  Monge gauge in differential geometry is generalized. The original Monge gauge
is based on a surface defined as a height function $h(x,y)$ above a flat
reference plane. The total curvature and the Gaussian curvature are found in
terms of the height function. Getting benefits from our mathematical knowledge
of general relativity, we shall extend the Monge gauge toward more complicated
surfaces. Here in this study we consider the height function above a curved
surface namely a sphere of radius $R$. The proposed height function is a
function of $\theta $ and $\varphi $ on a closed interval. We find the first,
second fundamental forms and the total and Gaussian curvatures in terms of the
new height function. Some specific limits are discussed and two illustrative
examples are given.
