Question

Using (10.15) show that${\mathit{g}}_{\mathbf{i}\mathbf{j}}$is a${\mathbf{2}}^{\mathbf{n}\mathbf{d}}\mathit{V}$-rank covariant tensor. Hint:Write the transformationequation for each$\mathit{d}\mathit{x}$, and set the scalar$\mathit{d}\mathit{s}{\mathbf{\text{'}}}^{\mathbf{2}}\mathbf{=}\mathit{d}{\mathit{s}}^{\mathbf{2}}$to find the transformationequation for${\mathit{g}}_{\mathbf{i}\mathbf{j}}$.

Short answer

The results are proved in the solution.

Step-by-step solution

Given information.

The scalar$ds{\text{'}}^2=d{s}^2$

Definition of a covariance and contravariance.

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

Write the general form of a differential.

Length is an invariant quantity. This implies that $ds{\text{'}}^2=d{s}^2$. Use$10.14,10.15$, and continue evaluation.

$\begin{array}{c}d{s}^2=g_{ij}d{x}_{i}d{x}_j\\ d{s}^2=g_{ij}\frac{\partial x_i}{\partial x_k^{\text{'}}}\frac{\partial x_j}{\partial x_l^{\text{'}}}d{x}_k^{\text{'}}d{x}_l^{\text{'}}\\ d{s}^{\text{'}2}=g_{kl}^{\text{'}}d{x}_k^{\text{'}}d{x}_l^{\text{'}}\end{array}$

Compare the equations and write the result.

Compare the deduced equations and write the result.

$g_{kl}^{\text{'}}=\frac{\partial x_i}{\partial x_k^{\text{'}}}\frac{\partial x_j}{\partial x_l^{\text{'}}}{g}_{ij}$

Thus, it can be concluded that $g_{ij}$is a second order tensor.