# Surface integrals

See also the graded exercises and additional exercises associated to this part of the course. If you want more, Sections 6.4 to 6.8 of OpenStax Calculus Volume 3 are a good option.

In this section we consider surfaces and how to define integral of vector fields over these surfaces. This is similar in many ways to line integrals but a higher dimensional version. Curves (for line integrals) are 1D subsets of higher dimensional space whereas surfaces are 2D subsets of higher dimensional space. Identically to line integrals, the first step is to understand a practical way to represent the surfaces, just like with curves we used paths as the parametric representation of the curve. Once we have clarified the parametric representation of surface we can define the surface integral (of a vector field) and show that it satisfies various properties which we would expect, including that the integral is independent of the choice of parametrization. Similar to how we were able to use a line integral (of a scalar) to calculate the length of a curve we can use a surface integral (of a scalar) to calculate the area of a surface.

We then introduce two important operators that act on vector fields, namely *curl* and *divergence*. Using these operators and the surface integral we introduce two theorems, Gauss' Theorem and Stokes' Theorem. These theorems connect line integrals with surface integrals and with volume integrals.

## Representation of a surface

Before developing parametric representations of surfaces let's recall an example of parametric representation of a curve (path). For example, the half circle

In a similar way, now in 2D we can have a parametric representation of a hemisphere.

The hemisphere

Observe that the second form above can be deduced from spherical coordinates (fixed distance from the origin).

The cone

Observe that the second form can be deduced from spherical coordinates (fixed angle from

### Fundamental vector product

A key notion for parametric surfaces and natural geometric object is the *fundamental vector product*. Consider the parametric surface, denoted

The vector-valued function defined as

is called the *fundamental vector product* of the representation

By definition, the vector-valued functions

As always we need to take some care about smoothness of the objects we work with. If *regular point* for that representation.

A surface

Just like we saw with paths to represent curves, there are many different ways we can find the parametric representation of a given surface. If the surface

The region

and consequently

An example of such a representation is as follows for the hemisphere. Let

The surface

In this case, all points are regular except the equator

Let

The surface

and so the fundamental vector product of this representation is

In this case many points map to the north pole

## Surface integral of scalar field

Mirroring the process for line integrals we will define surface integrals both for scalar fields and for vector fields. The surface integral of a scalar field is closely related to the area of a parametric surface, just like the length of a curve is closely related to the line integral of a scalar field.

Definition

The area of the parametric surface

Observe that the definition is in terms of a multiple integral over the region

Later we will show that *independent* of the choice of representation as we require for such a definition, it would be unreasonable if the area of a surface depended on the choice of representation.

We will check that this definition corresponds to a fact that we already know by computing the surface area of a hemisphere. Let, as before,

Taking the definition of area and evaluating the multiple integral, this means that

The surface integral of a scalar field is defined in a way similar to the area of a surface.

Definition (scalar surface integral)

Let

whenever the double integral on the right exists.

Observe that, if we choose

From the point of view of applications, we could take

## Change of surface parametrization

In order to validate the definition of a surface integral and consequently that of the area of a surface, we will now show that the the value of the evaluated integral doesn't depend on the choice of representation for any given surface.

Suppose that

Since

Observe that

so the definition does make sense.

## Surface integral of a vector field

In preparation for defining the surface integral of a vector field we need the notion of the *normal* vector of a surface. This is a natural geometric notion, for each point in the surface it is the unit vector field which is orthogonal to the surface.

Definition (unit normal)

Let

This definition makes *pair* of normals at any regular point, but which one is

If

Definition (vector surface integral)

Let

is said to be the *surface integral of with respect to the normal *.

For convenience let

and so for evaluating the surface integral of a vector field there is typically no need to evaluate the norm of the fundamental vector product. Also note that

## Curl and divergence

Suppose that

Definition (curl)

The *curl* of

Definition (divergence)

The *divergence* of

Often the notation

- If
then , , .

The quantity defined as

Some examples:

If