Surface integrals are the natural generalization of the multiple integral into the sum of some function over any arbitrary surface instead of just over a region on a plane.
Surface Integral of a Scalar Function
Formally the surface integral of a scalar function over the surface is defined as
Where is some parameterization of over the region in the -plane, and and are the partial derivatives of with respect to and respectively.
Derivation
We can use similar intuition to line integrals to derive the definition of surface integrals. Just as line integrals generalize Riemann sums from intervals to curves, surface integrals generalize double integrals from planar regions to curved surfaces.
For a double integral over a region in the plane, we have:
Where represents small rectangular patches in the plane. For a surface integral, instead of summing over planar patches, we sum over small patches on the curved surface . We represent these surface patches as :
Where is a point on each surface patch. To evaluate this surface integral, we need to express in terms of a parameterization of the surface. Let be a parameterization of over a region in the -plane.
Consider a small rectangular patch in the -plane with dimensions by . This patch maps to a small curved patch on the surface . The edges of this rectangular patch in the parameter space map to curves on the surface. The displacement along the -direction (holding constant) is approximately:
Similarly, the displacement along the -direction (holding constant) is approximately:
These two displacement vectors span a parallelogram on the surface, and the area of this parallelogram approximates the surface area of the curved patch. The area of a parallelogram spanned by two vectors is the magnitude of their cross product:
Where is the area of the rectangular patch in the parameter space. This gives us the relationship between the surface area element and the parameter space area element:
Substituting this back into our definition of the surface integral gives us:
Surface Integral of a Vector Field
The surface integral of a vector field over a surface is defined as
Where is some parameterization of over the region in the -plane. Note that is a vector representing both the magnitude and orientation of the surface element.
Derivation
To derive the surface integral of a vector field, we use a similar approach to the surface integral of a scalar function. We start by approximating the surface integral using a sum over surface patches:
Where is a small vectorial surface element. Unlike the scalar case where we summed over scalar surface areas , here we sum over vector surface elements that capture both the area and the normal direction of each surface patch.
From our earlier derivation, we found that a small patch in the parameter space with dimensions by maps to a parallelogram on the surface spanned by and .
The vector area of this parallelogram is given by the cross product of these displacement vectors:
This gives us the vector surface element in differential form:
Note that is a vector perpendicular to the surface (a normal vector) with magnitude equal to the area scaling factor from the scalar case.
Substituting this back into our definition of the surface integral gives us:
Examples
Example 1
Find the flux across the piece of the cone , oriented with upward pointing unit normal.
Example 2
The temperature in a metal ball is proportional to the square distance of the distance from the centre of the ball. Find the rate of heat flow across a sphere of radius with centred at the centre of the ball.