005张量的表示与张量的指标升降

$$\begin{gathered} \Phi \in {\mathcal{T}}^3(R^m) \\ \Phi (u,v,w)\in R \\ \Phi (u^i{g}_i,v_j{g}^j,w^k{g}_k)\in R(用\Phi (u^i{g}_i,v^j{g}_j,w^k{g}_k)表示也行) \\ 利用张量的性质： \\ \Phi (u^i{g}_i,v_j{g}^j,w^k{g}_k)=\Phi (g_i,g^j,g_k)u^i{v}_j{w}^k \\ 其中u^i{v}_j{w}^k=(u,g^i)(v,g_k)(w,g^k)=g^i\otimes g_j\otimes g^k(u,v,w) \\ \Phi (g_i,g^j,g_k)记为{{{\Phi }_i}^j}_k \\ \Phi ={{{\Phi }_i}^j}_k{g}^i\otimes g_j\otimes g^k \end{gathered}$$

$$\begin{gathered} 指标升降：{{{\Phi }^i}_j}^k,i降，j升 \\ {{{\Phi }^i}_j}^k=\Phi (g^i,g_j,g^k) \\ =\Phi (g^{ip}{g}_p,g_{jq}{g}^q,g^k) \\ =g^{ip}{g}_{jq}\Phi (g_p,g^q,g^k) \\ =g^{ip}{g}_{jq}{{{\Phi }_p}^q}^k \end{gathered}$$

$$3阶张量有8中表示，{\Phi }^{ijk}称为逆变分量，{\Phi }_{ijk}称为协变分量，其他称为混合分量$$