Portfolio Part A

MTH3008 Portfolio Part A

Original Documents: Problem Sheet

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1. Vector Expression Simplification and Tensor Derivatives

Question

1. Using suffix notation, simplify the vector expression $(\mathbf{a}\times \mathbf{b})\cdot (\mathbf{c}\times \mathbf{d})\mathbf{a}$ to remove all cross products, given that $\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}\in {\mathbb{R}}^3$.
2. Suppose $T_i$ is a first-rank tensor with non-zero components, expressed in an orthogonal coordinate system.
3. Show that the transformation rule for the derivative $\frac{\partial T_i}{\partial x^k}$ is ${\left(\frac{\partial T_i}{\partial x^k}\right)}^{\prime }=\frac{{\partial }^2{x}_m}{\partial x^{\prime i}\partial x^{\prime k}}{T}_m+L_{im}{L}_{kn}\frac{\partial T_m}{\partial x_n}$, where the transformation matrix is $L_{im}=\frac{\partial x_m}{\partial x^{\prime i}}$.
4. State the transformation rule that $\frac{\partial T_i}{\partial x^k}$ must satisfy to be considered a tensor. Using the result from the previous part, explain why this derivative is not a tensor. (You may use the result from part 2.1 even if you have not shown it.)
5. Prove that the quantity $A_{ik}=\frac{\partial T_i}{\partial x^k}-\frac{\partial T_k}{\partial x^i}$ transforms as a second-rank tensor. (You may use the result stated in part 2.1 even if you have not shown it.)