Section 12.1 Flux
At any give point along a curve, there is a natural vector, namely the (unit) tangent vector \(\TT\text{.}\) Therefore, it is natural to add up the tangential component of a given vector field along a curve. When the vector field represents force, this integral represents the work done by the force along the curve. But there is no natural tangential direction at a point on a surface, or rather there are too many of them. The natural vector at a point on a surface is the (unit) normal vector \(\nn\text{,}\) so on a surface it is natural to add up the normal component of a given vector field; this integral is known as the flux of the vector field through the surface.
We already know that the vector surface element is given by
Since \(d\rr_1\) and \(d\rr_2\) are both tangent to the surface, \(d\AA\) is perpendicular to the surface, and is therefore often written
Putting this all together, the flux of a vector field \(\FF\) through the surface is given by
We first consider a problem typical of those in calculus textbooks, namely finding the flux of the vector field \(\FF=z\,\zhat\) up through the part of the plane \(x+y+z=1\) lying in the first octant, as shown in Figure 12.1.1. We begin with the infinitesimal vector displacement in rectangular coordinates in 3 dimensions, namely
A natural choice of curves in this surface is given by setting \(y\) or \(x\) constant, so that \(dy=0\) or \(dx=0\text{,}\) respectively. We thus obtain
where we have used what we know (the equation of the plane) to determine each expression in terms of a single parameter. The surface element is thus
and the flux becomes 1
The limits were chosen by visualizing the projection of the surface into the \(xy\)-plane, which is a triangle bounded by the \(x\)-axis, the \(y\)-axis, and the line whose equation is \(x+y=1\text{.}\) Note that this latter equation is obtained from the equation of the surface by using what we know, namely that \(z=0\text{.}\)
Just as for line integrals, there is a rule of thumb which tells you when to stop using what you know to compute surface integrals: Don't start integrating until the integral is expressed in terms of two parameters, and the limits in terms of those parameters have been determined. Surfaces are two-dimensional!