# The Calculus of Variations in Several Dimensions. Fields and Their Geometric Invariants

• B. A. Dubrovin
• A. T. Fomenko
• S. P. Novikov
Part of the Graduate Texts in Mathematics book series (GTM, volume 93)

## Abstract

Let OD denote a region with piecewise smooth boundary D, of the Euclidean space I1’ with Euclidean co-ordinates x1,..., xn. Consider the linear space F of smooth vector-functions f (x1,..., xn) = (f1,..., fk) defined on D, i.e. with domain D. Let L(x β;p j ;q a i )be a smooth real-valued function of the three arguments xβ, 1 ≤β ≤ n; p j , 1jk; q a i , 1ik, 1an (making altogether n + k + nk real arguments); we call such a function a Lagrangian, and from a given such Lagrangian we construct a functional I[f] defined on F, as follows:
$$I\left[ f \right] = \int_D {L\left( {{x^\beta };{f^j}\left( {{x^\beta }} \right);f_{{x^\alpha }}^i\left( {{x^\beta }} \right)} \right)} d{x^1} \wedge \cdots \wedge d{x^n}$$
where the integral is the multiple integral (see §26) over the region D (which we shall later assume to be bounded), and where $$f_{{x^\alpha }}^i\left( {{x^\beta }} \right) = \left( {\partial /\partial {x^\alpha }} \right){f^i}\left( {{x^\beta }} \right)$$.

## Keywords

Electromagnetic Field Gauge Transformation Gravitational Field Spinor Representation Curvature Form
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Authors and Affiliations

• B. A. Dubrovin
• 1
• A. T. Fomenko
• 2
• S. P. Novikov
• 3
1. 1.c/o VAAP-Copyright Agency of the U.S.S.R.MoscowUSSR
2. 2.3 Ya KaracharavskayaMoscowUSSR
3. 3.L. D. Landau Institute for Theoretical PhysicsAcademy of Sciences of the U.S.S.R.MoscowUSSR