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^2 } \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^2 } \right ) $

where $ \boldsymbol \Phi $, $ \Psi $ are differentiable $ \epsilon $ and:


 * $ \displaystyle \boldsymbol \Phi \left ( { \mathbf x, u, \frac{ \partial u }{ \partial \mathbf x }; 0 } \right ) = \mathbf x $


 * $ \displaystyle \Psi \left ( { \mathbf x, u, \frac{ \partial u }{ \partial \mathbf x }; 0 } \right ) = u $


 * $ \displaystyle \boldsymbol \phi \left ( { \mathbf x, u, \frac{ \partial u }{ \partial \mathbf x } } \right ) = \frac{ \partial \boldsymbol \Phi }{ \partial \epsilon } \Bigg \vert_{ \epsilon = 0 } $


 * $ \displaystyle \psi \left ( { \mathbf x, u, \frac{ \partial u }{ \partial \mathbf x} } \right ) = \frac{ \partial \Psi }{ \partial \epsilon } \Bigg \vert_{ \epsilon = 0 } $

Then the variation of the functional $ J $ due to the original mapping being transformed by the aforementioned transformation reads:


 * $ \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:

From the definition of $ \mathbf x^* $ a Jacobian matrix can be constructed:


 * $ \displaystyle \frac{ \partial x_i^* }{ \partial x_j } = \delta_i^j + \epsilon \frac{ \partial \phi_i }{ \partial x_j } + \mathcal O \left ( { \epsilon^2 } \right ) $

By Taylor's theorem, The corresponding Jacobian determinant can be expanded $ \epsilon$.

The $ \epsilon^0 $ term is obtained from the diagonal product by opening brackets and collecting $\epsilon$-free terms.

The $ \epsilon^1 $ term is obtained from the same diagonal product by choosing non-constant term once.


 * $ \displaystyle \det \left ( { J_{ \mathbf x^* } } \right ) = 1 + \epsilon \frac{ \partial \boldsymbol \phi }{ \partial \mathbf x } + \mathcal O  \left ( { \epsilon^2 } \right ) $


 * $ \displaystyle \Delta J = \int_R \left [ { F \left ( { \mathbf x^*, u^* \frac{ \partial u^* }{ \partial \mathbf x^*} } \right ) \left ( { 1 + \epsilon \frac{ \partial \boldsymbol \phi }{ \partial \mathbf x } } \right ) - F \left ( { \mathbf x, u, \frac{ \partial u }{ \partial \mathbf x }  } \right )  } \right ] \mathrm d x_1 \dots \mathrm d x_n + \mathcal O \left ( { \epsilon^2 } \right ) $

By definition, the principal part is:


 * $ \displaystyle \delta J = \int_R \left [ { F_{ \mathbf x } \delta \mathbf x + F_u \delta u + F_{ u_{ \mathbf x } } \delta u_{ \mathbf x } + \epsilon F \frac{ \partial \boldsymbol \phi }{ \partial \mathbf x } } \right ] \mathrm d x_1 \dots \mathrm d x_n $

Introduce the following differences:


 * $ \Delta u = u^* \left ( { x^* } \right ) - u \left ( { x } \right ) $


 * $ \overline{ \Delta u } = u^* \left ( { x } \right ) - u \left ( { x } \right ) $


 * $ \Delta x = x^* - x $

Then:


 * $ \displaystyle \overline{ \Delta u } = \epsilon \overline{ \psi } + \mathcal O \left ( { \epsilon^2 } \right ) $


 * $ \overline{ \delta u } = \epsilon \overline{ \psi } $


 * $ \displaystyle \Delta u = \frac{ \partial u }{ \partial \mathbf x } \delta \mathbf x + \overline { \delta u } + \mathcal { O \left ( { \epsilon^2 } \right ) } $


 * $ \displaystyle \delta u = \frac{ \partial u }{ \partial \mathbf x } \delta \mathbf x + \overline{ \delta u } $


 * $ \left ( { \delta u } \right )_{ x_i } = \left ( { \overline{ \delta u } } \right )_{ x_i } + u_{ \mathbf x x_i } \left ( { \delta \mathbf x } \right )_{ x_i } $


 * $ \Delta \mathbf x = \epsilon \boldsymbol \phi + \mathcal { O \left ( { \epsilon^2 } \right ) } $


 * $ \delta \mathbf x = \epsilon \boldsymbol \phi $

Here $ \delta $ difference stands for principal part, which is constant or linear $ \epsilon $.

Thus, the variation of $ J $ reads:


 * $ \displaystyle \delta J = \int_R \left [ { F_{ \mathbf x } \delta \mathbf x + F_u \overline{ \delta u } + F_u u_{ \mathbf x } \delta \mathbf x + F_{ u_{ \mathbf x } } \left ( { \overline{ \delta u } } \right )_{ \mathbf x } + \sum_{ i, j \mathop = 1 }^n F_{ u_{ x_i } } u_{ x_i x_j } \delta x_j + F \left ( { \delta \mathbf x } \right )_{ \mathbf x } } \right ] \mathrm d x_1 \dots \mathrm d x_n $

A few terms can be rewritten using:


 * $ \displaystyle \frac{ \partial }{ \partial \mathbf x } \left ( { F \delta \mathbf x } \right ) = F_{ \mathbf x } \delta \mathbf x + F \left ( { \overline{ \delta \mathbf x } } \right )_{ \mathbf x } + F_u u_{ \mathbf x } \delta \mathbf x + \sum_{ i, j \mathop = 1 }^n F_{ u_{ x_i } } u_{ x_i x_j } \delta x_j $

and:


 * $ \displaystyle F_{ u_{ \mathbf x } } \left ( { \overline{ \delta u } } \right )_{ \mathbf x } = \frac{ \partial }{ \partial \mathbf x } \left ( { F_{ u_{ \mathbf x } } \overline{ \delta u } } \right ) - \frac{ \partial F_{ u_{

\mathbf x } }}{ \partial \mathbf x } \delta u $

Substitution of previous results into variation of $ J $ leads to:


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

Substitute expressions for $ \overline{ \delta u } $ and $ \delta \mathbf x $ to obtain the desired result.