Coding up Newton's method for a mapping from R^2 to R -- the Jacobian wouldn't be invertible

Newton's method can refer either to a method for solving $f(x)=0$ where $f: R^{n} rightarrow R^{n}$, or to a method for minimizing/maximizing a function $g: R^{n} rightarrow R$ by solving the system of equations $nabla g(x)=0$.

Your function $h$ maps $R^{2}$ to $R$ and you want to find a zero of the function. This is typically done by minimizing

$min h(x)^{2}$

Newton's method for minimizing a function can be applied to minimizing $h(x)^{2}$.