General Variation of Integral Functional/Dependent on n Variables

Theorem
Let $ \mathbf x $ be an n-dimensional vector.

Let $ u = u \left ( { \mathbf x } \right ) $ be a real mapping.

Let $ J $ be a functional such that:


 * $ \displaystyle J \left [ { u } \right ] = \int_R F \left ( { \mathbf x, u, \frac{ \partial u }{ \partial \mathbf x } } \right ) \mathrm d x_1 \dots \mathrm d x_n $

Let $ \mathbf x^* $, $ u^* $ be such that:


 * $ \displaystyle \mathbf x^* = \boldsymbol \Phi \left ( { \mathbf x, u, \frac{ \partial u }{ \partial \mathbf x }; \epsilon } \right ) = \mathbf x + \epsilon \boldsymbol \phi \left ( { \mathbf x, u, \frac{ \partial u }{ \partial \mathbf x }  } \right ) + \mathcal O \left ( { \epsilon^{ 1 + 0_+ } } \right ) $


 * $ \displaystyle u^* = \Psi \left ( { \mathbf x, u, \frac{ \partial u }{ \partial \mathbf x }; \epsilon } \right ) = \mathbf x + \epsilon \psi \left ( { \mathbf x, u, \frac{ \partial u }{ \partial \mathbf x }  } \right ) + \mathcal O \left ( { \epsilon^{ 1 + 0_+ } } \right ) $

Then:


 * $ \displaystyle \delta J = \epsilon \int_R \left ( { F_u - \frac{ \partial F_{ u_{ \mathbf x } } }{ \partial \mathbf x } } \right ) \overline{ \psi } \mathrm d x_1 \dots \mathrm d x_n + \epsilon \int_R \frac{ \partial }{ \partial \mathbf x } \left ( { F_{ u_{ x } } \overline{ \boldsymbol \psi } +F \boldsymbol \phi } \right ) \mathrm d x_1 \dots \mathrm d x_n $

where


 * $ \displaystyle \overline{ \psi } = \psi - u_{ \mathbf x } \boldsymbol{ \phi } $

Proof
By definition: