Chapter 10.10, Problem 12P

Show that in a general coordinate system with variables $x_1,x_2,x_3,$ the contravariant basis vectors are given by $a^i=\nabla x_i=\text{i}\frac{\partial x_i}{\partial x}+\text{j}\frac{\partial x_i}{\partial y}+\text{k}\frac{\partial x_i}{\partial z}$.

Hint: Write the gradient in terms of its covariant components and the $a^i$ basis vectors to get $\nabla u=a^j\partial u/\partial x_j$ and let $u=x_i$.