First variation
Encyclopedia
In applied mathematics
and the calculus of variations
, the first variation of a functional
J(y) is defined as the linear functional
mapping the function h to
where y and h are functions, and ε is a scalar.
From the definition above,
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and the calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
, the first variation of a functional
Functional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
J(y) is defined as the linear functional
mapping the function h towhere y and h are functions, and ε is a scalar.
Example
Compute the first variation ofFrom the definition above,
External links
- Exampleproblems.com has more examples of computing the first variation of functionals.