Cartesian tensor

A multidimensional array whose entries transform in a specific way under a rotation of axes.

A Cartesian tensor of rank $k$ is a multidimensional array of $3^k$ components that transforms under a rotation matrix $R$ according to a specific rule (one factor of $R$ per index). What makes an array a genuine tensor is exactly this transformation law -- a plain array of numbers with no such rule is not automatically a tensor, even if it happens to be called one.

Examples

The inertia tensor and the stress tensor are both rank-2 Cartesian tensors describing physical quantities that depend on direction.