# Implicit Mapping Theorem for a Surface

1. Apr 7, 2013

### Parmenides

The question, from Edward's Advanced Calculus (which is becoming a rather frustrating book), asks if the following surface can be represented as a function of $z(x,y)$ near the point $(0,2,1)$:

$$xy - ylog(z) + sin(xz) = 0$$
Naturally, this should invite me to use the Implicit Mapping Theorem, but I've seen this theorem presented mostly for systems of equations, rather than a single case. But anyhow, let the surface be called $G({\bf{x}},{\bf{y}})$, where ${\bf{x}} = (x,y)$ and ${\bf{y}} = z$. Then,
$$\frac{\partial{G}}{\partial{z}} = xcos(xz) - \frac{y}{z}$$
So at $(0,2,1)$, $G'((0,2),(1)) = -2$. Thus, since the invertible "matrix" is non-singular, there exists a neighborhood near $(0,2,1)$ for which the surface can be represented as a function of $z$.

I suspect this is correct, but being confused by some notation, I just wanted to make sure. Thanks.