91Ó°ÊÓ

A covariant tensor and a contravariant tensor are given.

The components of a vector relative to a tangent bundle basis are covariant in differential geometry if they change with the same linear transformation as the basis. If they change as a result of the inverse transformation, they are contravariant.

Raise the index of$V_i^{\text{'}}$since$V^i$is contravariant

$\begin{array}{c}{V}_i^{\text{'}}=g_{ij}^{\text{'}}{V}^{\text{'}j}\\ =g_{ij}^{\text{'}}\frac{∂x_j^{\text{'}}}{∂x_k}{V}^k\end{array}$

Continue evaluation considering$g_{ij}$is a second order covariant tensor.

$\begin{array}{c}{V}_i^{\text{'}}=g_{lm}\frac{∂x_l}{∂x_i^{\text{'}}}\frac{∂x_m}{∂x_j^{\text{'}}}\frac{∂x_j^{\text{'}}}{∂x_k}{V}^k\\ {\mathrm{δ}}_k^m=\frac{∂x_m}{∂x_j^{\text{'}}}\frac{∂x_j^{\text{'}}}{∂x_k}\\ V_i^{\text{'}}=\frac{∂x_l}{∂x_i^{\text{'}}}{g}_{lm}{V}^m\\ =\frac{∂x_l}{∂x_i^{\text{'}}}{V}_l\end{array}$

$\begin{array}{c}V{\text{'}}^i=g^{ij}{V}_j^{\text{'}}\\ =g^{ij}\frac{∂x_k}{∂x_j^{\text{'}}}{V}_k\\ g{\text{'}}^{ij}=g^{lm}\frac{∂x_i^{\text{'}}}{∂x_l}\frac{∂x_j^{\text{'}}}{∂x_m}\\ V^{\text{'}i}=g^{lm}\frac{∂x_i^{\text{'}}}{∂x_l}\frac{∂x_j^{\text{'}}}{∂x_m}\frac{∂x_k}{∂x_j^{\text{'}}}{V}_k\end{array}$

Continue the evaluation.

$\begin{array}{c}{\mathrm{δ}}_m^k=\frac{∂x_j^{\text{'}}}{∂x_m}\frac{∂x_k}{∂x_j^{\text{'}}}\\ V^i=\frac{∂x_i^{\text{'}}}{∂x_l}{g}^{lm}{V}_m\\ =\frac{∂x_i^{\text{'}}}{∂x_l}{V}^l\end{array}$