$ \newcommand{\Ua}{\mathrm{a}} \newcommand{\Ub}{\mathrm{b}} \newcommand{\Uc}{\mathrm{c}} \newcommand{\Ud}{\mathrm{d}} \newcommand{\Ue}{\mathrm{e}} \newcommand{\Uf}{\mathrm{f}} \newcommand{\Ug}{\mathrm{g}} \newcommand{\Uh}{\mathrm{h}} \newcommand{\Ui}{\mathrm{i}} \newcommand{\Uj}{\mathrm{j}} \newcommand{\Uk}{\mathrm{k}} \newcommand{\Ul}{\mathrm{l}} \newcommand{\Um}{\mathrm{m}} \newcommand{\Un}{\mathrm{n}} \newcommand{\Uo}{\mathrm{o}} \newcommand{\Up}{\mathrm{p}} \newcommand{\Uq}{\mathrm{q}} \newcommand{\Ur}{\mathrm{r}} \newcommand{\Us}{\mathrm{s}} \newcommand{\Ut}{\mathrm{t}} \newcommand{\Uu}{\mathrm{u}} \newcommand{\Uv}{\mathrm{v}} \newcommand{\Uw}{\mathrm{w}} \newcommand{\Ux}{\mathrm{x}} \newcommand{\Uy}{\mathrm{y}} \newcommand{\Uz}{\mathrm{z}} \newcommand{\UA}{\mathrm{A}} \newcommand{\UB}{\mathrm{B}} \newcommand{\UC}{\mathrm{C}} \newcommand{\UD}{\mathrm{D}} \newcommand{\UE}{\mathrm{E}} \newcommand{\UF}{\mathrm{F}} \newcommand{\UG}{\mathrm{G}} \newcommand{\UH}{\mathrm{H}} \newcommand{\UI}{\mathrm{I}} \newcommand{\UJ}{\mathrm{J}} \newcommand{\UK}{\mathrm{K}} \newcommand{\UL}{\mathrm{L}} \newcommand{\UM}{\mathrm{M}} \newcommand{\UN}{\mathrm{N}} \newcommand{\UO}{\mathrm{O}} \newcommand{\UP}{\mathrm{P}} \newcommand{\UQ}{\mathrm{Q}} \newcommand{\UR}{\mathrm{R}} \newcommand{\US}{\mathrm{S}} \newcommand{\UT}{\mathrm{T}} \newcommand{\UU}{\mathrm{U}} \newcommand{\UV}{\mathrm{V}} \newcommand{\UW}{\mathrm{W}} \newcommand{\UX}{\mathrm{X}} \newcommand{\UY}{\mathrm{Y}} \newcommand{\UZ}{\mathrm{Z}} % \newcommand{\Uzero }{\mathrm{0}} \newcommand{\Uone }{\mathrm{1}} \newcommand{\Utwo }{\mathrm{2}} \newcommand{\Uthree}{\mathrm{3}} \newcommand{\Ufour }{\mathrm{4}} \newcommand{\Ufive }{\mathrm{5}} \newcommand{\Usix }{\mathrm{6}} \newcommand{\Useven}{\mathrm{7}} \newcommand{\Ueight}{\mathrm{8}} \newcommand{\Unine }{\mathrm{9}} % \newcommand{\Ja}{\mathit{a}} \newcommand{\Jb}{\mathit{b}} \newcommand{\Jc}{\mathit{c}} \newcommand{\Jd}{\mathit{d}} \newcommand{\Je}{\mathit{e}} \newcommand{\Jf}{\mathit{f}} \newcommand{\Jg}{\mathit{g}} \newcommand{\Jh}{\mathit{h}} \newcommand{\Ji}{\mathit{i}} \newcommand{\Jj}{\mathit{j}} \newcommand{\Jk}{\mathit{k}} \newcommand{\Jl}{\mathit{l}} \newcommand{\Jm}{\mathit{m}} \newcommand{\Jn}{\mathit{n}} \newcommand{\Jo}{\mathit{o}} \newcommand{\Jp}{\mathit{p}} \newcommand{\Jq}{\mathit{q}} \newcommand{\Jr}{\mathit{r}} \newcommand{\Js}{\mathit{s}} \newcommand{\Jt}{\mathit{t}} \newcommand{\Ju}{\mathit{u}} \newcommand{\Jv}{\mathit{v}} \newcommand{\Jw}{\mathit{w}} \newcommand{\Jx}{\mathit{x}} \newcommand{\Jy}{\mathit{y}} \newcommand{\Jz}{\mathit{z}} \newcommand{\JA}{\mathit{A}} \newcommand{\JB}{\mathit{B}} \newcommand{\JC}{\mathit{C}} \newcommand{\JD}{\mathit{D}} \newcommand{\JE}{\mathit{E}} \newcommand{\JF}{\mathit{F}} \newcommand{\JG}{\mathit{G}} \newcommand{\JH}{\mathit{H}} \newcommand{\JI}{\mathit{I}} \newcommand{\JJ}{\mathit{J}} \newcommand{\JK}{\mathit{K}} \newcommand{\JL}{\mathit{L}} \newcommand{\JM}{\mathit{M}} \newcommand{\JN}{\mathit{N}} \newcommand{\JO}{\mathit{O}} \newcommand{\JP}{\mathit{P}} \newcommand{\JQ}{\mathit{Q}} \newcommand{\JR}{\mathit{R}} \newcommand{\JS}{\mathit{S}} \newcommand{\JT}{\mathit{T}} \newcommand{\JU}{\mathit{U}} \newcommand{\JV}{\mathit{V}} \newcommand{\JW}{\mathit{W}} \newcommand{\JX}{\mathit{X}} \newcommand{\JY}{\mathit{Y}} \newcommand{\JZ}{\mathit{Z}} % \newcommand{\Jzero }{\mathit{0}} \newcommand{\Jone }{\mathit{1}} \newcommand{\Jtwo }{\mathit{2}} \newcommand{\Jthree}{\mathit{3}} \newcommand{\Jfour }{\mathit{4}} \newcommand{\Jfive }{\mathit{5}} \newcommand{\Jsix }{\mathit{6}} \newcommand{\Jseven}{\mathit{7}} \newcommand{\Jeight}{\mathit{8}} \newcommand{\Jnine }{\mathit{9}} % \newcommand{\BA}{\boldsymbol{A}} \newcommand{\BB}{\boldsymbol{B}} \newcommand{\BC}{\boldsymbol{C}} \newcommand{\BD}{\boldsymbol{D}} \newcommand{\BE}{\boldsymbol{E}} \newcommand{\BF}{\boldsymbol{F}} \newcommand{\BG}{\boldsymbol{G}} \newcommand{\BH}{\boldsymbol{H}} \newcommand{\BI}{\boldsymbol{I}} \newcommand{\BJ}{\boldsymbol{J}} \newcommand{\BK}{\boldsymbol{K}} \newcommand{\BL}{\boldsymbol{L}} \newcommand{\BM}{\boldsymbol{M}} \newcommand{\BN}{\boldsymbol{N}} \newcommand{\BO}{\boldsymbol{O}} \newcommand{\BP}{\boldsymbol{P}} \newcommand{\BQ}{\boldsymbol{Q}} \newcommand{\BR}{\boldsymbol{R}} \newcommand{\BS}{\boldsymbol{S}} \newcommand{\BT}{\boldsymbol{T}} \newcommand{\BU}{\boldsymbol{U}} \newcommand{\BV}{\boldsymbol{V}} \newcommand{\BW}{\boldsymbol{W}} \newcommand{\BX}{\boldsymbol{X}} \newcommand{\BY}{\boldsymbol{Y}} \newcommand{\BZ}{\boldsymbol{Z}} \newcommand{\Ba}{\boldsymbol{a}} \newcommand{\Bb}{\boldsymbol{b}} \newcommand{\Bc}{\boldsymbol{c}} \newcommand{\Bd}{\boldsymbol{d}} \newcommand{\Be}{\boldsymbol{e}} \newcommand{\Bf}{\boldsymbol{f}} \newcommand{\Bg}{\boldsymbol{g}} \newcommand{\Bh}{\boldsymbol{h}} \newcommand{\Bi}{\boldsymbol{i}} \newcommand{\Bj}{\boldsymbol{j}} \newcommand{\Bk}{\boldsymbol{k}} \newcommand{\Bl}{\boldsymbol{l}} \newcommand{\Bm}{\boldsymbol{m}} \newcommand{\Bn}{\boldsymbol{n}} \newcommand{\Bo}{\boldsymbol{o}} \newcommand{\Bp}{\boldsymbol{p}} \newcommand{\Bq}{\boldsymbol{q}} \newcommand{\Br}{\boldsymbol{r}} \newcommand{\Bs}{\boldsymbol{s}} \newcommand{\Bt}{\boldsymbol{t}} \newcommand{\Bu}{\boldsymbol{u}} \newcommand{\Bv}{\boldsymbol{v}} \newcommand{\Bw}{\boldsymbol{w}} \newcommand{\Bx}{\boldsymbol{x}} \newcommand{\By}{\boldsymbol{y}} \newcommand{\Bz}{\boldsymbol{z}} % \newcommand{\Bzero }{\boldsymbol{0}} \newcommand{\Bone }{\boldsymbol{1}} \newcommand{\Btwo }{\boldsymbol{2}} \newcommand{\Bthree}{\boldsymbol{3}} \newcommand{\Bfour }{\boldsymbol{4}} \newcommand{\Bfive }{\boldsymbol{5}} \newcommand{\Bsix }{\boldsymbol{6}} \newcommand{\Bseven}{\boldsymbol{7}} \newcommand{\Beight}{\boldsymbol{8}} \newcommand{\Bnine }{\boldsymbol{9}} % \newcommand{\Balpha }{\boldsymbol{\alpha} } \newcommand{\Bbeta }{\boldsymbol{\beta} } \newcommand{\Bgamma }{\boldsymbol{\gamma} } \newcommand{\Bdelta }{\boldsymbol{\delta} } \newcommand{\Bepsilon}{\boldsymbol{\epsilon} } \newcommand{\Bvareps }{\boldsymbol{\varepsilon} } \newcommand{\Bvarepsilon}{\boldsymbol{\varepsilon}} \newcommand{\Bzeta }{\boldsymbol{\zeta} } \newcommand{\Beta }{\boldsymbol{\eta} } \newcommand{\Btheta }{\boldsymbol{\theta} } \newcommand{\Bvarthe }{\boldsymbol{\vartheta} } \newcommand{\Biota }{\boldsymbol{\iota} } \newcommand{\Bkappa }{\boldsymbol{\kappa} } \newcommand{\Blambda }{\boldsymbol{\lambda} } \newcommand{\Bmu }{\boldsymbol{\mu} } \newcommand{\Bnu }{\boldsymbol{\nu} } \newcommand{\Bxi }{\boldsymbol{\xi} } \newcommand{\Bpi }{\boldsymbol{\pi} } \newcommand{\Brho }{\boldsymbol{\rho} } \newcommand{\Bvrho }{\boldsymbol{\varrho} } \newcommand{\Bsigma }{\boldsymbol{\sigma} } \newcommand{\Bvsigma }{\boldsymbol{\varsigma} } \newcommand{\Btau }{\boldsymbol{\tau} } \newcommand{\Bupsilon}{\boldsymbol{\upsilon} } \newcommand{\Bphi }{\boldsymbol{\phi} } \newcommand{\Bvarphi }{\boldsymbol{\varphi} } \newcommand{\Bchi }{\boldsymbol{\chi} } \newcommand{\Bpsi }{\boldsymbol{\psi} } \newcommand{\Bomega }{\boldsymbol{\omega} } \newcommand{\BGamma }{\boldsymbol{\Gamma} } \newcommand{\BDelta }{\boldsymbol{\Delta} } \newcommand{\BTheta }{\boldsymbol{\Theta} } \newcommand{\BLambda }{\boldsymbol{\Lambda} } \newcommand{\BXi }{\boldsymbol{\Xi} } \newcommand{\BPi }{\boldsymbol{\Pi} } \newcommand{\BSigma }{\boldsymbol{\Sigma} } \newcommand{\BUpsilon}{\boldsymbol{\Upsilon} } \newcommand{\BPhi }{\boldsymbol{\Phi} } \newcommand{\BPsi }{\boldsymbol{\Psi} } \newcommand{\BOmega }{\boldsymbol{\Omega} } % \newcommand{\IA}{\mathbb{A}} \newcommand{\IB}{\mathbb{B}} \newcommand{\IC}{\mathbb{C}} \newcommand{\ID}{\mathbb{D}} \newcommand{\IE}{\mathbb{E}} \newcommand{\IF}{\mathbb{F}} \newcommand{\IG}{\mathbb{G}} \newcommand{\IH}{\mathbb{H}} \newcommand{\II}{\mathbb{I}} \renewcommand{\IJ}{\mathbb{J}} \newcommand{\IK}{\mathbb{K}} \newcommand{\IL}{\mathbb{L}} \newcommand{\IM}{\mathbb{M}} \newcommand{\IN}{\mathbb{N}} \newcommand{\IO}{\mathbb{O}} \newcommand{\IP}{\mathbb{P}} \newcommand{\IQ}{\mathbb{Q}} \newcommand{\IR}{\mathbb{R}} \newcommand{\IS}{\mathbb{S}} \newcommand{\IT}{\mathbb{T}} \newcommand{\IU}{\mathbb{U}} \newcommand{\IV}{\mathbb{V}} \newcommand{\IW}{\mathbb{W}} \newcommand{\IX}{\mathbb{X}} \newcommand{\IY}{\mathbb{Y}} \newcommand{\IZ}{\mathbb{Z}} % \newcommand{\FA}{\mathsf{A}} \newcommand{\FB}{\mathsf{B}} \newcommand{\FC}{\mathsf{C}} \newcommand{\FD}{\mathsf{D}} \newcommand{\FE}{\mathsf{E}} \newcommand{\FF}{\mathsf{F}} \newcommand{\FG}{\mathsf{G}} \newcommand{\FH}{\mathsf{H}} \newcommand{\FI}{\mathsf{I}} \newcommand{\FJ}{\mathsf{J}} \newcommand{\FK}{\mathsf{K}} \newcommand{\FL}{\mathsf{L}} \newcommand{\FM}{\mathsf{M}} \newcommand{\FN}{\mathsf{N}} \newcommand{\FO}{\mathsf{O}} \newcommand{\FP}{\mathsf{P}} \newcommand{\FQ}{\mathsf{Q}} \newcommand{\FR}{\mathsf{R}} \newcommand{\FS}{\mathsf{S}} \newcommand{\FT}{\mathsf{T}} \newcommand{\FU}{\mathsf{U}} \newcommand{\FV}{\mathsf{V}} \newcommand{\FW}{\mathsf{W}} \newcommand{\FX}{\mathsf{X}} \newcommand{\FY}{\mathsf{Y}} \newcommand{\FZ}{\mathsf{Z}} \newcommand{\Fa}{\mathsf{a}} \newcommand{\Fb}{\mathsf{b}} \newcommand{\Fc}{\mathsf{c}} \newcommand{\Fd}{\mathsf{d}} \newcommand{\Fe}{\mathsf{e}} \newcommand{\Ff}{\mathsf{f}} \newcommand{\Fg}{\mathsf{g}} \newcommand{\Fh}{\mathsf{h}} \newcommand{\Fi}{\mathsf{i}} \newcommand{\Fj}{\mathsf{j}} \newcommand{\Fk}{\mathsf{k}} \newcommand{\Fl}{\mathsf{l}} \newcommand{\Fm}{\mathsf{m}} \newcommand{\Fn}{\mathsf{n}} \newcommand{\Fo}{\mathsf{o}} \newcommand{\Fp}{\mathsf{p}} \newcommand{\Fq}{\mathsf{q}} \newcommand{\Fr}{\mathsf{r}} \newcommand{\Fs}{\mathsf{s}} \newcommand{\Ft}{\mathsf{t}} \newcommand{\Fu}{\mathsf{u}} \newcommand{\Fv}{\mathsf{v}} \newcommand{\Fw}{\mathsf{w}} \newcommand{\Fx}{\mathsf{x}} \newcommand{\Fy}{\mathsf{y}} \newcommand{\Fz}{\mathsf{z}} % \newcommand{\Fzero }{\mathsf{0}} \newcommand{\Fone }{\mathsf{1}} \newcommand{\Ftwo }{\mathsf{2}} \newcommand{\Fthree}{\mathsf{3}} \newcommand{\Ffour }{\mathsf{4}} \newcommand{\Ffive }{\mathsf{5}} \newcommand{\Fsix }{\mathsf{6}} \newcommand{\Fseven}{\mathsf{7}} \newcommand{\Feight}{\mathsf{8}} \newcommand{\Fnine }{\mathsf{9}} % \newcommand{\CA}{\mathcal{A}} \newcommand{\CB}{\mathcal{B}} \newcommand{\CC}{\mathcal{C}} \newcommand{\CD}{\mathcal{D}} \newcommand{\CE}{\mathcal{E}} \newcommand{\CF}{\mathcal{F}} \newcommand{\CG}{\mathcal{G}} \newcommand{\CH}{\mathcal{H}} \newcommand{\CI}{\mathcal{I}} \newcommand{\CJ}{\mathcal{J}} \newcommand{\CK}{\mathcal{K}} \newcommand{\CL}{\mathcal{L}} \newcommand{\CM}{\mathcal{M}} \newcommand{\CN}{\mathcal{N}} \newcommand{\CO}{\mathcal{O}} \newcommand{\CP}{\mathcal{P}} \newcommand{\CQ}{\mathcal{Q}} \newcommand{\CR}{\mathcal{R}} \newcommand{\CS}{\mathcal{S}} \newcommand{\CT}{\mathcal{T}} \newcommand{\CU}{\mathcal{U}} \newcommand{\CV}{\mathcal{V}} \newcommand{\CW}{\mathcal{W}} \newcommand{\CX}{\mathcal{X}} \newcommand{\CY}{\mathcal{Y}} \newcommand{\CZ}{\mathcal{Z}} % \newcommand{\KA}{\mathfrak{A}} \newcommand{\KB}{\mathfrak{B}} \newcommand{\KC}{\mathfrak{C}} \newcommand{\KD}{\mathfrak{D}} \newcommand{\KE}{\mathfrak{E}} \newcommand{\KF}{\mathfrak{F}} \newcommand{\KG}{\mathfrak{G}} \newcommand{\KH}{\mathfrak{H}} \newcommand{\KI}{\mathfrak{I}} \newcommand{\KJ}{\mathfrak{J}} \newcommand{\KK}{\mathfrak{K}} \newcommand{\KL}{\mathfrak{L}} \newcommand{\KM}{\mathfrak{M}} \newcommand{\KN}{\mathfrak{N}} \newcommand{\KO}{\mathfrak{O}} \newcommand{\KP}{\mathfrak{P}} \newcommand{\KQ}{\mathfrak{Q}} \newcommand{\KR}{\mathfrak{R}} \newcommand{\KS}{\mathfrak{S}} \newcommand{\KT}{\mathfrak{T}} \newcommand{\KU}{\mathfrak{U}} \newcommand{\KV}{\mathfrak{V}} \newcommand{\KW}{\mathfrak{W}} \newcommand{\KX}{\mathfrak{X}} \newcommand{\KY}{\mathfrak{Y}} \newcommand{\KZ}{\mathfrak{Z}} \newcommand{\Ka}{\mathfrak{a}} \newcommand{\Kb}{\mathfrak{b}} \newcommand{\Kc}{\mathfrak{c}} \newcommand{\Kd}{\mathfrak{d}} \newcommand{\Ke}{\mathfrak{e}} \newcommand{\Kf}{\mathfrak{f}} \newcommand{\Kg}{\mathfrak{g}} \newcommand{\Kh}{\mathfrak{h}} \newcommand{\Ki}{\mathfrak{i}} \newcommand{\Kj}{\mathfrak{j}} \newcommand{\Kk}{\mathfrak{k}} \newcommand{\Kl}{\mathfrak{l}} \newcommand{\Km}{\mathfrak{m}} \newcommand{\Kn}{\mathfrak{n}} \newcommand{\Ko}{\mathfrak{o}} \newcommand{\Kp}{\mathfrak{p}} \newcommand{\Kq}{\mathfrak{q}} \newcommand{\Kr}{\mathfrak{r}} \newcommand{\Ks}{\mathfrak{s}} \newcommand{\Kt}{\mathfrak{t}} \newcommand{\Ku}{\mathfrak{u}} \newcommand{\Kv}{\mathfrak{v}} \newcommand{\Kw}{\mathfrak{w}} \newcommand{\Kx}{\mathfrak{x}} \newcommand{\Ky}{\mathfrak{y}} \newcommand{\Kz}{\mathfrak{z}} % \newcommand{\Kzero }{\mathfrak{0}} \newcommand{\Kone }{\mathfrak{1}} \newcommand{\Ktwo }{\mathfrak{2}} \newcommand{\Kthree}{\mathfrak{3}} \newcommand{\Kfour }{\mathfrak{4}} \newcommand{\Kfive }{\mathfrak{5}} \newcommand{\Ksix }{\mathfrak{6}} \newcommand{\Kseven}{\mathfrak{7}} \newcommand{\Keight}{\mathfrak{8}} \newcommand{\Knine }{\mathfrak{9}} % $

