What can happen if R is not simply connected?
The easiest way to answer is to think of the gradient of the simplest non-function we know. This is the anglein polar cylindric or spherical coordinates. This angle (whose tangent is y / x) is not a function: each time you wander around the origin, it increases by 2. It is a "multiple-valued function" whose values differ by integral multiples of 2.