Potential Function of the Vector Field

For functions of one variable, the Fundamental Theorem of Calculus shows that the integral is the opposite of the derivative. In this section we shall see that the line integral is the opposite of the gradient.

By a vector field we mean a vector valued function

F(x,y) = P(x,y)i + Q(x,y)j

where P and Q are smooth functions on an open rectangle D.

For example, if f(x, y) has continuous second partials on D then its gradient grad f is a vector field.

Many vector fields are found in physics. Examples are gravitational force fields and magnetic force fields, in which a force vector F(x, y) is associated with each point (x, y). Another example is the flow velocity V(x, y) of a fluid. A vector field in economics is the demand vector

D(x,y) = D₁(x,y)i + D₂(x,y)j,

where D₁(x, y) is the demand for commodity one and D₂(x, y) is the demand for commodity two at the prices x for commodity one and y for commodity two. All of the examples above have analogues for three variables and three dimensions (and the demand vector for n commodities has n variables and n dimensions).

Not every vector field has a potential function. Theorem 1 below shows which vector fields have potential functions, and Theorem 2 tells how to find a potential function when there is one.

Using the equality of mixed partials, we see that if the vector field Pi + Qj has a potential function, then ∂P/∂y = ∂Q/∂x. If f is a potential function of Pi + Qj, we have