$ \newcommand{\Lin}{\mathop{\rm Lin}\nolimits} \newcommand{\modop}{\mathop{\rm mod}\nolimits} \renewcommand{\div}{\mathop{\rm div}\nolimits} \newcommand{\Var}{\Delta} \newcommand{\evat}{\bigg|} \newcommand\varn[3]{D_{#2}#1\cdot #3} \newcommand{\dtp}{\cdot} \newcommand{\dyd}{\otimes} \newcommand{\tra}{^T} \newcommand{\del}{\partial} \newcommand{\dif}{d} \newcommand{\rbr}[1]{\left(#1\right)} \newcommand{\sbr}[1]{\left[#1\right]} \newcommand{\cbr}[1]{\left\{#1\right\}} \newcommand{\cbrn}[1]{\{#1\}} \newcommand{\abr}[1]{\left\langle #1 \right\rangle} \newcommand{\abrn}[1]{\langle #1 \rangle} \newcommand{\deriv}[2]{\frac{d #1}{d #2}} \newcommand{\dderiv}[2]{\frac{d^2 #1}{d {#2}^2}} \newcommand{\partd}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\nnode}{n_n} \newcommand{\ndim}{n_d} \newcommand{\suml}[2]{\sum\limits_{#1}^{#2}} \newcommand{\Aelid}[2]{A^{#1}_{#2}} \newcommand{\dv}{\, dv} \newcommand{\dx}{\, dx} \newcommand{\ds}{\, ds} \newcommand{\da}{\, da} \newcommand{\dV}{\, dV} \newcommand{\dA}{\, dA} \newcommand{\eqand}{\quad\text{and}\quad} \newcommand{\eqor}{\quad\text{or}\quad} \newcommand{\eqwith}{\quad\text{and}\quad} \newcommand{\inv}{^{-1}} \newcommand{\veci}[1]{#1_1,\ldots,#1_n} \newcommand{\var}{\delta} \newcommand{\Var}{\Delta} \newcommand{\eps}{\epsilon} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\Norm}[1]{\left\lVert#1\right\rVert} \newcommand{\Abs}[1]{\left|#1\right|} \newcommand{\dabr}[1]{\left\langle\!\left\langle #1 \right\rangle\!\right\rangle} \newcommand{\dabrn}[1]{\langle\!\langle #1 \rangle\!\rangle} \newcommand{\idxsep}{\,} $

