Null Lagrangians in Cosserat Elasticity

Abstract

In the framework of nonlinear theory of Cosserat elasticity, also called micropolar elasticity, we provide the complete characterization of null Lagrangians for three dimensional bodies as well as for shells. Using the Gibb’s rotation vector for description of the microrotation, this task is possible by an application of a theorem stated by Olver and Sivaloganathan in ‘the structure of null Lagrangians’ (Nonlinearity 1:389–398, 1988). A set of necessary and sufficient conditions is also provided for the elasticity tensors to correspond to a null Lagrangian in linearized micropolar theory.

Appendices

Appendix A: Expansion of $\omega$

For the purpose of convenience of derivation, we assume that $\det \mathbf{G}\ne 0$, let $\nabla _{\theta }\boldsymbol{\chi }=\mathbf{T}=\mathbf{F} \mathbf{G}^{-1}$. With (3.2), then

$$ \begin{aligned} \omega &= \mathscr{A} \Omega _{v}+\frac{1}{2}\upepsilon _{ABC} \mathrm{B}_{Ai}F_{iM}dx_{M} {\wedge } dx_{B} {\wedge } dx_{C} \\ &+ \frac{1}{2}\upepsilon _{ijk}\mathrm{C}_{iA}dy_{j}{\wedge } dy_{k} { \wedge } (\mathbf{F}^{-1})_{Al}dy_{l}+\mathscr{D}dy_{1} {\wedge } dy_{2} {\wedge } dy_{3} \\ &+\frac{1}{2}\upepsilon _{ABC}\widetilde{\mathrm{B}}_{A\alpha }G_{ \alpha M}dx_{M} {\wedge } dx_{B} {\wedge } dx_{C}+\frac{1}{2} \upepsilon _{i\beta \gamma }\widetilde{\mathrm{C}}_{iA}d\theta _{\beta }{\wedge } d\theta _{\gamma }{\wedge } (\mathbf{G}^{-1})_{A\alpha }d \theta _{\alpha }+\widetilde{\mathscr{D}}d\theta _{1} {\wedge } d \theta _{2} {\wedge } d\theta _{3} \\ &+\frac{1}{2}\upepsilon _{\alpha \beta \gamma }\widehat{\mathrm{B}}_{ \alpha i}T_{i\alpha }d\theta _{\alpha }{\wedge } d\theta _{\beta }{\wedge } d\theta _{\gamma }+\frac{1}{2}\upepsilon _{ijk}\widehat{\mathrm{C}}_{i \alpha }dy_{j}{\wedge } dy_{k} {\wedge } (\mathbf{T}^{-1})_{\alpha l}dy_{l} \\ &+J_{\alpha j C}G_{\alpha A}F_{jB}dx_{A} {\wedge } dx_{B} {\wedge } dx_{C}, \end{aligned} $$

(A.1)

where

$$ \Omega _{v}:=dx_{1} {\wedge } dx_{2} {\wedge } dx_{3}. $$

(A.2)

Expanding further,

$$ \begin{aligned} \omega &= \mathscr{A} \Omega _{v} +\frac{1}{2}\upepsilon _{ABC} \upepsilon _{MBC}\mathrm{B}_{Ai}F_{iM}\Omega _{v}+\frac{1}{2} \upepsilon _{ijk}\mathrm{C}_{iA} (\mathbf{F}^{-1})_{Al} \upepsilon _{jkl}( \det \mathbf{F})\Omega _{v}+\mathscr{D}(\det \mathbf{F})\Omega _{v} \\ &+\frac{1}{2}\upepsilon _{ABC}\upepsilon _{MBC} \widetilde{\mathrm{B}}_{A\alpha }G_{\alpha M}\Omega _{v}+\frac{1}{2} \upepsilon _{i\beta \gamma }\widetilde{\mathrm{C}}_{iA} (\mathbf{G}^{-1})_{A \alpha } \upepsilon _{\alpha \beta \gamma }(\det \mathbf{G})\Omega _{v}+ \widetilde{\mathscr{D}}(\det \mathbf{G})\Omega _{v} \\ &+\frac{1}{2}\upepsilon _{\delta \beta \gamma }\upepsilon _{\alpha \beta \gamma }\widehat{\mathrm{B}}_{\delta i}T_{i \alpha }(\det \mathbf{G})\Omega _{v}+\frac{1}{2}\upepsilon _{ijk} \widehat{\mathrm{C}}_{i\alpha } (\mathbf{T}^{-1})_{\alpha l} \upepsilon _{jkl}(\det \mathbf{F})\Omega _{v} \\ &+\upepsilon _{ABC} J_{\alpha j C}G_{\alpha A}F_{jB}\Omega _{v}. \end{aligned} $$

