Math Problems Vault

Problem IX.2.1

In a basis $\mathbf{e}$ the metric tensor is given by a matrix $\Gamma$ , and the tensor $a_{i j}$ by a matrix $A$ :

\Gamma =\left\| \begin{array}{ll} 1 & 1 \\ 1 & 2 \end{array}\right\| , \quad A=\left\| \begin{array}{ll} 3 & 4 \\ 5 & 7 \end{array}\right\| .

Find the matrices $B$ , $C$ and $D$ of the tensors $a_{. j}^{k}$ , $a_{j}^{k}$ . and $a^{j k}$ .

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Problem IX.2.2

One may notice that in problem 1 the determinants of the matrices, $A, B, C$ and $D$ coincide, and the traces of $B$ and $C$ are the same. Explain this.

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Problem IX.2.3

Simplify the expression

\left(\delta _{r}^{m} a_{i j}^{r} g^{j k}+\delta _{i}^{j} a_{l j}^{m} g^{l k}\right) g_{k s}+a_{[j k]}^{m} g^{j k} g_{i s} .

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