the meaning of flux

What do we mean by Flux? In the following we look at its meaning. Recall that we said that

Let's consider an example to try and give this a slightly more precise physical meaning. Suppose that we have a vector field, F = -y i + x j, which is the velocity field of a fluid with density 1. This means that F has units of velocity, say m/s, and the density is in mass/volume, say kg/m³. Now, if we ask for "the amount of fluid passing through the surface," we could mean "the mass passing through per time unit," that is, mass/time, say, kg/s.

figure 1: a surface and the flux through it.

So, how do we get this? We get mass from density times volume. The volume will be the velocity times the surface area of the surface, which gives the volume of fluid coming through the surface per second. To see this, suppose that the velocity field were constant: say, F = 2 j, a constant flow in the j direction. Then think about a 1x1 square on the xz-plane: in 1 second the fluid particles that are on the 1x1 square will have moved 2m to the right, forming the outer edge of the volume of fluid that has come through the surface. This is shown in the figure to the right. The volume that has passed through the surface is therefore 2x1x1 = 2 m³.

So, the volume of fluid passing through the surface is (velocity perpendicular to the surface)(surface area). Thus the flux through a little patch of surface area, dS, is . Here, N is a unit normal, which we've seen we can find in a couple of different ways. If the surface is given parametrically as r(u,v), then . Therefore the differential mass/second passing through the small patch of surface given by dS is .

Of course, for the case in figure 1 it's even easier than that: the normal vector is N = j, and because dS is just an area in the xz-plane, dS = dx dz.

This gives the surface integral of the vector field over the surface. We define so that we can write the flux as the first expression in the definition in the first equation on the page. This is directly analagous to what we did with line integrals, saying that

Different Ways of Representing Surfaces
How we approach finding N largely depends on how the surface S is defined. We have two common ways this is done:

1. As a function: z = f(x, y). In this case we can find a normal by re-writing the surface as the zero level surface of the function G(x, y, z) = z - f(x, y) and remembering that the gradient of a level curve or surface is a vector perpendicular to the curve or surface: thus N = <-f_x, -f_y, 1> / sqrt(1 + f_x² + f_y²), and we've derived the expression for dS = sqrt(1 + f_x² + f_y²) dx dy, so dS = <-f_x, -f_y, 1> dx dy.
2. Parametrically: r(u, v) = x(u, v) i + y(u, v) j + z(u, v) k. In this case, we use our parametric derivation of N and dS, and find dS = (r_u x r_v) du dv.