The Calculus of Variations is a course about optimization. In a begining Calculus course, we learn to find minimum and maximum values of a given function. That is, we learn to solve the following type of problem:

Find numbers such that some function is optimized.

What if the arguments aren't numbers, but functions instead? Here are examples of such problems:

• What is the shortest path across a bumpy surface?
• What is the shortest path across a smooth surface if there are obstacles?
• What path should a rocket follow to its target in order to expend the least fuel?

In each example, the argument isn't a number, but a path. These paths are described by functions. In general, variational problems take the following form:

Find functions such that some functional is optimized.

The techniques used to solve these path problems extend to the study of bubble shapes, vibrations of elastic materials, deflections of solid structures, shapes of molecular orbitals, bending of light beams, and the control of electrical systems. Variational techniques also lead to the derivation and numerical solution of some ODE and PDE.