




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.
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.)What can happen if R is not simply connected?
The easiest way to answer is to think of the gradient of the simplest nonfunction
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 "multiplevalued function" whose values differ by integral
multiples of 2.