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 mapping.

Let $J$ be a functional such that:


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

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


 * $\displaystyle \mathbf x^*=\map {\boldsymbol\Phi} {\mathbf x,u,\dfrac{\partial u}{\partial\mathbf x};\epsilon}=\mathbf x+\epsilon\map {\boldsymbol\phi} {\mathbf x,u,\dfrac{\partial u}{\partial\mathbf x} }+\map {\mathcal O} {\epsilon^2}$


 * $\displaystyle u^*=\map \Psi {\mathbf x,u,\dfrac{\partial u}{\partial\mathbf x};\epsilon}=\mathbf x+\epsilon\map \psi {\mathbf x,u,\dfrac{\partial u}{\partial\mathbf x } }+\map {\mathcal O} {\epsilon^2}$

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


 * $\map{\boldsymbol \Phi} {\mathbf x,u,\dfrac {\partial u}{\partial\mathbf x};0}=\mathbf x$


 * $\map \Psi {\mathbf x,u,\dfrac {\partial u}{\partial\mathbf x};0}=u$


 * $\displaystyle \map {\boldsymbol \phi} {\mathbf x,u,\frac{\partial u}{\partial\mathbf x} }=\dfrac{\partial\boldsymbol\Phi}{\partial\epsilon}\Bigg\vert_{\epsilon=0}$


 * $\displaystyle\map \psi {\mathbf x,u,\frac{\partial u}{\partial\mathbf x} }=\dfrac{\partial\Psi}{\partial\epsilon}\Bigg\vert_{\epsilon=0}$

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


 * $\displaystyle\delta J=\epsilon\int_R\paren{F_u-\dfrac{\partial F_{u_{\mathbf x} } }{\partial\mathbf x} }\overline{\psi}\rd x_1 \dots\rd x_n+\epsilon\int_R\dfrac{\partial}{\partial\mathbf x}\paren{ F_{ u_{ x } } \overline{\boldsymbol\psi}+F\boldsymbol\phi}\mathrm d x_1\dots\rd x_n$

where


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

Proof
By definition:

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


 * $\displaystyle\dfrac{\partial x_i^*}{\partial x_j}=\delta_i^j+\epsilon\dfrac{\partial\phi_i}{\partial x_j}+\map {\mathcal O} {\epsilon^2}$

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.


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


 * $\displaystyle\Delta J=\int_R\sqbrk{\map F {\mathbf x^*,u^*\dfrac{\partial u^*}{\partial\mathbf x^*} }\paren{1+\epsilon\dfrac{\partial\boldsymbol\phi}{\partial\mathbf x} }-\map F {\mathbf x,u,\dfrac{\partial u}{\partial\mathbf x} } }\rd x_1\dots\rd x_n+\map {\mathcal O} {\epsilon^2}$

By definition, the principal part is:


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

Introduce the following differences:


 * $\Delta u=\map {u^*} {x^*}-\map u x$


 * $\overline{\Delta u}=\map {u^*} x-\map u x$


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

Then:


 * $\displaystyle\overline{\Delta u}=\epsilon\overline{\psi}+\map {\mathcal O} {\epsilon^2}$


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


 * $\displaystyle\Delta u=\dfrac{\partial u}{\partial\mathbf x}\delta\mathbf x+\overline {\delta u}+\map {\mathcal O} {\epsilon^2}$


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


 * $\paren{\delta u}_{x_i}=\paren{\overline{\delta u} }_{x_i}+u_{\mathbf x x_i}{\delta\mathbf x}_{x_i}$


 * $\Delta\mathbf x=\epsilon\boldsymbol\phi+\map {\mathcal O} {\epsilon^2}$


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

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

Thus, the variation of $J$ reads:


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

A few terms can be rewritten using:


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

and:


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

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


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

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