| Tensor Notation | Name | Vector Notation | Expanded |
A. | \(a_i b_i\) | Vector Dot Product | \({\bf a} \cdot {\bf b}\) | \(a_x b_x + a_y b_y + a_z b_z\) |
B. | \(\epsilon_{rst} a_r b_t\) | Vector Cross Product | \({\bf b} \times {\bf a}\) | No need to expand |
C. | \(A_{rs} B_{ts}\) | Matrix Multiplication | \({\bf A} \cdot {\bf B}^T\) | No need to expand |
D. | \(A_{ii}\) | Trace of Matrix | \(\text{tr}({\bf A})\) | \(A_{11} + A_{22} + A_{33}\) |
E. | \(\boldsymbol{\sigma}_{ij,j}\) | Divergence of Stress Tensor | \(\nabla \cdot \boldsymbol{\sigma}\) | No need to expand |
F. | \(f_{,kk}\) | Laplacian | \(\nabla^2 f\) | \({\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}\) |