In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

In the following we assume that the surfaces are smooth, that is, they are assumed to be images of parameterised surfaces for which:

- is a non-empty, compact and Jordan-measurable subset of ;
- the mapping is one-to-one;
- is continuously differentiable
- the normal vector except possibly at a finite number of points;

(Notice, the condition that is compact can also be replaced by the condition that the surface is compact.)

**Surface area**

In a previous post we discussed parameterised surfaces. Now we calculate the area of parameterised surfaces.

Recall that the area of a parallelogram in spanned by two vectors and is given by the Euclidean norm of the vector obtained by taking the cross product of these two vectors, that is, by

From the parameterisation we obtain two tangent vectors and We can approximate a piece of the surface at some point () by a parallelogram spanned by the vectors and whose area can be approximated by

By summing over all pieces which approximate the whole surface, i.e. forming the sum

and considering the limit when the size of the pieces goes to zero we obtain the integral

(Here, is the normal vector defined here.) We call the surface area of the surface .

Definition

Let be a parameterisation of a surface Then the surface area of is defined by

The last formula can also be written as

If the surface is a graph of a function then and hence the surface of the graph is given by

**Example**

Consider a sphere of radius To calculate its surface area, notice that, because of symmetry, we can calculate the surface area of the upper hemisphere and multiply the result by to obtain the surface area of the whole sphere. The upper hemisphere is given by the equation where We can set and use the parameterisation of surfaces for functions as shown in Section 1. The parameter domain is in this case and the normal vector is

The length of this vector is

Hence the surface area of the hemisphere (which we shall denote by ) is given by

The last integral can be evaluated using polar coordinates by which we obtain

Hence the area of the sphere is given by

**Exercise**

Calculate the surface area of a cone parameterised by and where and

**Scalar surface integrals**

We now integrate scalar fields over surfaces. This is in analogy to scalar line integrals considered in Chapter 3, Section 1.

Definition

Let be a parameterisation of the surface and let be continuous. Then the integral of over is given by

The last formula can also be written as

If the surface is the graph of a function then we also have

**Example**

Let a surface be given by with and let Then set and Then

Hence

**Exercise**

The surface in the previous example is the graph of a function. Use this to parameterise the surface and calculate the scalar surface integral using this approach.

**Surface integrals of scalar valued functions over graphs**

Suppose the surface is the graph of a function defined on a domain Then we can use the parameterisation Then the normal vector is Hence and therefore

where is the unit normal vector to the surface

**Example**

Let a surface be given by and and Let Then a normal vector to the surface is Since the unit normal vector is Hence . We can describe the surface as a graph of the function where the domain is Therefore

**Applications**

Scalar surface integrals can be used to calculate mass, center of mass and moments of inertia of thin shells. Let be the density function of a very thin shell.

- Mass
- Center of mass and
- Moment of inertia and