Chapter 10.10, Problem 16P

Continue Problem 8.15 to find the $g^{ij}$ matrix and the contravariant basis vectors. Check your result by solving the given equations for $u$ and $v$ in terms of $x$ and $y,$ and finding the contravariant basis vectors using Problem 12. On your Problem 8.15 sketches of the lines $u=$ const. and $v=$ const., also sketch the covariant and contravariant basis vectors. Observe that the covariant basis vectors lie along the lines $u=$ const. and $v=$ const. and the contravariant basis vectors lie along the normals to these lines.