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# 10. Tensors ## Transformations ### (a) Vectors as lists of components $\vec{v} = (v_1, v_2, v_3)$ (in a given coordinate system). ### (b) Coordinate transformation $v_i \rightarrow v'_i = \sum_j A_{ij} v_j$ $\begin{aligned} A: \text{ transformation matrix } \\ \vec{v'} = A\vec{v} \end{aligne...

# 10. Tensors ## Transformations ### (a) Vectors as lists of components $\vec{v} = (v_1, v_2, v_3)$ (in a given coordinate system). ### (b) Coordinate transformation $v_i \rightarrow v'_i = \sum_j A_{ij} v_j$ $\begin{aligned} A: \text{ transformation matrix } \\ \vec{v'} = A\vec{v} \end{aligned}$ ### (c) Scalars Invariant under coordinate transformations. Examples: mass, energy, electric charge. ## Definition of Tensors ### (a) $(0,0)$ tensor = scalar ### (b) $(1,0)$ tensor = vector Transformation law: $v_i \rightarrow v'_i = \sum_j A_{ij} v_j$. ### (c) $(0,1)$ tensor = dual vector Transformation law: $\tilde{v}_i \rightarrow \tilde{v'}_i = \sum_j (A^{-1})_{ji} \tilde{v}_j$. ### (d) $(n,m)$ tensor - $n$ indices transform like vectors - $m$ indices transform like dual vectors Transformation law: $T_{i_1... i_n j_1... j_m} \rightarrow T'_{i_1... i_n j_1... j_m} = \sum_{k_1... k_n l_1... l_m} A_{i_1 k_1}... A_{i_n k_n} (A^{-1})_{l_1 j_1}... (A^{-1})_{l_m j_m} T_{k_1... k_n l_1... l_m}$ ### (e) Examples $(0,0)$ scalar $(1,0)$ vector $(0,1)$ dual vector $(2,0)$ $T_{ij} \rightarrow T'_{ij} = \sum_{kl} A_{ik} A_{jl} T_{kl}$ $(0,2)$ $T_{ij} \rightarrow T'_{ij} = \sum_{kl} (A^{-1})_{ki} (A^{-1})_{lj} T_{kl}$ $(1,1)$ $T_{ij} \rightarrow T'_{ij} = \sum_{kl} A_{ik} (A^{-1})_{lj} T_{kl}$