(Solved): 4. Show that the following functions of a second-order tensor A. \[ A_{i}^{i} ...

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4. Show that the following functions of a second-order tensor A. \[ A_{i}^{i}, \quad A_{i}^{j} A_{j}^{i}, \quad A_{i}^{j} A_{k}^{i} A_{j}^{k} \] are invariants. 5. Let \( W \) be a scalar function of a second-order tensor \( A \). Show that the partia! derivatives \[ B^{i j} \equiv \frac{\partial W}{\partial A_{i j}} \] are contravariant components of a second-order tensor. 6. Verify the following relation between the Christoffel symbols and the metric tensor \[ \Gamma_{i j s}=\frac{1}{2}\left(g_{i s, j}+g_{j s, i}-g_{i j, s}\right) \] 7. Derive the expressions for the compunents of \( \Gamma_{i j}^{k} \) in: \( ( \) a \( ) \) cylinuifical coordinates, and (b) spherical coordinates. Note: The answers are: (a) \( \Gamma_{12}^{2}=\Gamma_{21}^{2}=\frac{1}{r}, \quad \Gamma_{22}^{1}=-r \), other \( \Gamma_{i j}^{k}=0 \) (b) \( \Gamma_{22}^{1}=-r, \quad \Gamma_{33}^{1}=-r \sin ^{2} \theta, \quad \Gamma_{12}^{2}=\frac{1}{r} \) \[ \Gamma_{33}^{2}=-\sin \theta \cos \theta, \quad \Gamma_{13}^{3}=\frac{1}{r}, \quad \Gamma_{23}^{3}=\cot \theta, \quad \text { other } \quad \Gamma_{i j}^{k}=0 \]