\[\newcommand{\u}[1]{\boldsymbol{\mathsf{#1}}} \renewcommand{\b}[1]{\boldsymbol{#1}} \newcommand{\t}[1]{\textsf{#1}} \newcommand{\m}[1]{\mathbb{#1}} \def\RR{\bf R} \def\bold#1{\bf #1} \def\mbf#1{\mathbf #1} \def\uv#1{\hat{\usf {#1}}} \def\dl#1{\underline{\underline{#1}}} \newcommand{\usf}[1]{\boldsymbol{\mathsf{#1}}} \def\bs#1{\usf #1}\]

Mechanics using the language of differential geometry

Metric co-efficients

The metric components \(g^{ij}\) and $g_{ij}$ are symmetric with respect to their indices. That is, $g^{ij}=g^{ji}$ and $g_{ij}=g_{ij}$.

Christoffel symbols

The differentiation of the covariant base vectors gives \(\left( Green \space and \space Zerna \right)\)

\[\begin{align} \label{eq:ChristoffelSymbolsFirstKind} \Gamma_{ijs}&=\frac{\partial^2 x^r }{\partial \theta^i \partial \theta^j}\frac{\partial x^r}{\partial \theta^s} \\ \label{eq:ChristoffelSymbolsSecondKind} \Gamma_{ij}^{r} &= g^{rs} \Gamma_{ijs} \end{align}\]

The symbols \(\Gamma_{ijs}\) and \(\Gamma_{ij}^s\) are the Christoffel Symbols of the first and the second kind, respectively. As can be seen by \(\eqref{eq:ChristoffelSymbolsFirstKind}\)–\(\eqref{eq:ChristoffelSymbolsSecondKind}\) The Christoffel Symbols \(\Gamma_{ij}^{s}\) are symmetric with respect to their indices \(i\) and \(j\).

\[\begin{align} \boldsymbol{g}^{i}_{,j} &=-\Gamma^{i}_{jr}\boldsymbol{g}^r \end{align}\]

Co-variant derivative of the strain tensor

In this section we compute the co-variant first and second derivatives of the strain tensor \(\gamma=\gamma_{ij}\boldsymbol{g}^{i}\otimes \boldsymbol{g}^j\). These derivatives are denoted as \(\mid\gamma_{ij}\mid_{p}\) and \(\mid\gamma_{ij}\mid_{pq}\). They are defined such that

\[\begin{align} \boldsymbol{\gamma}_{,p}&=\frac{\partial \boldsymbol{\gamma}}{\partial x^p}= \left|\gamma_{ij}\right|_{p} \boldsymbol{g}^{i}\otimes \boldsymbol{g}^j \\ \boldsymbol{\gamma}_{,pq}&=\frac{\partial \boldsymbol{\gamma}}{\partial x^p \partial x^q}= \left|\gamma_{ij}\right|_{,p,q} \boldsymbol{g}^{i}\otimes \boldsymbol{g}^j \end{align}\] \[\begin{align} \boldsymbol{\gamma} &= \gamma_{ij}\boldsymbol{g}^{i}\otimes\boldsymbol{g}^{j} \\ \boldsymbol{\gamma}_{,p}&=\gamma_{ij,p}\boldsymbol{g}^{i}\otimes\boldsymbol{g}^{j}+ \gamma_{ij}\boldsymbol{g}^{i}_{,p}\otimes\boldsymbol{g}^{j}+ \gamma_{ij}\boldsymbol{g}^{i}\otimes\boldsymbol{g}^{j}_{,p} \\ &=\gamma_{ij,p}\boldsymbol{g}^{i}\otimes\boldsymbol{g}^{j} -\gamma_{ij}\Gamma^{i}_{pr}\boldsymbol{g}^r\otimes\boldsymbol{g}^{j}- \gamma_{ij}\boldsymbol{g}^{i}\otimes\Gamma^{j}_{pr}\boldsymbol{g}^r \\ &=\gamma_{ij,p}\boldsymbol{g}^{i}\otimes\boldsymbol{g}^{j} -\gamma_{mj}\Gamma^{m}_{pr}\boldsymbol{g}^r\otimes\boldsymbol{g}^{j}- \gamma_{in}\Gamma^{n}_{pr}\boldsymbol{g}^{i}\otimes\boldsymbol{g}^r \\ &=\gamma_{ij,p}\boldsymbol{g}^{i}\otimes\boldsymbol{g}^{j} -\gamma_{mj}\Gamma^{m}_{pi}\boldsymbol{g}^i\otimes\boldsymbol{g}^{j}- \gamma_{in}\Gamma^{n}_{pj}\boldsymbol{g}^{i}\otimes\boldsymbol{g}^j \\ &=\left(\gamma_{ij,p}-\gamma_{mj}\Gamma^{m}_{pi}-\gamma_{in}\Gamma^{n}_{pj}\right)\boldsymbol{g}^{i}\otimes\boldsymbol{g}^j \\ \end{align}\]

We can write the last equation as \(\begin{align} \boldsymbol{\gamma}_{,p} &=A_{ij,p} \end{align}\)

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