Page 497 - Maxwell House

APPENDIX 477

of a non-orientable surface is the Möbius strip depicted in Figure A18a. It follows from this
theorem that the normal component of the curl is

∮ ∘
(curl ) = ( × ) = lim (A.56)

→0
The first and most fundamental consequence following from Stokes’ theorem that the work can
be done by the vector force () along the closed path if this force has the “rotating”
component, i.e., curl ≠ 0 (see FigureA.18b).
The idea behind Stokes’ theorem is quite evident. The integral in the right side of (A.55) can
be presented as the superposition of multiple elementary curls in some area as roughly shown
in Figure A19. Evidently, the adjacent forces marked red mutually cancel, and the only non-
zero components of the force along the curve L contribute to the integral in the left side of
(A.55).

Figure A19 Stokes’ theorem illustration

Gauss’ or Divergent Theorem

Surface Integral of Vector Fields (i.e., flux) or integral of vector fields F over surfaces A is
defined as

= ∬ ∘ (A.57)

Here the infinitesimal vector element of surface is oriented positively meaning outward.
In Cartesian coordinate system

= + + = [ , , ] � (A.58)
0
0

0
= + + = [, , ]
0
0
0
Therefore,
= ∬ � + + � (A.59)

Gauss’ or Divergent Theorem tells us how much flux can be created by some vector force ()
crossing the closed surface and builds the bridges between the flux value and the divergence of
this field inside the volume bounded by this surface
∯ ∘ = ∫ ∘ (A.60)