(A.3)

Simplifying above expression, we find that

$$ \begin{aligned} \omega &= (\mathscr{A}+\mathrm{B}_{Ai}F_{iA}+\mathrm{C}_{iA} ( \mathbf{F}^{-1})_{Ai} (\det \mathbf{F})+\mathscr{D}(\det \mathbf{F}) \\ &+\widetilde{\mathrm{B}}_{A\alpha }G_{\alpha A}+ \widetilde{\mathrm{C}}_{\alpha A} (\mathbf{G}^{-1})_{A\alpha } ( \det \mathbf{G})+\widetilde{\mathscr{D}}(\det \mathbf{G}) \\ &+\widehat{\mathrm{B}}_{\alpha i}T_{i\alpha }\det \mathbf{G}+ \widehat{\mathrm{C}}_{i\alpha } (\mathbf{T}^{-1})_{\alpha i} (\det \mathbf{F}) +J_{\alpha j C}G_{\alpha A}F_{jB}\upepsilon _{CAB}) \Omega _{v}, \end{aligned} $$

(A.4)

which can be written as

$$ \begin{aligned} \omega &=(\mathscr{A}+\mathbf{B}^{\top }\cdot {\mathbf{F}}+ \mathbf{C}\cdot {\mathrm {cof~}} {\mathbf{F}}+\mathscr{D}\det {\mathbf{F}}+ \widetilde{\mathbf{B}}^{\top }\cdot {\mathbf{G}}+ \widetilde{\mathbf{C}}\cdot {\mathrm {cof~}} {\mathbf{G}}+ \widetilde{\mathscr{D}}\det {\mathbf{G}} \\ &+\widehat{\mathbf{B}}^{\top }\cdot {\mathbf{F}\mathbf{G}^{-1}}( \det \mathbf{G})+\widehat{\mathbf{C}}\cdot {\mathrm {cof~}}({\mathbf{F}}{ \mathbf{G}}^{-1})(\det \mathbf{G})+J_{\alpha j C}(.)_{\alpha j C}) \Omega _{v},\end{aligned} $$

(A.5)

where $(.)_{\alpha j C}=G_{\alpha A}F_{jB}\upepsilon _{CAB}=\mathbf{G}^{ \top }\boldsymbol{e}_{\alpha }\wedge \mathbf{F}^{\top }\boldsymbol{e}_{j} \cdot \boldsymbol{e}_{C}$, $(.)=\boldsymbol{e}_{\alpha }\otimes \boldsymbol{e}_{j} \otimes (\mathbf{G}^{\top }\boldsymbol{e}_{\alpha }\wedge \mathbf{F}^{\top }\boldsymbol{e}_{j})$. Replacing the inverse of $\mathbf{G}$ by the cofactor, we get the form (3.3) which does not depend on the invertibility of $\mathbf{G}$. The components of the cofactor of $\mathbf{A}$ are given by

$$ (\mathrm {cof~}\mathbf{A})_{ij}=\frac{1}{2}\upepsilon _{imn}\upepsilon _{jpq} \mathrm{A}_{mp}\mathrm{A}_{nq}, $$

(A.6)

Appendix B: Expansion of $d\zeta$

The expression (3.5) leads to its exterior derivative

$$ \begin{aligned} d\zeta &=\frac{1}{2}\upepsilon _{ABC}d\mathrm{L}_{A}{\wedge } dx_{B}{ \wedge } dx_{C}+d\mathrm{K}_{iA}{\wedge } dy_{i} {\wedge } dx_{A}+ \frac{1}{2}\upepsilon _{ijk}d\mathrm{M}_{i} {\wedge } dy_{j} { \wedge } dy_{k} \\ &+d\widetilde{\mathrm{K}}_{\alpha A}\wedge d\theta _{\alpha }{\wedge } dx_{A} +\frac{1}{2}\upepsilon _{\alpha \beta \gamma }d \widetilde{\mathrm{M}}_{\alpha }\wedge d\theta _{\beta }{\wedge } d \theta _{\gamma }+d\mathrm{H}_{\alpha j}\wedge d\theta _{\alpha }{\wedge } dy_{j}, \end{aligned} $$

