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.
Observe that the second form above can be deduced from spherical coordinates (fixed distance from the origin).
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
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
An example of such a representation is as follows for the hemisphere. Let
In this case, all points are regular except the equator
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.
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
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)
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.
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)
This definition makes
Definition (vector surface integral)
is said to be the surface integral of
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
The curl of
The divergence of
Often the notation
then , , .
The quantity defined as