Previously we have seen parameterised curves. We now generalise this idea to surfaces.
Let be Jordan measurable. By a parameterised surface we mean a continuously differentiable function . The image is called surface.
Notice that we included the condition that is continuously differentiable in the definition above since we only consider such parameterisations. One could of course consider more general surfaces.
As a first example, consider a continuously differentiable function . By setting
we obtain a parameterisation of the graph of .
Not all surfaces can be parameterised as in (1) since not all surfaces can be represented by a function. For instance a sphere cannot be represented by a function since there are points for which one would have to assign two values (for a sphere of radius at this would be and ).
A sphere of radius centered at given by can be parameterised by
Consider the cone with We can parameterise this surface by with and
Hence, in the following we deal with functions of the form
where are continuously differentiable functions. The derivatives of are given by
Notice that the vectors and are tangent to the surface.
The following notation is convenient to find normal vectors to the surface:
Indeed, it can be verified that the normal vector is given by
Let where Then the normal vector is given by
The surfaces which we consider in the following sections are assumed to be piecewise smooth, that is, they are assumed to be images of parameterised surfaces for which:
- is a Jordan-measurable subset of ;
- the mappings are one-to-one;
- the normal vector except possibly at a finite number of points;
A geometric explanation of parameterised surfaces is given in the following video.