First variation

In applied mathematics and the calculus of variations, the first variation of the functional J (y ) is defined as

With a functional, as well as functions in the function space, and ε is a scalar (pronounced: the first variation of J with respect to y ).

Alternative definition

An alternative definition, which is also more common in theoretical physics, and especially to be found in field theory is as follows:

Be a space of test functions and

A linear functional.

The first variation ( also sometimes referred to as a functional derivative ) of F is defined as the distribution of which

Met. The functional derivative plays the part of a gradient, which is to be expressed by the notation that the discharge takes place in the direction of a test function.

Properties

Example

The first variation of

According to the definition above is