Vector analysis is the multi-dimensional analogue of single-variable calculus. It is used to represent (and analyze) the motions of the solar system, fluid flows, the flow of electric charge, and other phenomena.

In single-variable calculus, the derivative of a function f from the real numbers to the real numbers is its rate of change. For example, if f(t) represents the distance traveled at time t, the derivative of f at t represents speed. Vector analysis on the other hand typically concerns functions from three-dimensional space to the real numbers. Derivatives of such functions are taken with respect to a direction. For example, suppose that the function f represents the temperature of a room. Then its derivative in the upwards direction is the rate of temperature change as we move upwards. Often this direction is given a name like positive z-direction. Then the derivative of f...