Tensors encode multilinear maps on a vector space as single algebraic objects that remain unchanged under any basis switch. Rank-0 tensors are scalars, rank-1 are vectors. A rank-2 case is the matrix
P = โกPโโ Pโแตง Pโ๐โค
โขPแตงโ Pแตงแตง Pแตง๐โฅ
โฃP๐โ P๐แตง P๐๐โฆ
whose nine entries give the force-per-area vectors on the six faces of a material cube. Rank equals the number of indices; a rank-3 tensor in three dimensions therefore stores exactly 3ยณ = 27 components.
Finite-element codes for turbine blades feed the Cauchy stress tensor into the equilibrium equations to compute local strain and remaining fatigue life under cyclic centrifugal loads.
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