Tensor Analysis

Quantities with many directions. Index notation, Kronecker delta and Levi-Civita symbols, tensor transformation and invariants, then the stress and strain tensors, principal values, Mohr's circle, Hooke's law and von Mises stress.

• Index Notation & Tensors — Tensors generalise scalars and vectors to quantities that need several directions to describe — stress, strain, inertia, conductivity. Learn index (Einstein) notation, the Kronecker delta and Levi-Civita symbol, tensor order, transformation rules under rotation, contraction, symmetric/antisymmetric parts, and how to translate between index form and matrix form.
• Stress & Strain Tensors — The stress tensor σᵢⱼ and strain tensor εᵢⱼ are the most-used tensors in engineering. Learn Cauchy's traction formula, principal stresses as eigenvalues, invariants, hydrostatic and deviatoric parts, Mohr's circle, the small-strain tensor from displacement gradients, generalised Hooke's law and the von Mises criterion — with a NumPy eigen-analysis.