MTH3008 Final Exam 2023-24

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1.1. Outer Product, Rank-Four Transformation, and Triple Cross Product

Question

(a) Compute the outer product $M\otimes N$ of the matrices

$$M=\left(\begin{array}{cc}0& 1\\ 7& 2\end{array}\right),N=\left(\begin{array}{cc}1& 0\\ 1& -1\\ -1& 2\end{array}\right).$$

(b) Suppose $T_{ijk\ell }$ is a rank-four tensor.

1. Write down the transformation rule for $T_{ijk\ell }$ under a change of basis.
2. Using this transformation rule, show that the contracted quantity $T_{ijij}$ is a scalar.

(c) Let $u,v,w,z$ be vectors in ${\mathbb{R}}^3$. Using suffix notation, find an expression involving no cross products for $(u\times (v\times w))\cdot z$. Write your final answer in vector notation, and show all intermediate steps of your working.

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2.1. Dual Basis, Components, and Tensor Transformation in a New Frame

Question

In this question, let $K$ be the Cartesian coordinate system with orthonormal basis vectors $i_1,i_2,i_3$ for ${\mathbb{R}}^3$. Let $K^{\prime }$ be a new coordinate system with basis vectors $e_1,e_2,e_3$ defined by

$$e_1=2i_1+i_3,e_2=i_2,e_3=i_1-i_2+i_3.$$

(a) Find the dual basis $e^1,e^2,e^3$ corresponding to $e_1,e_2,e_3$.

(b) Consider the vector $V=3i_1-2i_2+i_3$. Find both the covariant and contravariant components of $V$ with respect to the basis $e_1,e_2,e_3$ and the dual basis $e^1,e^2,e^3$.

(c) In the coordinate system $K$, consider the second-order tensor with components

$$[P^i{}_k]=[P_i{}^k]=[P_{\cdot k}^i]=[P_i{}_{\cdot }{}^k]=\left(\begin{array}{ccc}0& 2& 0\\ 3& 0& -1\\ -1& 0& 1\end{array}\right).$$

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

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3.1. Parabolic Coordinates, Metric, and Christoffel Symbols

Question

Consider the three-dimensional orthogonal coordinate system $(x^1,x^2,x^3)=(\sigma ,\tau ,\varphi )$ with position vector

$$r=\sigma \tau i_1+\frac{1}{2}({\sigma }^2-{\tau }^2)i_2+\varphi i_3,$$

where $i_1,i_2,i_3$ are the Cartesian basis vectors.

(a) Compute the (covariant) basis vectors $e_1,e_2,e_3$ of this coordinate system.

(b) Compute the metric coefficients of the arc length and the components $g_{ii}$ of the covariant metric tensor for this parabolic coordinate system.

(c) Determine the following Christoffel symbols of the first kind for this coordinate system: ${\Gamma }_{112},{\Gamma }_{123},{\Gamma }_{122},{\Gamma }_{233}$.

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4.1. Symmetry Properties, Non-Tensorial Derivative, and Curl of Gradient

Question

(a) Let $T_{ijk}$ be a third-rank tensor that is symmetric in its last two indices and antisymmetric in its first and second indices, that is $T_{ijk}=T_{ikj}$ and $T_{ijk}=-T_{jik}$ for all $i,j,k$. Show that all components of this tensor are zero, i.e. $T_{ijk}=0$ for every choice of indices $i,j,k$.

(b) Let $V_i$ be a (non-constant) covariant tensor field transforming according to $V_i^{\prime }=L^k{}_i{}^{\prime }{V}_k$. Prove that the partial derivative $\frac{\partial V_i}{\partial x^j}$ is not a tensor, that is, it does not satisfy the tensor transformation law. You may use the relations

$$L^k{}_i{}^{\prime }=\frac{\partial x^k}{\partial x^{\prime i}},L^{i^{\prime }}{}_k=\frac{\partial x^{\prime i}}{\partial x^k}.$$

(c) Let $f$ be a scalar field. Using suffix notation, evaluate the vector expression $\nabla \times (\nabla f)$.

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