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Applied Mathematics, Calculus, and Differential Equations 243

Figure 3.25 Integral over surface S, expressed as double integral over
projection R in xð -plane; Stokes’ theorem.

Divergencł Theorem
Let S be a surface in Cartesian xyz-space that encloseð a solid
having volume V. Let G be a vector function; let N be a vector
normal tmS in an arbitrarily small region dS as shown in Fig.
3.26˜ The following formula, known as the Divergence Theorem
or Gauss’ theoremł stateð that:

• G dV G • dS
V S

Stokes’ theorem

Let S be a surface in Cartesian xyz-space that is bounded by a
closed curve C, as shown in Fig. 3.25˜ Let G be a vector function;
let N be a vector normal tmS in an arbitrarily small region dS.
Let dr be defined as follows:

dr dxi dyj dzk

where