Volume Integrals

Figure 2.9 Sketch for surface integrals. The

three-dimensional surface area S is bounded by the closed curve C.

Figure 2.10 Volume V enclosed by the closed surface S.

Let A be a vector field in space. The volume integral over the volume V of the quantity A is written as

A dV = volume integral of a vector A over the volume V (the result is a vector)

2.2.11 Relations Between Line, Surface, and Volume Integrals

Consider again the open area S bounded by the closed curve C, as shown in Figure 2.9. Let A be a vector field. The line integral of A over C is related to the surface integral of A over S by Stokes ’ theorem:

[2.25]

Consider again the volume V enclosed by the closed surface S, as shown in Figure 2.10. The surface and volume integrals of the vector field A are related through the divergence theorem:

s

(V. A) dV

[2.26]

v

If p represents a scalar field, a vector relationship analogous to Equation (2.26) is given by the gradient theorem:

[2.27]

VpdV