# Homework #2

1. Identify the following tensor notation quantities. The first one is done to demonstrate what's requested.

Tensor NotationNameVector NotationExpanded
A.$$a_i b_i$$Vector Dot Product$${\bf a} \cdot {\bf b}$$$$a_x b_x + a_y b_y + a_z b_z$$
B.$$\epsilon_{rst} a_r b_t$$No need to expand
C.$$A_{rs} B_{ts}$$No need to expand
D.$$A_{ii}$$
E.$$\boldsymbol{\sigma}_{ij,j}$$No need to expand
F.$$f_{,kk}$$

2. Demonstrate that starting with: $$\qquad \epsilon_{ijk} \epsilon_{lmn} = \delta_{il} (\delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km}) + \delta_{im} (\delta_{jn} \delta_{kl} - \delta_{jl} \delta_{kn}) + \delta_{in} (\delta_{jl} \delta_{km} - \delta_{jm} \delta_{kl})$$

and multiplying both sides through by $$\delta_{il}$$ produces: $$\qquad \epsilon_{ijk} \epsilon_{imn} = \delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km}$$

3. Show that $\nabla \times ({\bf u} \times {\bf v}) = ({\bf v} \cdot \nabla){\bf u} - {\bf v} (\nabla \cdot {\bf u}) + {\bf u} (\nabla \cdot {\bf v}) - ({\bf u} \cdot \nabla){\bf v}$

4. We've done the simpler version of this in class. This time, invert Hooke's Law for strain as a function of stress, to get stress as a function of strain, but with thermal expansion thrown into the mix.

The starting point is

$\epsilon_{ij} = {1 \over E} \left[ (1+\nu) \sigma_{ij} - \delta_{ij} \, \nu \, \sigma_{kk} \right] + \delta_{ij} \, \alpha (T - T_{ref})$
where $$\alpha$$ is the thermal expansion coefficient, and $$T$$ and $$T_{ref}$$ are temperatures.