Chapter 10.9, Problem 1P

Prove (9.4) in the following way. Using (9.2) with $u=x_1,$ show that $\nabla x_1=e_1/h_1$.

Similarly, show that $\nabla x_2=e_2/h_2$ and $\nabla x_3=e_3/h_3.$ Let $e_1,e_2,e_3$ in that order form a right-handed triad (so that and show that $\nabla x_1\times \nabla x_2=$ $e_3/\left(h_1{h}_2\right).$ Take the divergence of this equation and, using the vector identities (h) and (b) in the table at the end of Chapter 6, show that $\nabla \cdot \left(e_3/h_1{h}_2\right)=0$.

The other parts of (9.4) are proved similarly.