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Rank

  • Number of indices
  • Basis vectors per dimension/component
  • 0
    • Scalar
  • 1
    • Column Vector
  • 2
    • Square Matrix
  • 3
    • Cube matrix

Matrices are not inherently rank-2 tensors. Matrices are just the formatting structure. The tensor described by the matrix must follow the transformation rules to be a tensor !tensor.png

Transformation Rules

  1. Transforms like a tensor
  2. Invariant to a change in coordinate system
    • Components change according to mathematical formulae

Dimension

  • Dimensionality to the rank = number of components

An $n$-rank tensor in $m$-dimensional space is a mathematical object that has n indices and m^n components and obeys certain transformation rules

From <wolfram>