**Definition of surface**: A surface is defined as the locus of a point whose Cartesian coordinates (x, y, z) are functions of two independent parameters u and v.

Thus we can write

x = *f *(u, v) y = g(u, v) z = h(u, v)

or, x =x(u, v) y = y(u, v) z = z(u, v)

**Parametric equation of a surface (vector form):** *A surface is defined as the locus of a point whose position* vector x can be expressed as functions of two independent parameters. Thus an equation of the form

__x__ =__ x __(u, v) ———————————- (i)

represents a surface.

The above equation is known as the Gaussian form of the surface. The parameters u and v are called curvilinear coordinates or surface coordinates of the current point on the surface.

**Monge’s form of the surface:** If the equation of a surface can be written in the form __z__ = *f *(x, y), then it is called Monge’s form of the equation of the surface. In this form x and y themselves can be regarded as independent parameters.

i.e., __x__ = __x__,__ y __=__ y__, __z __= *f* (x, y)

In vector form __x__ = (__x__, __y__, __z__) = (__x__, __y__,* f* (x, y)) is the equation of a surface with parameters x, y.