Equipping a vector space with an inner product results in a natural isomorphism $\CV\to\CV^\ast$, where the metric tensor can be interpreted as the linear mapping $\Bg:\CV\to\CV^\ast$ and its inverse $\Bg\inv:\CV^\ast\to\CV$.

Notation: Given two real vector spaces $\CV$ and $\CW$, we denote their inner products as \(\dabrn{\cdot,\cdot}_{\CV}\) and \(\dabrn{\cdot,\cdot}_{\CW}\) respectively. Given vectors $\Bv\in\CV$ and $\Bw\in\CW$, we define their lengths as

\[\begin{equation} \Norm{\Bv}_\CV = \sqrt{\dabrn{\Bv,\Bv}_\CV} \eqand \Norm{\Bw}_\CW = \sqrt{\dabrn{\Bw,\Bw}_\CW}. \end{equation}\]

Regarding $\CV$ and $\CW$,

1. their bases are denoted $\cbrn{\BE_A}$ and $\cbrn{\Be_a}$,
2. their dual bases are denoted $\cbrn{\BE^A}$ and $\cbrn{\Be^a}$,
3. their metrics are denoted $\BG$ and $\Bg$ with the components \(G_{AB}=\dabrn{\BE_A,\BE_B}_\CV\) and \(g_{ab}=\dabrn{\Be_a,\Be_b}_\CW\),

respectively. Here, the indices pertaining to $\CV$ are uppercase $(ABC\dots)$ and the indices pertaining to $\CW$ are lowercase $(abc\dots)$.

Definition: Let $\BP:\CV\to\CW$ be a linear mapping. Then the transpose, or adjoint of $\BP$, written $\BP\tra$, is the linear mapping

\[\begin{equation} \boxed{ \BP\tra: \CW\to\CV \quad\text{such that}\quad \dabrn{\Bv,\BP\tra\Bw}_\CV = \dabrn{\BP\Bv,\Bw}_\CW } \end{equation}\]

for all $\Bv\in\CV$ and $\Bw\in\CW$. Carrying out the products,

\[\begin{equation} G_{BA} v^B (P\tra){}^{A}{}_{d} w^d = g_{ab} P{}^{b}{}_{C}v^Cw^a. \end{equation}\]

For arbitrary $\Bv$ and $\Bw$,

\[\begin{equation} G_{BA} (P\tra){}^{A}{}_{a} = g_{ab} P{}^{b}{}_{A} \end{equation}\]

from which we can obtain the components of the transpose as

\[\begin{equation} \boxed{ (P\tra){}^{A}{}_{a} = g_{ab} P{}^{b}{}_{B} G^{AB} \eqwith \BP\tra = (P\tra){}^{A}{}_{a} \BE_A\dyd\Be^a . } \end{equation}\]

If $\BB:\CV\to\CV$ is a linear mapping, it is called symmetric if $\BB=\BB\tra$.

