Tensor Calculus

Our journey here prioritizes ideas that help with understanding other pages.

Constants move with Impunity

Let a,b,c be math objects allowed by our rules:

abc3 = ab3c = a3bc = 3abc

Kronecker Delta

\delta_{\: \: \alpha}^\beta x^\alpha = x^\beta

\delta_{\: \: \alpha}^\beta x_\beta = x_\alpha

Index Notation and Matrix Multiplication

A^i_{:\ :\ j} B^j_{\: \: k}= C^k_{ :\ :\ i}

Tensor Multiplications

a^i b_i = a^1 b_1 + a^2 b_2 + a^3 b_3

a^i b_j = \begin{bmatrix}a^1 b_1 & a^1 b_2 & a^1 b_3 \\ a^2 b_1 & a^2 b_2 & a^2 b_3 \\ a^3 b_1 & a^3 b_2 & a^3 b_3 \end{bmatrix}

Tensor Definition

A^{a_1}_{u_1} A^{a_2}_{u_2} A^{a_3}_{u_3} ... A^{a_N}_{u_N} \: (A^{-1})_{b_1}^{v_1} (A^{-1}) _{b_2}^{v_2} (A^{-1}) _{b_3}^{v_3} ... (A^{-1}) _{b_M}^{v_M} \: t^{u_1 u_2 u_3 ... u_N}_{v_1 v_2 v_3 ... v_M} \: \: = \: \: t'^{a_1 a_2 a_3 ... a_N}_{b_1 b_2 b_3 ... b_M}

this can be compared to something on another page…

Notice that A takes us from ‘u’ to ‘a’ and from ‘v’ to ‘b’.