MTH3008 Weekly Problems 6

Original Documents: Problem Sheet / My Handwritten Solutions / Provided Solutions

Vibes: …

Used Techniques:

• …

---

6.1. Composition Of Coordinate Transformation Matrices

Question

Using the alternative definitions of the transformation coefficients $L^k{}_{i^{\prime }}$ and $L^{i^{\prime }}{}_k$, namely

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

show that

$$L^i{}_{k^{\prime }}{L}^{k^{\prime }}{}_j={\delta }^i{}_j,$$

where ${\delta }^i{}_j$ is the usual Kronecker delta written in the notation appropriate for generalised coordinate systems. [file:1]

…

---

6.2. Contraction Of Covariant And Contravariant Tensors

Question

Let $A_{ik\ell }$ be a covariant tensor of order $3$ and $B^{pqmn}$ a contravariant tensor of order $4$. Prove that the object with components

$$A_{ik\ell }{B}^{k\ell mn}$$

is a mixed tensor of order $3$ with one covariant index and two contravariant indices. [file:1] In particular, verify that it transforms with one factor of the inverse transformation for the lower index and two factors of the direct transformation for the upper indices.

…

---

6.3. Valid Relations Between Associated Tensors

Question

For each of the following proposed relations between associated tensors, decide whether it is correct, and justify your answer in each case. [file:1] (You may assume $g_{ij}$ and $g^{ij}$ are the components of the metric tensor and its inverse, and that repeated indices are summed.)

1. $T^p{}_q=g^r{}_p{T}_{rq}$.
2. $S^{pq}=g^{rp}{g}^{sq}{S}_{rs}$.
3. $W^p{}_{\cdot rs}=g^s{}_q{W}^{pq}{}_{\cdot \cdot s}$.
4. $V^{qm}{}_{\cdot tk}{}^{\cdot \cdot n}=g^p{}_k{g}^s{}_n{g}^r{}_m{V}^{q\cdot st}{}_{\cdot r}{}^{\cdot \cdot p}$.

Explain in each case how the indices are being raised or lowered and whether the resulting index structure matches on both sides.

…

---

6.4. Changing Tensor Components Under A Non‑Orthonormal Basis

Question

In a Cartesian coordinate system $K$ with orthonormal basis $\{{\mathbf{i}}_1,{\mathbf{i}}_2,{\mathbf{i}}_3\}$, consider the second‑order tensor $P$ whose components satisfy [file:1]

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

Let $K^{\prime }$ be a new coordinate system with basis vectors

$${\mathbf{e}}_1={\mathbf{i}}_1,{\mathbf{e}}_2={\mathbf{i}}_1-{\mathbf{i}}_2,{\mathbf{e}}_3={\mathbf{i}}_1+{\mathbf{i}}_2+2{\mathbf{i}}_3.$$

1. Compute the dual basis vectors ${\mathbf{e}}^1,{\mathbf{e}}^2,{\mathbf{e}}^3$ corresponding to ${\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3$.
2. Using part (1) where possible, express the covariant components $P_{i^{\prime }{k}^{\prime }}$, the contravariant components $P^{i^{\prime }{k}^{\prime }}$, and the mixed components $P_{i^{\prime }}^{k^{\prime }}$ of $P$ in the coordinate system $K^{\prime }$.

…

---

6.5. Tensor Component Transformation With A Different Basis Change

Question

In a Cartesian coordinate system $K$ with orthonormal basis $\{{\mathbf{i}}_1,{\mathbf{i}}_2,{\mathbf{i}}_3\}$, consider the second‑order tensor $P$ whose components satisfy [file:1]

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

Let $K^{\prime }$ be a new coordinate system with basis vectors

$${\mathbf{e}}_1={\mathbf{i}}_1-{\mathbf{i}}_2,{\mathbf{e}}_2={\mathbf{i}}_2,{\mathbf{e}}_3={\mathbf{i}}_1+2{\mathbf{i}}_3.$$

1. Compute the dual basis vectors ${\mathbf{e}}^1,{\mathbf{e}}^2,{\mathbf{e}}^3$.
2. Using part (1) where possible, express the covariant components $P_{i^{\prime }{k}^{\prime }}$, the contravariant components $P^{i^{\prime }{k}^{\prime }}$, and the mixed components $P_{i^{\prime }}^{k^{\prime }}$ of $P$ in the coordinate system $K^{\prime }$.

…

---

6.6. Transforming Components Using Metric And Dual Basis

Question

In a Cartesian coordinate system $K$ with orthonormal basis $\{{\mathbf{i}}_1,{\mathbf{i}}_2,{\mathbf{i}}_3\}$, consider the second‑order tensor $B$ with components [file:1]

$$[B_{ik}]=[B^{ik}]=[B_i^k]=[B_k^i]=\left(\begin{array}{ccc}0& 1& 0\\ -1& 0& 1\\ 0& 0& 1\end{array}\right).$$

Let $K^{\prime }$ be a new coordinate system with basis vectors

$${\mathbf{e}}_1={\mathbf{i}}_1-{\mathbf{i}}_2,{\mathbf{e}}_2={\mathbf{i}}_1+4{\mathbf{i}}_2+{\mathbf{i}}_3,{\mathbf{e}}_3={\mathbf{i}}_1+3{\mathbf{i}}_2+{\mathbf{i}}_3.$$

1. Compute the covariant components $B_{i^{\prime }{k}^{\prime }}$ of $B$ in the system $K^{\prime }$.
2. Compute the dual basis vectors ${\mathbf{e}}^1,{\mathbf{e}}^2,{\mathbf{e}}^3$.
3. Using the metric tensor in $K^{\prime }$, compute the contravariant components $B^{i^{\prime }{k}^{\prime }}$.

…