Tensors generalize vectors as lists and matrices as grids into higher-order arrays like 3D cubes.
Comparisons include a vector as a list, a matrix as a grid, and a 3-tensor as a cube, along with the stress tensor σij on a cube featuring arrows for components like σ11 and σ23, the product derivative rule, the Riemann curvature tensor R(u,v)w = ∇u ∇_v w − ∇_v ∇_u w − ∇[u,v]w, quantum superposition 1/√2(|00⟩ + |11⟩), and algebraic operations like the tensor product.
It is used to analyze internal forces and deformations in engineering materials and to describe the geometry of spacetime under gravity in general relativity.