Chapter 10.3, Problem 2P

Show that the fourth expression in (3.1) is equal to $\partial u/\partial x_i^{\prime }.$ By equations (2.6) and (2.10), show that $\partial x_j/\partial x_i^{\prime }=a_{ij},$ so $\frac{\partial u}{\partial x_i^{\prime }}=\frac{\partial u}{\partial x_j}\frac{\partial x_j}{\partial x_i^{\prime }}=a_{ij}\frac{\partial u}{\partial x_j}$.

Compare this with equation (2.12) to show that $\nabla u$ is a Cartesian vector.Hint: Watch the summation indices carefully and if it helps, put back the summation signs or write sums out in detail as in (3.1) until you get used to summation convention.