(B.1)

which can be expanded further given that $\boldsymbol{L}, \boldsymbol{M}, \widetilde{\boldsymbol{M}}, \mathbf{K}, \widetilde{\mathbf{K}}, \mathbf{H}$ are functions of $(\boldsymbol{x},\boldsymbol{\chi },\boldsymbol{\theta })$ so that

$$ \begin{aligned} d\zeta &=\frac{1}{2}\upepsilon _{ABC}\mathrm{L}_{A,D}dx_{D} {\wedge } dx_{B}{\wedge } dx_{C}+\frac{1}{2}\upepsilon _{ABC}\mathrm{L}_{A,i}dy_{i} {\wedge } dx_{B}{\wedge } dx_{C} \\ &+\frac{1}{2}\upepsilon _{ABC} \mathrm{L}_{A,\alpha }d\theta _{\alpha }{\wedge } dx_{B}{\wedge } dx_{C} +\mathrm{K}_{iA,B}dx_{B}{\wedge } dy_{i} {\wedge } dx_{A} \\ &+ \mathrm{K}_{iA,j}dy_{j}{\wedge } dy_{i} {\wedge } dx_{A}+\mathrm{K}_{iA, \beta }d\theta _{\beta }{\wedge } dy_{i} {\wedge } dx_{A} \\ &+\frac{1}{2}\upepsilon _{ijk}\mathrm{M}_{i,A}dx_{A} {\wedge } dy_{j} {\wedge } dy_{k}+\frac{1}{2}\upepsilon _{ijk}\mathrm{M}_{i,l}dy_{l} { \wedge } dy_{j} {\wedge } dy_{k}+\frac{1}{2}\upepsilon _{ijk} \mathrm{M}_{i,\alpha }d\theta _{\alpha }{\wedge } dy_{j} {\wedge } dy_{k} \\ &+\widetilde{\mathrm{K}}_{\alpha A,B}dx_{B}{\wedge } d\theta _{\alpha }{\wedge } dx_{A}+\widetilde{\mathrm{K}}_{\alpha A,j}dy_{j}{ \wedge } d\theta _{\alpha }{\wedge } dx_{A}+\widetilde{\mathrm{K}}_{ \alpha A,\beta }d\theta _{\beta }{\wedge } d\theta _{\alpha }{\wedge } dx_{A} \\ &+\frac{1}{2}\upepsilon _{\alpha \beta \gamma } \widetilde{\mathrm{M}}_{\alpha ,A}dx_{A} {\wedge } d\theta _{\beta }{ \wedge } d\theta _{\gamma }+\frac{1}{2}\upepsilon _{\alpha \beta \gamma } \widetilde{\mathrm{M}}_{\alpha ,l}dy_{l} {\wedge } d\theta _{\beta }{ \wedge } d\theta _{\gamma }+\frac{1}{2}\upepsilon _{\alpha \beta \gamma } \widetilde{\mathrm{M}}_{\alpha ,\delta }d\theta _{\delta }{\wedge } d \theta _{\beta }{\wedge } d\theta _{\gamma }\\ &+\mathrm{H}_{\alpha j,A}dx_{A}\wedge d\theta _{\alpha }{\wedge } dy_{j}+ \mathrm{H}_{\alpha j,k}dy_{k}\wedge d\theta _{\alpha }{\wedge } dy_{j}+ \mathrm{H}_{\alpha j,\beta }d\theta _{\beta }\wedge d\theta _{\alpha }{ \wedge } dy_{j}. \end{aligned} $$

(B.2)

Collecting the terms accompanying the same exterior product of differentials, we get

