General Variation of Integral Functional/Dependent on n Variables

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

Let $u = \map u {\mathbf x}$ be a real-valued function.

Let $J$ be a functional such that:


 * $\ds J \sqbrk u = \int_R \map F {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } \rd x_1 \dotsm \rd x_n$

Let $\mathbf x^*, u^*$ be defined by the following transformations $\paren \star$:

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

and:

Then the variation of the functional $J$ due to the original mapping being transformed by $\paren \star$ reads:


 * $\ds \delta J = \epsilon \int_R \paren {F_u - \dfrac {\partial F_{u_{\mathbf x} } } {\partial \mathbf x} } \overline \psi \rd x_1 \dotsm \rd x_n + \epsilon \int_R \map {\dfrac {\partial} {\partial \mathbf x} } {F_{u_x} \overline {\boldsymbol \psi} + F \boldsymbol \phi} \rd x_1 \dotsm \rd x_n$

where:


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

Proof
By definition:

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


 * $\ds \dfrac {\partial x_i^*} {\partial x_j} = \delta_i^j + \epsilon \dfrac {\partial \phi_i} {\partial x_j} + \map \OO {\epsilon^2}$

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.


 * $\ds \det \paren {J_{\mathbf x^*} } = 1 + \epsilon \dfrac {\partial \boldsymbol \phi} {\partial \mathbf x} + \map \OO {\epsilon^2}$


 * $\ds \Delta J = \int_R \paren {\map F {\mathbf x^*, u^* \dfrac {\partial u^*} {\partial \mathbf x^*} } \paren {1 + \epsilon \dfrac {\partial \boldsymbol \phi} {\partial \mathbf x} } - \map F {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } } \rd x_1 \dotsm \rd x_n + \map \OO {\epsilon^2}$

By definition, the principal part is:


 * $\ds \delta J = \int_R \paren {F_{\mathbf x} \delta \mathbf x + F_u \delta u + F_{u_{\mathbf x} } \delta u_{\mathbf x} + \epsilon F \dfrac {\partial \boldsymbol \phi} {\partial \mathbf x} } \rd x_1 \dotsm \rd x_n$

Introduce the following differences:

Then:

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

Thus, the variation of $J$ reads:


 * $\ds \delta J = \int_R \paren {F_{\mathbf x} \delta \mathbf x + F_u \overline {\delta u} + F_u u_{\mathbf x} \delta \mathbf x + F_{u_{\mathbf x} } {\overline {\delta u} }_{\mathbf x} + \sum_{i, j \mathop = 1}^n F_{u_{x_i} } u_{x_i x_j} \delta x_j + \map F {\delta \mathbf x}_{\mathbf x} } \rd x_1 \dotsm \rd x_n$

A few terms can be rewritten using:


 * $\ds \dfrac {\partial} {\partial \mathbf x} \paren {F \delta \mathbf x} = F_{\mathbf x} \delta \mathbf x + \map F {\overline {\delta \mathbf x} }_{\mathbf x} + F_u u_{\mathbf x} \delta \mathbf x + \sum_{i, j\mathop = 1}^n F_{u_{x_i} } u_{x_i x_j} \delta x_j$

and:


 * $\ds \map {F_{u_{\mathbf x} } } {\overline{\delta u} }_{\mathbf x} = \dfrac {\partial} {\partial \mathbf x} {F_{u_{\mathbf x} } \overline {\delta u} } - \dfrac {\partial F_{u_{\mathbf x} } } {\partial \mathbf x} \delta u$

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


 * $\ds \delta J = \int_R \paren {F_u - \dfrac {\partial F_{u_{\mathbf x} } } {\partial \mathbf x} } \overline {\delta u} \rd x_1 \dotsm \rd x_n + \int_R \map {\dfrac \partial {\partial \mathbf x} } {F_{u_{\mathbf x} } \overline {\delta u} + F \delta \mathbf x} \rd x_1 \dotsm \rd x_n$

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