Computing Surface Integrals in Mathcad
Last month's article, "Computing Line Integrals in Mathcad," explained how to compute line integrals of vector fields. This month's article explains how to compute surface integrals, which have many applications to electromagnetism, fluid dynamics, and other areas of science.
Surface Integrals of Scalar Functions
To calculate the surface integral of a function,
The hemisphere is shown in the graph below.
Spherical coordinates used in the plots:
You can combine the three coordinate functions, X, Y, and Z, into a single vector-valued function Φ.
Now, suppose g is a scalar-valued function of three variables. The surface integral of g over the hemisphere is defined by the following double integral.
The region of integration is the unit disk D defined by the limits on u and v.
For example, if g is the constant function 1, the integral equals the surface area of the hemisphere.
To evaluate the surface integral, first define a function that computes the cross product symbolically.
Notice that this expression is undefined when u2 + v2 = 1 - that is, on the boundary of the disk D - so the surface integral is an improper integral. Nevertheless, the integral has a finite value.
To evaluate the integral, substitute the cross product function into the integrand and insert the limits of integration.
The result is the area of the hemisphere.
In this example, you can simplify the integrand by rewriting the cross product as follows.
Since Φ(u,v) lies on the unit sphere, its magnitude is 1. So the integrand simplifies to
You can then evaluate the integral symbolically
Surface Integrals of Vector Fields
A vector field is a function that assigns a vector to each point in n-dimensional space. For example, the following function F defines a 3-dimensional vector field:
The surface integral of F over the hemisphere is defined to be
where n is a unit vector that is perpendicular (or normal) to the sphere. The dot product F · n is the component of F normal to the surface.
There are actually two unit normal vectors to a surface, n and -n, which point in opposite directions. Which normal you choose for the integral depends on the specific application.
You can compute a unit normal vector by the following formula.
Since the partial derivatives are tangent vectors to the surface, and their cross product is normal to both of them,
The graph below shows the unit normal n and the vector F(Φ(u,v)), attached at the point corresponding to u = -0.2 and v = 0.8.
The green vectors are the partial derivatives of Φ with respect to u and v. The blue vector is the unit normal n, which points outward from the hemisphere. The red vector is the field vector F(Φ(u,v)).
If you substitute the above formula for n into the surface integral and cancel the norm of the cross product, the integral becomes
To evaluate this expression, substitute the coordinate functions X, Y, and Z for Φ(u,v).
Here's a physical interpretation for the surface integral. Think of the hemisphere as a porous surface through which a fluid is flowing, and F(x,y,z) as the velocity vector of the fluid. Then F
Besides fluid dynamics, surface integrals are also central to the theory of electromagnetism. For example, Gauss' law states that the sum of electric charges inside a closed surface S (multiplied by a suitable constant) is equal to the surface integral over S of the electric field E produced by the charges. Gauss's law can be expressed by the following equation.
The single integral with the subscript S represents a surface integral over S (without specifying a parameterization for the surface). Q is the sum of the charges inside S, and ε0 is a constant.
Look for the continuation of this article in the December issue of PTC Express.