$$ \begin{aligned} d\zeta &=\mathrm{L}_{A,A}\Omega _{v}+(\frac{1}{2}\upepsilon _{ABC} \mathrm{L}_{A,i}- \mathrm{K}_{iC,B})dy_{i} {\wedge }dx_{B} {\wedge } dx_{C} \\ &+(\mathrm{K}_{kA,j}+\frac{1}{2}\upepsilon _{ijk}\mathrm{M}_{i,A})dy_{j} {\wedge } dy_{k}\wedge dx_{A}+\mathrm{M}_{i,i}\Omega _{y}+ \widetilde{\mathrm{M}}_{\alpha ,\alpha }\Omega _{\theta }\\ &+(\frac{1}{2}\upepsilon _{ijk}\mathrm{M}_{i,\alpha }+\mathrm{H}_{ \alpha j,k})dy_{j}\wedge dy_{k}\wedge d\theta _{\alpha }+(\frac{1}{2} \upepsilon _{ABC}\mathrm{L}_{A,\alpha }-\widetilde{\mathrm{K}}_{ \alpha C,B})d\theta _{\alpha }{\wedge } dx_{B}{\wedge } dx_{C} \\ &+(\widetilde{\mathrm{K}}_{\gamma A,\beta }+\frac{1}{2}\upepsilon _{ \alpha \beta \gamma }\widetilde{\mathrm{M}}_{\alpha ,A})d\theta _{\beta }{\wedge } d\theta _{\gamma }\wedge dx_{A}+(\mathrm{K}_{jC,\alpha }- \widetilde{\mathrm{K}}_{\alpha C,j}+\mathrm{H}_{\alpha j,C})d \theta _{\alpha }{\wedge } dy_{j}\wedge dx_{C} \\ &+(\frac{1}{2}\upepsilon _{\alpha \beta \gamma } \widetilde{\mathrm{M}}_{\alpha ,i}+\mathrm{H}_{\gamma i,\beta })dy_{i}{ \wedge } d\theta _{\beta }\wedge d\theta _{\gamma }, \end{aligned} $$

(B.3)

where (A.2) is used.

Appendix C: Expansion of $\nabla \cdot \boldsymbol{P}$

The expression (3.14) is equivalent to

$$ \begin{aligned} \boldsymbol{P} &= \boldsymbol{L}+({\mathbf{F}}^{\top }\mathbf{K})^{ \times }+({\mathrm {cof~}}{\mathbf{F}})^{\top }\boldsymbol{M}+({\mathbf{G}}^{ \top }\widetilde{\mathbf{K}})^{\times } \\ &+({\mathrm {cof~}}{\mathbf{G}})^{\top }\widetilde{\boldsymbol{M}}+ \upepsilon _{ABC}G_{\alpha B}F_{iC}\mathrm{H}_{\alpha i} \boldsymbol{e}_{A}, \end{aligned} $$

(C.1)

as

$$\begin{aligned} \upepsilon _{ABC}G_{\alpha B}F_{iC}\mathrm{H}_{\alpha i} \boldsymbol{e}_{A} &=\boldsymbol{e}_{A}(\boldsymbol{e}_{A}\cdot ( \boldsymbol{e}_{B}\wedge \boldsymbol{e}_{C}))G_{\alpha B}F_{iC} \mathrm{H}_{\alpha i} \\ &=\boldsymbol{e}_{A}\otimes \boldsymbol{e}_{A}(\boldsymbol{e}_{B} \wedge \boldsymbol{e}_{C})G_{\alpha B}F_{iC}\mathrm{H}_{\alpha i} \\ &=(G_{\alpha B}\boldsymbol{e}_{B}\wedge F_{iC}\boldsymbol{e}_{C}) \mathrm{H}_{\alpha i} \\ &=(\mathbf{G}^{\top }\boldsymbol{e}_{\alpha }\wedge \mathbf{F}^{\top } \boldsymbol{e}_{j})\mathrm{H}_{\alpha j}=(\mathbf{G}^{\top } \boldsymbol{e}_{\alpha }\wedge \mathbf{F}^{\top }\boldsymbol{e}_{j}) \mathbf{H}\cdot \boldsymbol{e}_{\alpha }\otimes \boldsymbol{e}_{j} \\ &=(\mathbf{G}^{\top }\boldsymbol{e}_{\alpha }\wedge \mathbf{F}^{\top } \mathrm{H}_{\alpha j}\boldsymbol{e}_{j})=\mathbf{G}^{\top } \boldsymbol{e}_{\alpha }\wedge \mathbf{F}^{\top }\mathbf{H}^{\top } \boldsymbol{e}_{\alpha }\\ &=\mathrm{axl}(\mathbf{F}^{\top }\mathbf{H}^{\top }\boldsymbol{e}_{\alpha }\otimes \mathbf{G}^{\top }\boldsymbol{e}_{\alpha }-\mathbf{G}^{ \top }\boldsymbol{e}_{\alpha }\otimes \mathbf{F}^{\top }\mathbf{H}^{ \top }\boldsymbol{e}_{\alpha }) \\ &=\mathrm{axl}(\mathbf{F}^{\top }\mathbf{H}^{\top }\mathbf{G} - \mathbf{G}^{\top }\mathbf{H}\mathbf{F}) \\ &=-\frac{1}{2}(\mathbf{F}^{\top }\mathbf{H}^{\top }\mathbf{G} - \mathbf{G}^{\top }\mathbf{H}\mathbf{F})^{\times } \end{aligned}$$

