Tensor - Examples

Examples

This table shows important examples of tensors, including both tensors on vector spaces and tensor fields on manifolds. The tensors are classified according to their type (n, m). For example, a bilinear form is the same thing as a (0, 2)-tensor; an inner product is an example of a (0, 2)-tensor, but not all (0, 2)-tensors are inner products. In the (0, M)-entry of the table, M denotes the dimension of the underlying vector space or manifold.

n, m	n = 0	n = 1	n = 2	...	n = N	...
m = 0	scalar, e.g. scalar curvature	vector (e.g. direction vector)	bivector, e.g. inverse metric tensor		N-vector, a sum of N-blades
m = 1	covector, linear functional, 1-form	linear transformation, Kronecker delta
m = 2	bilinear form, e.g. inner product, metric tensor, Ricci curvature, 2-form, symplectic form	e.g. cross product in three dimensions	e.g. elasticity tensor
m = 3	e.g. 3-form	e.g. Riemann curvature tensor
...
m = M	e.g. M-form i.e. volume form
...

Raising an index on an (n, m)-tensor produces an (n + 1, m − 1)-tensor; this can be visualized as moving diagonally up and to the right on the table. Symmetrically, lowering an index can be visualized as moving diagonally down and to the left on the table. Contraction of an upper with a lower index of an (n, m)-tensor produces an (n − 1, m − 1)-tensor; this can be visualized as moving diagonally up and to the left on the table.