Definition: Let $\BP:\CV\to\CW$ be a linear mapping. Then the dual of $\BP$ is a metric independent mapping

\[\begin{equation} \boxed{ \BP^\ast: \CW^\ast\to\CV^\ast \quad\text{such that}\quad \abrn{\Bv,\BP^\ast\Bbeta}_\CV = \abrn{\BP\Bv,\Bbeta}_\CW } \end{equation}\]

defined through natural pairings for all $\Bv\in\CV$ and $\Bbeta\in\CW^\ast$. Carrying out the products,

\[\begin{equation} v^A (P^\ast){}_{A}{}^{a} \beta_a = P{}^{b}{}_{B} v^B \beta_b. \end{equation}\]

For arbitrary $\Bv$ and $\Bbeta$, we obtain the components of the dual mapping as

\[\begin{equation} \boxed{ (P^\ast){}_{A}{}^{a} = P{}^{a}{}_{A} \eqwith \BP^\ast = (P^\ast){}_{A}{}^{a} \BE^A\dyd\Be_a = P{}^{a}{}_{A} \BE^A\dyd\Be_a . } \end{equation}\]

To fully appreciate the symmetry that originates from the duality, we can think of not just the mappings between $\CV$ and $\CW$, but also between their dual spaces. To this end we can enumerate four mappings corresponding to $\cbr{\CV,\CV^\ast}\to\cbr{\CW,\CW^\ast}$ and their duals, corresponding to $\cbr{\CW,\CW^\ast}\to\cbr{\CV,\CV^\ast}$. Their definitions can be found in the table below.

Tensors $\BP$, $\BQ$, $\BR$ and $\BS$ as linear mappings (top), and their duals $\BP^\ast$, $\BQ^\ast$, $\BR^\ast$ and $\BS^\ast$ (bottom). In the respective tables, the first row displays the tensor spaces, basis vectors and components of the subsequent mappings, and the second and third row display the representations of the tensor as linear and bilinear mappings respectively. The results of the mappings are given in the mapping, matrix and index representations respectively. The mappings are over vectors $\Bv\in\CV$, $\Bw\in\CW$ and one-forms $\Balpha\in\CV^\ast$, $\Bbeta\in\CW^\ast$.

