MTH3008 Final Exam 2024–25

Original Documents: [[mth3008 final exam 2024-25.pdf|Exam Paper]] / Provided Solutions

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1. Inner Products, Tensor Transformation, Determinant Zero from Repeated Rows

Question

We consider the tensor with components $[T_{ik}]=\left(\begin{array}{ccc}1& 0& 2\\ 0& 2& 1\\ 1& -1& 4\end{array}\right)$, and the vectors $A={\mathbf{i}}_1-{\mathbf{i}}_2+{\mathbf{i}}_3$, $B={\mathbf{i}}_1-2{\mathbf{i}}_2$.

1. Compute the inner product $T_{ik}{A}_k$.
2. Compute the scalar $T_{ik}{A}_k{B}_i$.

Next, let $T_{ijk}$ and $R_{ijk}$ be rank–3 tensors.

1. Using the formal transformation rule (definition) of a tensor, show that $T^i{}_p{}^j{R}_{ij}{}^q$ is a rank–2 tensor. You must explicitly use the tensor transformation rule; solutions relying only on counting free indices are not accepted.

Finally, let $A$ be a $3\times 3$ real matrix.

1. Using suffix notation, show that if $A$ has two identical rows, then its determinant $|A|$ is zero. You must give a suffix–notation proof; general linear algebra arguments (e.g. “row operations” reasoning) are not accepted.

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2. Dual Basis, Components in a Non-Orthonormal Basis, Tensor Components under Rotation

Question

Let $K$ be the Cartesian coordinate system with orthonormal basis ${\mathbf{i}}_1,{\mathbf{i}}_2,{\mathbf{i}}_3$ in ${\mathbb{R}}^3$. Consider another coordinate system $K^{\prime }$ with basis vectors ${\mathbf{e}}_1=3{\mathbf{i}}_1+{\mathbf{i}}_2+{\mathbf{i}}_3,{\mathbf{e}}_2={\mathbf{i}}_2,{\mathbf{e}}_3=2{\mathbf{i}}_1+{\mathbf{i}}_3.$

1. Find the dual basis ${\mathbf{e}}^1,{\mathbf{e}}^2,{\mathbf{e}}^3$ corresponding to ${\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3$.

2. Let $V={\mathbf{i}}_1+{\mathbf{i}}_2-2{\mathbf{i}}_3$.

   1. Find the covariant components $V_i$ of $V$ with respect to the basis ${\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3$.
   2. Find the contravariant components $V^i$ of $V$ with respect to the basis ${\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3$.

3. Let $L=(L^j{}_{i^{\prime }})$ denote the rotation matrix relating $K$ and $K^{\prime }$.

   1. Determine the components $L^j{}_{i^{\prime }}$.
   2. In the coordinate system $K$, consider the second–order tensor with components $[P_{ik}]=\left(\begin{array}{ccc}1& 0& 0\\ 0& 2& -1\\ 0& 0& 3\end{array}\right),$ where we may also write $P_{ik}=P_{ki}$ or in mixed index positions as $P^{\cdot k}{}_i$, etc. Express the covariant components of this tensor in the coordinate system $K^{\prime }$. Show all working and justify each step in your tensor–transformation formula.

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3. Symmetry Properties of a Constructed Tensor, Scalar Triple Products in Suffix Notation, Radial Gradient Formula

Question

Let $B_{ik}$ be a second–rank tensor with $B_{ik}\ne 0$ for all $i,k\in \{1,2,3\}$. Define $A_{ijk}={\epsilon }_{ij\ell }{B}_{k\ell },$ where ${\epsilon }_{ij\ell }$ is the Levi–Civita symbol. The quantity $A_{ijk}$ is a third–rank tensor (you do not need to prove this).

1. Investigate the symmetry of $A_{ijk}$ with respect to the first and second indices. Decide whether $A_{ijk}$ is symmetric, antisymmetric, or neither in these indices, and justify your answer.
2. Investigate the symmetry of $A_{ijk}$ with respect to the first and third indices. Decide whether $A_{ijk}$ is symmetric, antisymmetric, or neither in these indices, and justify your answer.

Next, let $\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}$ be vectors in ${\mathbb{R}}^3$.

1. Using suffix notation and the Levi–Civita symbol, derive an expression without any cross products for $(\mathbf{a}\times (\mathbf{b}\times \mathbf{c}))\cdot ((\mathbf{a}\cdot \mathbf{c})\mathbf{d}).$ Show all intermediate steps in suffix notation, then write your final simplified result back in vector notation.

Finally, let $\mathbf{r}=(x_1,x_2,x_3)$ denote the position vector in some coordinate system on ${\mathbb{R}}^3$, let $r=|\mathbf{r}|$, and let $f$ be a scalar field depending only on $r$. Let $\nabla$ denote the gradient operator.

1. Show that $\nabla f(r)=f^{\prime }(r)\frac{\mathbf{r}}{r}.$ You may use the fact that $f^{\prime }(r)=\frac{\partial f}{\partial r}$, but you must derive the gradient expression explicitly from the chain rule and the relationship between $r$ and the Cartesian coordinates.

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4. Orthogonal Curvilinear Coordinates, Metric Tensor, Arc Length, Christoffel Symbols

Question

Consider the three–dimensional orthogonal coordinate system $(x^1,x^2,x^3)=(\rho ,\phi ,\theta )$ with position vector $\displaystyle \mathbf{r} = e^{\rho} \cos \varphi \cos \theta, \mathbf{i}_1

• e^{\rho} \sin \varphi \cos \theta, \mathbf{i}_2

• e^{\rho} \sin \theta, \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 ${\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3$ of this coordinate system (i.e. the vectors tangent to coordinate lines).

2. Using your results:

   1. Compute the components $g_{ik}$ of the metric tensor for this coordinate system.
   2. Write down the line element $(\mathrm{d}s{)}^2$ in terms of the metric coefficients, and identify the associated scale factors (Lamé coefficients) $h_1,h_2,h_3$.

3. Determine the following Christoffel symbols of the first kind for this coordinate system: ${\Gamma }_{111},{\Gamma }_{122},{\Gamma }_{223},{\Gamma }_{233}.$ Express your answers in terms of $\rho$ and $\theta$ and the metric coefficients, and show all intermediate steps (including any required derivatives of the metric components).

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