# Derivative of functional on path space

## Definition

Let be a differential manifold and be a path space on (viz, the space of piecewise smooth paths from to in .

Let be an element of the tangent space at a path and let be a functional from to . Then the derivative of at the point with respect to the tangent direction is defined as follows:

Let be any variation whose variation vector field is . The derivative is then:

A critical path of a functional is defined as a path (element of the path space) at which the derivative of the functional is zero.