$$ v'_i = Q_{ij}v_j, \qquad [\v'] = [\Q][\v] $$

$$ A'_{ij} = Q_{ik}Q_{jl}A_{kl}, \qquad [\A'] = [\Q][\A][\Q^T] $$

$$ \A = \A^K + \A^D = \frac{1}{3}(\mathrm{tr}\A)\I + \left(\A - \frac{1}{3}(\mathrm{tr}\A)\I\right) $$

$$ \A = A_{\hspace{-3mm}\underbrace{\scriptstyle ij\ldots p}_{N\;\textrm{indeksów}}\hspace{-3mm}}\,\e_i \otimes \e_j \otimes \cdots \otimes \e_p $$

$$ \f: \mathcal{T}_M \rightarrow \mathcal{T}_N,\quad \Y = \f(\Z), \quad Y\!\underbrace{_{ij\ldots k}}_N = f_{ij\ldots k}(Z\!\underbrace{_{lm\ldots n}}_M) $$

$$ \f: E^3 \rightarrow \mathcal{T}_N, \quad\Y = \f(\x), \quad Y\!\underbrace{_{ij\ldots n}}_N = f_{ij\ldots n}(x_p) $$

$$ \grad \A = \A\otimes \nabla=A_{ij\ldots kl,m}\e_i\otimes\e_j\otimes\cdots\otimes\e_k\otimes\e_l\otimes\e_m\in\mathcal{T}_{N+1} $$

$$ \div \A=\A\nabla=A_{ij\ldots kl,l} \e_i\otimes\e_j \otimes \cdots \otimes\e_k\in\mathcal{T}_{N-1} $$

$$ \nabla^2\A = (\A\otimes\nabla)\nabla=A_{ij\ldots kl,mm}\e_i\otimes\e_j\otimes\cdots\otimes\e_k\otimes\e_l\in\mathcal{T}_N $$

$$ \rot\A=\nabla\times\A=\epsilon_{mip}A_{ij\ldots kl,m}\e_p\otimes\e_j\otimes\cdots\otimes\e_k\otimes\e_l\in\mathcal{T}_N $$

$$ \x = \R(t)\X + \u(t), \quad x_i = R_{ij}(t)X_j +u_i(t) $$

$$ \u(\X, t) = \x(\X, t)-\X, \qquad u_i(X_j, t) = x_i(X_j, t) - X_i $$

$$ \F = \mathrm{Grad} \x = \x \otimes\nabla_{X} = \cfrac{\partial\x}{\partial\X}, \quad F_{ij} = \cfrac{\partial x_i}{\partial X_j} $$

$$ [\F] =\mat{\pp{x_1}{X_1}}{\pp{x_1}{X_2}}{\pp{x_1}{X_3}}{\pp{x_2}{X_1}}{\pp{x_2}{X_2}}{\pp{x_2}{X_3}}{\pp{x_3}{X_1}}{\pp{x_3}{X_2}}{\pp{x_3}{X_3}} $$

$$ \f = \mathrm{grad} \X = \X \otimes\nabla_{x} = \pp{\X}{\x}, \quad f_{ij} = \pp{X_i}{x_j} $$

$$ \H=\mathrm{Grad}\u = \u \otimes \nabla_X, \, H_{ij} = \pp{u_i}{X_j} $$

$$ \h=\mathrm{grad}\u= \u \otimes \nabla_x,\, h_{ij} = \pp{u_i}{x_j} $$

$$ C_{jk} = F_{ij}F_{ik}, \, \C = \F^T\F $$

$$ E_{ij} = \cfrac{1}{2}(C_{ij} - \delta_{ij}),\, \E = \cfrac{1}{2}(\C-\I) $$

$$ e_{ij} = \cfrac{1}{2}(\delta_{ij} - c_{ij}),\, \e = \cfrac{1}{2}(\I-\c) $$

$$ \E = \cfrac{1}{2}(\H + \H^T + \H^T\H) $$

$$ \ee = \cfrac{1}{2}(\H + \H^T) \approx \cfrac{1}{2}(\h + \h^T) $$

$$ dV_t = \det\F dV_0 $$

$$ \rr{dA_t\n = J\F^{-T} \N dA_0}{dA_t n_j = J F_{ji}^{-1} N_j dA_0} $$

$$ \dot{\overline{\int_{\Omega} (\cdot)dV_t}} =\int_{\Omega} \dot{\overline{(\cdot)}}dV_t + (\cdot)\dot{\overline{dV_t}}= $$

$$ \rho c\dot T = \div(\ll\nabla T) + \rho r = (\lambda_{ij}T_{,j})_{,i} + \rho r $$

$$ \vc{\sigma_{11}}{\sigma_{22}}{\sigma_{12}} = \frac E{1-\nu^2} \left[\begin{array}{ccc}1&\nu&0\\\nu&1&0\\0&0&(1-\nu)/2 \end{array}\right]\vc{\varepsilon_{11}}{\varepsilon_{22}}{2\varepsilon_{12}}, \quad\varepsilon_{33} = \frac{-\nu}{1-\nu}(\varepsilon_{11} + \varepsilon_{22}) $$

$$ \vc{\varepsilon_{11}}{\varepsilon_{22}}{2\varepsilon_{12}}=\left[\begin{array}{ccc}1/E&-\nu/E&0\\ -\nu/E&1/E&0\\0&0&1/G \end{array}\right]\vc{\sigma_{11}}{\sigma_{22}}{\sigma_{12}},\quad \varepsilon_{33} = \frac{-\nu}{E}(\sigma_{11} + \sigma_{22}) $$

$$ \vc{\sigma_{11}}{\sigma_{22}}{\sigma_{12}}=\frac{E}{(1+\nu)(1-2\nu)}\left[\begin{array}{ccc}1-\nu&\nu&0\\ \nu&1-\nu&0\\ 0&0&\frac{1-2\nu}2\end{array}\right]\vc{\varepsilon_{11}}{\varepsilon_{22}}{2\varepsilon_{12}}, \quad\sigma_{33} = \frac{E\nu(\varepsilon_{11}+\varepsilon_{22})}{(1+\nu)(1-2\nu)} $$

$$ \vc{\varepsilon_{11}}{\varepsilon_{22}}{2\varepsilon_{12}}=\frac{1+\nu}E \left[\begin{array}{ccc}1-\nu&-\nu&0\\-\nu&1-\nu&0\\0&0&2 \end{array}\right]\vc{\sigma_{11}}{\sigma_{22}}{\sigma_{12}},\quad \sigma_{33} =\nu(\sigma_{11}+\sigma_{22}) $$