Tensor type ($(r,s)$)

A tensor of type $(r,s)$ has $r$ contravariant and $s$ covariant indices. Taking a covariant derivative adds one covariant index, while an outer tensor product adds the corresponding contravariant and covariant index counts.

Under a coordinate change $x\mapsto x^{\prime }$, the components of a type-$(r,s)$ tensor acquire one forward Jacobian for each contravariant index and one inverse Jacobian for each covariant index. For a type-$(1,1)$ tensor,

A tensor density of weight $w$ transforms like a tensor with an additional factor $|\det (\partial x/\partial x^{\prime }){|}^w$ under $x\mapsto x^{\prime }$. A scalar density of weight one can be integrated against the coordinate volume element to give an invariant integral. On a spatial hypersurface, $\sqrt{h}$ is such a scalar density; hence ${\pi }^{ij}=\sqrt{h}{p}^{ij}$ is a tensor density when $p^{ij}$ is an ordinary tensor. The Levi-Civita connection extends to densities and satisfies $D_i\sqrt{h}=0$.

The identity tensor is the canonical type-$(1,1)$ tensor that maps each tangent vector to itself. It pairs a covector $\eta$ and vector $V$ as $\eta (V)$, and its components are the Kronecker delta ${\delta }^{\mu }{}_{\nu }$ in every coordinate system.