(C.2)

(recall $\mathrm{axl}(\boldsymbol{b}\otimes \boldsymbol{c}-\boldsymbol{c} \otimes \boldsymbol{b})=\boldsymbol{c}\wedge \boldsymbol{b}$). Thus, using the conditions stated in §3,

$$ \begin{aligned} \nabla \cdot \boldsymbol{P} &=\mathscr{A}+({\nabla _{y}}\boldsymbol{L})^{\top } \cdot {\mathbf{F}}+({\nabla _{\theta }}\boldsymbol{L})^{\top }\cdot { \mathbf{G}} \\ &+(\mathrm {Curl}_{x}\mathbf{K}\cdot {\mathbf{F}}-{\mathrm {cof~}}{\mathbf{F}}\cdot ({ \mathrm {Curl}_{y}}\mathbf{K}^{\top })^{\top })+\upepsilon _{ABC}{F}_{iB}\mathrm{K}_{iC, \alpha }G_{\alpha A} \\ &+{\mathrm {cof~}}{\mathbf{F}}\cdot (\nabla _{x}\boldsymbol{M}+(\nabla _{y}\boldsymbol{M})\mathbf{F}+(\nabla _{\theta }\boldsymbol{M})\mathbf{G}) \\ &+(\mathrm {Curl}_{x}\widetilde{\mathbf{K}}\cdot {\mathbf{G}}-{\mathrm {cof~}}{ \mathbf{G}}\cdot ({\mathrm {Curl}_{\theta }}\widetilde{\mathbf{K}}^{\top })^{\top })+ \upepsilon _{ABC}{G}_{\alpha B}\widetilde{\mathrm{K}}_{\alpha C,i}F_{iA} \\ &+({\mathrm {cof~}}{\mathbf{G}})\cdot (\nabla _{x}\widetilde{\boldsymbol{M}}+( \nabla _{y}\widetilde{\boldsymbol{M}})\mathbf{F}+(\nabla _{\theta }\widetilde{\boldsymbol{M}})\mathbf{G}) \\ &+\upepsilon _{ABC}G_{\alpha B}F_{iC}\mathrm{H}_{\alpha i,A}+ \upepsilon _{ABC}G_{\alpha B}F_{iC}\mathrm{H}_{\alpha i,\beta }G_{ \beta A} \\ &+\upepsilon _{ABC}G_{\alpha B}F_{iC}\mathrm{H}_{\alpha i,j}F_{jA}, \end{aligned} $$

(C.3)

i.e.,

$$ \begin{aligned} \nabla \cdot \boldsymbol{P} & =\mathscr{A}+\mathbf{B} \cdot {\mathbf{F}}+ \mathbf{C}\cdot {\mathrm {cof~}} {\mathbf{F}}+\mathscr{D}\det {\mathbf{F}}+ \widetilde{\mathbf{B}} \cdot {\mathbf{G}}+\widetilde{\mathbf{C}} \cdot {\mathrm {cof~}} {\mathbf{G}}+\widetilde{\mathscr{D}}\det {\mathbf{G}} \\ &+({\mathrm {cof~}}{\mathbf{F}})\mathbf{G}^{\top }\cdot (\nabla _{\theta }\boldsymbol{M})+(\nabla _{y}\widetilde{\boldsymbol{M}})^{\top }\cdot \mathbf{F}({\mathrm {cof~}}{\mathbf{G}})^{\top } \\ &+\upepsilon _{ABC}{F}_{iB}\mathrm{K}_{iC,\alpha }G_{\alpha A}+ \upepsilon _{ABC}{G}_{\alpha B}\widetilde{\mathrm{K}}_{\alpha C,i}F_{iA}+ \upepsilon _{ABC}G_{\alpha B}F_{iC}\mathrm{H}_{\alpha i,A} \\ &+\upepsilon _{ABC}G_{\beta A}G_{\alpha B}F_{iC}\mathrm{H}_{\alpha i, \beta }-\upepsilon _{ABC}F_{jA}F_{iB}G_{\alpha C}\mathrm{H}_{\alpha i,j}, \end{aligned} $$

