By definition: {{begin-eqn}} {{eqn | l = \Delta J       | r = J \sqbrk {\map {u^*} {x^*} } - J \sqbrk {\map u x} }} {{eqn | r = \int_{R^*} \map F {\mathbf x^*, u^*, \frac {\partial u^*} {\partial \mathbf x^*} } \rd x_1^* \dotsm \rd x_n^* - \int_R \map F {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } \rd x_1 \dotsm \rd x_n }} {{eqn | r = \int_R \paren {\map F {\mathbf x^*, u^*, \dfrac {\partial u^*} {\partial \mathbf x^*} } \det \paren {J_{\mathbf x^*} } - \map F {\mathbf x, u, \frac {\partial u} {\partial \mathbf x} } } \rd x_1 \dotsm \rd x_n       | c = Volume element transformation }} {{end-eqn}} 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 {{WRT}} $\epsilon$. 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. :$\ds \det \paren {J_{\mathbf x^*} } = 1 + \epsilon \dfrac {\partial \boldsymbol \phi} {\partial \mathbf x} + \map \OO {\epsilon^2}$ {{explain|make explanation more precise}} :$\ds \Delta J = \int_R \paren {\map F {\mathbf x^*, u^* \dfrac {\partial u^*} {\partial \mathbf x^*} } \paren {1 + \epsilon \dfrac {\partial \boldsymbol \phi} {\partial \mathbf x} } - \map F {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } } \rd x_1 \dotsm \rd x_n + \map \OO {\epsilon^2}$ By definition, the principal part is: :$\ds \delta J = \int_R \paren {F_{\mathbf x} \delta \mathbf x + F_u \delta u + F_{u_{\mathbf x} } \delta u_{\mathbf x} + \epsilon F \dfrac {\partial \boldsymbol \phi} {\partial \mathbf x} } \rd x_1 \dotsm \rd x_n$ {{explain|If use of square brackets has a particular significance and meaning here, explain it. If it is just a variant on conventional parenthesis, then best if $\paren \cdot$ is used instead. It's ultimately more clear to use one style of parenthesis than mix and match various different types.}} Introduce the following differences: {{begin-eqn}} {{eqn | l = \Delta u       | r = \map {u^*} {x^*} - \map u x }} {{eqn | l = \overline {\Delta u}       | r = \map {u^*} x - \map u x }} {{eqn | l = \Delta x       | r = x^*-x }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \overline {\Delta u}       | r = \epsilon \overline \psi + \map \OO {\epsilon^2} }} {{eqn | l = \overline {\delta u}       | r = \epsilon \overline \psi }} {{eqn | l = \Delta u       | r = \dfrac {\partial u} {\partial \mathbf x} \delta \mathbf x + \overline {\delta u} + \map \OO {\epsilon^2} }} {{eqn | l = \delta u       | r = \dfrac {\partial u} {\partial \mathbf x} \delta \mathbf x + \overline {\delta u} }} {{eqn | l = \paren {\delta u}_{x_i}       | r = \paren {\overline {\delta u} }_{x_i} + u_{\mathbf x x_i} {\delta \mathbf x}_{x_i} }} {{eqn | l = \Delta \mathbf x       | r = \epsilon \boldsymbol \phi + \map \OO {\epsilon^2} }} {{eqn | l = \delta \mathbf x       | r = \epsilon \boldsymbol \phi }} {{end-eqn}} Here $\delta$ difference stands for principal part, which is constant or linear {{WRT}} $\epsilon$. 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. {{qed}}