Differentiable functions

Abstract

Let f be a real function of two real variables, x and y say. This means that, to each pair x, y in some region in the plane, there corresponds a real number f(x, y). This number may, but need not, be given by a formula, e.g.

$$ f(x,y) = {x^{2}} - 3xy - {y^{3}}\;or\;f(x,y) = \cos (2x + 3y) $$

. If x, y, z are Cartesian coordinates in three-dimensional space, then the equation z = f(x, y) represents, geometrically, a surface. Some examples are as follows.