Connections

Proposition 10.4. Suppose $\Phi:(M, g) \rightarrow(\widetilde{M}, \widetilde{g})$ is an isometry, then we have $$ \Phi_*\left(\nabla_X Y\right)=\widetilde{\nabla}_{\Phi_* X}\left(\Phi_* Y\right) > $$ for any $X, Y \in \Gamma^{\infty}(T M)$. Here $\nabla$ and $\widetilde{\nabla}$ are the Levi-Civita connections for $g$ and $\widetilde{g}$ respectively. That is, the Levi-Civita connection is isometric-invariant connection satisfies $\nabla(f \sigma)=d f \otimes \sigma+f \nabla \sigma$ and $d\left\langle\sigma_2, \sigma_2\right\rangle=\left\langle\nabla \sigma_1, \sigma_2\right\rangle+\left\langle\sigma_1, \nabla \sigma_2\right\rangle$, where $f: M \rightarrow R$, and $\langle$, $\rangle$ is the Riemannian metric on $E=L_e \oplus \varepsilon$ which was described above.

Coordinates

Configuration $\mathbf{g}_i\in T\mathcal{B}$ and $\mathbf{g}^i\in T^*\mathcal{B}$ Coordinates Reference The covariant derivative associated with the Levi-Civita connection on $\mathcal{B}_0$ is written as $$ \stackrel{G}{\nabla}_A(\cdot)=(\cdot) ; A . $$ Contravariant reference basis vectors can be written symbolically as gradients of coordinates in Euclidean space: $$ \stackrel{G}{\nabla} X^A=\partial_B X^A \mathbf{G}^B=\delta_B^A \mathbf{G}^B=\mathbf{G}^A . $$ $$ \nabla=\stackrel{G}{\nabla} \Rightarrow \nabla_{\partial_A} \partial_B=\nabla_{\partial_B} \partial_A=\partial_B \partial_A=\partial_A \bm{\partial}_B, $$ Current SOURCE: Clayton 2015 $$ \mathbf{G}_A\triangleq\frac{\partial}{\partial X^A}\equiv\partial_A, \qquad \mathbf{G}^A \equiv \bm{d} X^A $$ $$ \mathbf{a}_a\triangleq\frac{\partial}{\partial x^a}\equiv\partial_a, \qquad \mathbf{a}^a \equiv \bm{d} x^a $$ $$ \frac{\partial(\cdot)}{\partial X^A}=\partial_A(\cdot)=(\cdot)_{, A}, \quad \frac{\partial(\cdot)}{\partial x^a}=\partial_a(\cdot)=(\cdot)_{, a} $$ $$ \begin{aligned} \mathbf{g}^{A}(x, t)&=(F^{-1})_{~~a}^{A} \, \mathbf{a}^a \\ \end{aligned} \quad \mathbf{g}_A = F_{~~ A}^a \mathbf{a}_a $$ $$ \mathbf{G}_A(X)= \partial_A \bm{X}, \quad \mathbf{a}_a(x)= \partial_a \bm{x} \qquad\text{BAD, bold in }\partial(~\cdot~) $$ $$ \begin{array}{lcrlcrlcrlcrl} & & & \mathbf{G}_I & & & \mathbf{a}_i & & & \mathbf{g}_I \\ \hline \mathbf{G}_I & = & & \bm{\partial}_I & = & & & & \bm{F}^{-1} & \mathbf{g}_I \\ \mathbf{a}_i & = & & & = & & \bm{\partial}_i \\ \mathbf{g}_I & = & \bm{F} & \mathbf{G}_I & = & \partial_I x^i &\bm{\partial}_i \\ \mathbf{A}_i & = & \partial_i X^I & \mathbf{G}_I & = & \\[0.

Energy

One says that balance of energy holds if, for every nice open set $\mathcal{U} \subset \mathcal{B}$, $$ \frac{d}{\mathrm{~d} t} \int_{\varphi_t(\mathcal{U})} \rho\left(e+\frac{1}{2}\langle\langle\mathbf{v}, \mathbf{v}\rangle\rangle\right) \mathrm{d} v=\int_{\varphi_t(\mathcal{U})} \rho(\langle\langle\mathbf{b}, \mathbf{v}\rangle\rangle+r) \mathrm{d} v+\int_{\partial \varphi_l(\mathcal{U})}(\langle\langle\mathbf{t}, \mathbf{v}\rangle\rangle+h) \mathrm{d} a, $$ where $e=e(\mathbf{x}, t), r=r(\mathbf{x}, t)$ and $h=h(\mathbf{x}, \hat{\mathbf{n}}, t)$ are internal energy per unit mass, heat supply per unit mass and heat flux, respectively. Balance of energy. For an elastic material one assumes that there exists a strain energy function $e$ per unit mass whose change represents the change in the internal energy due to mechanical deformations.