Surface Integral

Thus a change in orientation of S changes the sign of the surface integral.

JUSTIFICATION

We show that this definition corresponds to the intuitive concept of flux, or net rate of fluid flow, across a surface. Suppose S is oriented so that the top surface of S is positive.

Let B(D) be the flux across the part of S over a region D. Consider an element of area ΔD and let ΔS be the area of S over ΔD. Then ΔS is almost a piece of the tangent plane. The component of fluid flow perpendicular to ΔS is given by the scalar product F · N where N is the unit normal vector on the top side of ΔS (Figure 13.5.7). Thus the flux across ΔS is

ΔB ≈ F · N ΔS (compared to ΔA).

Figure 13.5.7

This suggests the surface integral notation

Let us find F, N, and ΔS. The vector F at (x, y, z) is

(1)

F(x,y,z) = Pi + Qj + Rk.

From Section 13.1, one normal vector at (x, y, z) is

The unit normal vector N on the top side of ΔS has positive k component and length one, so

(2)

From our study of surface areas,

(3)

(compared to ΔA).

When we substitute Equations 1-3 into F · N ΔS, the radicals cancel out and we have

(compared to ΔA).

Using the Infinite Sum Theorem we get the surface integral formula