Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR AND TENSOR ANALYSIS
while the mass of ¯uid leaving V per unit time is
Z
Z
v ^nds ˆ r…v†dV;
S
where Gauss' theorem is used in changing the surface integral to volume integral.
Since there is neither a source nor a sink, mass conservation requires an exact
balance between these e€ects:
Z
V
@
@t
ZV
dV ˆ r…v†dV;
V
or
Z
V
@
@t ‡r…v† dV ˆ 0:
Also since V is arbitrary, mass conservation requires that the continuity equation
@
‡r…v† ˆ@
@t @t rj ˆ 0
must be satis®ed everywhere in the region.
Stokes' theorem
This theorem relates the line integral of a vector function and the surface integral
of the curl of that vector. It was ®rst discovered by Lord Kelvin in 1850 and
rediscovered by George Gabriel Stokes four years later.
If a continuous, di€erentiable vector ®eld A is de®ned a three-dimensional
region V, and S is a regular open surface embedded in V bounded by a simple
closed curve , the theorem states that
Z
I
rA da ˆ A dl;
…1:80†
S
where the line integral is to be taken completely around the curve and dl is an
element of line (Fig. 1.21).
Figure 1.21.
Relation between da and dl in de®ning curl.
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