Mappings
$\BP\in\CW \dyd\CV^\ast$ $P^a_{\idxsep A}=\BP(\Be^a,\BE_A)$ $\BP = P^{a}_{\idxsep A}\, \Be_a \dyd \BE^A$	$\BQ\in\CW^\ast \dyd\CV^\ast$ $Q_{aA}=\BQ(\Be_a,\BE_A)$ $\BQ = Q_{aA}\, \Be^a \dyd \BE^A$	$\BR\in\CW \dyd\CV$ $R^{aA}=\BR(\Be^a,\BE^A)$ $\BR = R^{aA}\, \Be_a \dyd \BE_A$	$\BS\in\CW^\ast \dyd\CV$ $S_a^{\idxsep A}=\BS(\Be_a,\BE^A)$ $\BS = S_{a}^{\idxsep A}\, \Be^a \dyd \BE_A$
$\BP: \CV \to \CW$ $\begin{aligned} \Bv &\mapsto \BP(\Be^a,\Bv) \Be_a \\ &= \BP\Bv \\ v^A\BE_A &\mapsto P^a_{\idxsep A} v^A \Be_a \end{aligned}$	$\BQ: \CV \to \CW^\ast$ $\begin{aligned} \Bv &\mapsto \BQ(\Be_a,\Bv) \Be^a \\ &= \BQ\Bv \\ v^A\BE_A &\mapsto Q_{aA} v^A \Be^a \end{aligned}$	$\BR: \CV^\ast \to \CW$ $\begin{aligned} \Balpha &\mapsto \BR(\Be^a,\Balpha) \Be_a \\ &= \BR\Balpha\tra \\ \alpha_A \BE^A &\mapsto R^{aA} \alpha_A \Be_a \end{aligned}$	$\BS: \CV^\ast \to \CW^\ast$ $\begin{aligned} \Balpha &\mapsto \BS(\Be_a,\Balpha) \Be^a \\ &= \BS\Balpha\tra \\ \alpha_A \BE^A &\mapsto S_a^{\idxsep A} \alpha_A \Be^a \end{aligned}$
$\BP: \CW^\ast \times \CV \to \IR$ $\begin{aligned} (\Bbeta,\Bv) &\mapsto \BP(\Bbeta,\Bv) \\ &=\Bbeta\BP\Bv \\ &= \beta_a P^{a}_{\idxsep A} v^A \end{aligned}$	$\BQ: \CW \times \CV \to \IR$ $\begin{aligned} (\Bw,\Bv) &\mapsto \BQ(\Bw,\Bv) \\ &= \Bw\tra\BQ\Bv \\ &= w^a Q_{aA} v^A \end{aligned}$	$\BR: \CW^\ast \times \CV^\ast \to \IR$ $\begin{aligned} (\Bbeta,\Balpha) &\mapsto \BR(\Bbeta,\Balpha) \\ &= \Bbeta\BR\Balpha\tra \\ &= \beta_a R^{aA} \alpha_A \end{aligned}$	$\BS: \CW \times \CV^\ast \to \IR$ $\begin{aligned} (\Bw,\Balpha) &\mapsto \BS(\Bw,\Balpha) \\ &= \Bw\tra\BS\Balpha\tra \\ &= w^a S_a^{\idxsep A} \alpha_A \end{aligned}$
Duals
$\BP^\ast\in \CV^\ast \dyd\CW$ $P^{\ast \, a}_A=\BP^\ast(\BE_A,\Be^a)$ $\BP^\ast = P^{\ast \, a}_A \, \BE^A \dyd \Be_a$	$\BQ^\ast\in \CV^\ast \dyd\CW^\ast$ $Q^\ast_{Aa}=\BQ^\ast(\BE_A,\Be_a)$ $\BQ^\ast = Q^\ast_{Aa}\, \BE^A \dyd \Be^a$	$\BR^\ast\in\CV \dyd \CW$ $R^{\ast Aa}=\BR^\ast(\BE^A,\Be^a)$ $\BR^\ast = R^{\ast Aa}\, \BE_A \dyd \Be_a$	$\BS^\ast\in\CV \dyd\CW^\ast$ $S^{\ast A}{}_{a}=\BS^\ast(\BE^A,\Be_a)$ $\BS^\ast = S^{\ast A}{}_{a}\, \BE_A \dyd \Be^a$
$\BP^\ast: \CW^\ast \to \CV^\ast $ $\begin{aligned} \Bbeta &\mapsto \BP^\ast(\BE_A,\Bbeta) \BE^A \\ &= \BP^\ast\Bbeta\tra \\ \beta_a\Be^a &\mapsto P^{\ast \, a}_A \beta_a \BE^A \end{aligned}$	$\BQ^\ast: \CW \to\CV^\ast$ $\begin{aligned} \Bw &\mapsto \BQ^\ast(\BE_A,\Bw) \BE^A \\ &= \BQ^\ast\Bw \\ w^a\Be_a &\mapsto Q^\ast_{Aa} w^a \BE^A \end{aligned}$	$\BR^\ast: \CW^\ast \to \CV$ $\begin{aligned} \Bbeta &\mapsto \BR^\ast(\BE^A,\Bbeta) \BE_A \\ &= \BR^\ast\Bbeta\tra \\ \beta_a\Be^a &\mapsto R^{\ast Aa} w^a \BE_A \end{aligned}$	$\BS^\ast: \CW \to\CV$ $\begin{aligned} \Bw &\mapsto \BS^\ast(\BE^A,\Bw) \BE_A \\ &= \BS^\ast\Bw \\ w^a\Be_a &\mapsto S^{\ast A}{}_{a} w^a \BE_A \end{aligned}$
$\BP^\ast: \CV \times \CW^\ast \to \IR$ $\begin{aligned} (\Bv,\Bbeta) &\mapsto \BP^\ast(\Bv,\Bbeta) \\ &= \Bv\tra\BP^\ast\Bbeta\tra \\ &= v^A P^{\ast \, a}_A \beta_a \end{aligned}$	$\BQ^\ast: \CV \times \CW \to \IR$ $\begin{aligned} (\Bv,\Bw) &\mapsto \BR^\ast(\Bv,\Bw) \\ &= \Bv\tra\BQ^\ast\Bw \\ &= v^A Q^\ast_{Aa} w^a \end{aligned}$	$\BR^\ast: \CV^\ast \times \CW^\ast \to \IR$ $\begin{aligned} (\Balpha,\Bbeta) &\mapsto \BS^\ast(\Balpha,\Bbeta) \\ &= \Balpha\BR^\ast\Bbeta\tra \\ &= \alpha_A R^{\ast Aa} \beta_a \end{aligned}$	$\BS^\ast: \CV^\ast \times \CW \to \IR$ $\begin{aligned} (\Balpha,\Bw) &\mapsto \BS^\ast(\Balpha,\Bw) \\ &= \Balpha\BS^\ast\Bw \\ &= \alpha_A S^{\ast A}{}_{a} w^a \end{aligned}$

The commutative diagrams pertaining to these mappings can be found in the figure below

Commutative diagrams involving the linear mappings $\BP,\BQ,\BR,\BS$ and their dual $\BP^\ast,\BQ^\ast,\BR^\ast,\BS^\ast$ based on the metrics $\BG$ and $\Bg$ of $\CV$ and $\CW$.