(C.4)

i.e.,

$$ \begin{aligned} \nabla \cdot \boldsymbol{P} & =\mathscr{A}+\mathbf{B} \cdot {\mathbf{F}}+ \mathbf{C}\cdot {\mathrm {cof~}} {\mathbf{F}}+\mathscr{D}\det {\mathbf{F}}+ \widetilde{\mathbf{B}} \cdot {\mathbf{G}}+\widetilde{\mathbf{C}} \cdot {\mathrm {cof~}} {\mathbf{G}}+\widetilde{\mathscr{D}}\det {\mathbf{G}} \\ &+\widehat{\mathbf{B}}\cdot ({\mathrm {cof~}}\mathbf{G})\mathbf{F}^{\top }+ \widehat{\mathbf{C}}\cdot ({\mathrm {cof~}}{\mathbf{F}}){\mathbf{G}}^{\top } \\ &-({\mathrm {cof~}}{\mathbf{F}})\mathbf{G}^{\top }\cdot (\mathrm {Curl}_{y}\mathbf{H})^{ \top }+(\mathrm {Curl}_{\theta }\mathbf{H}^{\top })\cdot \mathbf{F}({\mathrm {cof~}}{ \mathbf{G}})^{\top } \\ &+\mathtt{J}\cdot \boldsymbol{e}_{\alpha }\otimes \boldsymbol{e}_{j} \otimes (\mathbf{G}^{\top }\boldsymbol{e}_{\alpha }\wedge \mathbf{F}^{ \top }\boldsymbol{e}_{j}) \\ &+\upepsilon _{ABC}{F}_{iB}\mathrm{K}_{iC,\alpha }G_{\alpha A}+ \upepsilon _{ABC}{G}_{\alpha B}\widetilde{\mathrm{K}}_{\alpha C,i}F_{iA} \\ &-\upepsilon _{ABC}(\mathrm{K}_{iC,\alpha }-\widetilde{\mathrm{K}}_{ \alpha C,i}+\mathrm{H}_{\alpha i,C})G_{\alpha A}F_{iB}+\upepsilon _{CAB}G_{ \alpha A}F_{iB}\mathrm{H}_{\alpha i,C} \\ &+\upepsilon _{ABC}G_{\alpha A}G_{\beta B}F_{iC}\mathrm{H}_{\beta i, \alpha }-\upepsilon _{ABC}F_{iA}F_{jB}G_{\alpha C}\mathrm{H}_{\alpha j,i}, \end{aligned} $$

(C.5)

so that finally,

$$ \begin{aligned} \nabla \cdot \boldsymbol{P} &=\mathscr{L}-({\mathrm {cof~}}{\mathbf{F}})\mathbf{G}^{ \top }\cdot (\mathrm {Curl}_{y}\mathbf{H})^{\top }+(\mathrm {Curl}_{\theta }\mathbf{H}^{\top }) \cdot \mathbf{F}({\mathrm {cof~}}{\mathbf{G}})^{\top } \\ &+\upepsilon _{\alpha \beta \gamma }(\mathrm {cof~}\mathbf{G})_{\gamma C}F_{iC}H^{ \top }_{i\beta ,\alpha }-\upepsilon _{ijk}(\mathrm {cof~}\mathbf{F})_{kC}G_{ \alpha C}\mathrm{H}_{\alpha j,i} \\ &=\mathscr{L}. \end{aligned} $$

(C.6)

Note that

$$\begin{aligned} \upepsilon _{ABC}\mathrm{A}_{i A}\mathrm{A}_{j B}B_{ij C}&= \upepsilon _{ABC}\mathrm{A}_{i A}\mathrm{A}_{j B}\delta _{CD}B_{ij D} \\ &=\upepsilon _{ABC}\mathrm{A}_{i A}\mathrm{A}_{j B}A^{\top }_{Ck }( \mathbf{A}^{-\top })_{kD}B_{ij D} \\ &=\upepsilon _{ijk}(\mathrm {cof~}\mathbf{A})_{kD}{B}_{ijD}. \end{aligned}$$

(C.7)