Tensor component transformation law
= Tensor component transformation law
Under a coordinate change $x\mapsto x'$, 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,
$$
T^{\mu'}{}_{\nu'}
=\frac{\partial x^{\mu'}}{\partial x^\alpha}
\frac{\partial x^\beta}{\partial x^{\nu'}}
T^\alpha{}_{\beta}.
$$