MTH3008 Final Exam 2022-23

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1. Outer Product, Scalar Contraction, and Cross Products in Suffix Notation

Question

1. Compute the outer product $M\otimes N$ for the matrices

$$M=\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right),N=\left(\begin{array}{ccc}1& 0& 3\\ 0& 1& -1\end{array}\right).$$

1. Let $T_{ijk\ell }$ be a rank–four tensor. Using the tensor transformation rule, show that the contracted quantity $T_{ijij}$ is a scalar.
2. Using suffix notation, derive an expression for $(\mathbf{u}\times \mathbf{v})\cdot (\mathbf{w}\times \mathbf{z})$ that involves no cross products.

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2. Dual Basis, Components, and Tensor Transformation between Bases

Question

Work in ${\mathbb{R}}^3$ with the Cartesian coordinate system $K$ having orthonormal basis vectors ${\mathbf{i}}_1,{\mathbf{i}}_2,{\mathbf{i}}_3$.

1. Consider the coordinate system $K^{\prime }$ with (not necessarily orthonormal) basis vectors

$${\mathbf{e}}_1={\mathbf{i}}_1,{\mathbf{e}}_2={\mathbf{i}}_1-{\mathbf{i}}_3,{\mathbf{e}}_3={\mathbf{i}}_1+{\mathbf{i}}_2+{\mathbf{i}}_3.$$

Find the dual basis ${\mathbf{e}}^1,{\mathbf{e}}^2,{\mathbf{e}}^3$. 2. For the vector $\mathbf{V}=2{\mathbf{i}}_1+{\mathbf{i}}_2+2{\mathbf{i}}_3$, compute its contravariant and covariant components with respect to the basis $\{{\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3\}$ and the dual basis $\{{\mathbf{e}}^1,{\mathbf{e}}^2,{\mathbf{e}}^3\}$. 3. In the Cartesian system $K$, consider the second–order tensor with components

$$[P^i{}_k]=[P_i{}^k]=[P_k^i]=[P_k^i]=\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& -1\\ -1& 1& 1\end{array}\right).$$

Express the covariant components of this tensor in the coordinate system $K^{\prime }$.

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3. Transformation Rules and Tensorial Inner Products

Question

Let $A_{ij}$ and $B_{ijk}$ be tensors in a general three–dimensional coordinate system.

1. Write down the transformation rules for the components $A_{ij}$ and $B_{ijk}$ under a change of coordinates.
2. Using these transformation rules, prove that the contracted inner product $A_{ij}{B}_{ikl}$ is a tensor, and state its rank.
   1. Show that the Levi–Civita symbols satisfy the identity ${ϵ}_{ijk}{ϵ}_{ij\ell }=2{\delta }_{k\ell }$.
   2. Suppose $A_{ij}$ and $B_{ijk}$ are related by

B_{ijk} = \epsilon_{ij\ell} A_{\ell k}. $$ Using the identity from part (3)(i), derive an explicit expression for $A_{\ell k}$ in terms of $B_{ijk}$.

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4. Parabolic Cylindrical Coordinates: Basis, Metric, and Christoffel Symbols

Question

Consider the parabolic cylindrical coordinate system $(x^1,x^2,x^3)=(\sigma ,\tau ,\varphi )$, with position vector

$$\mathbf{r}=\sigma \tau \cos \varphi {\mathbf{i}}_1+\sigma \tau \sin \varphi {\mathbf{i}}_2+\frac{1}{2}({\sigma }^2-{\tau }^2){\mathbf{i}}_3,$$

where ${\mathbf{i}}_1,{\mathbf{i}}_2,{\mathbf{i}}_3$ are the usual Cartesian basis vectors.

1. Compute the coordinate basis vectors ${\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3$ associated with the parabolic coordinates $(\sigma ,\tau ,\varphi )$.
2. Compute the metric coefficients for the arc length, and hence find the components of the covariant metric tensor $g_{ij}$ (in particular, $g_{ii}$) for this parabolic coordinate system.
3. Determine the following Christoffel symbols of the first kind for the parabolic coordinates: ${\Gamma }_{112},{\Gamma }_{123},{\Gamma }_{122},{\Gamma }_{233}$.

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