Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR INTEGRATION AND INTEGRAL THEOREMS
arises since the x 1 component of da at P is in the direction of negative x 1 .Wecan
now rewrite the above integral as
Z
Z
Z
@A 1
dV ˆ A
V @x 1 …Q†da 1 ‡ A 1 …P†da 1 ;
1 S Q S P
where S Q denotes that portion of the surface for which the x 1 component of the
outward normal to the surface element da 1 is in the positive x 1 -direction, and S P
denotes that portion of the surface for which da 1 is in the negative direction. The
two surface integrals then combine to yield the surface integral over the entire
surface S (if the surface is suciently concave, there may be several such as right
hand and left hand portions of the surfaces):
Z
V
@A 1
@x 1
dV ˆ
I
S
A 1 da 1 :
Similarly we can evaluate the x 2 and x 3 components. Summing all these together,
we have Gauss' theorem:
Z
X
I
@A i
X
Z
I
dV ˆ A
@x i da i or rAdV ˆ A da:
i
V
i
S
i
We have proved Gauss' theorem for a simply connected region (a volume
bounded by a single surface), but we can extend the proof to a multiply connected
region (a region bounded by several surfaces, such as a hollow ball). For interested
readers, we recommend the book Electromagnetic Fields, Roald K.
Wangsness, John Wiley, New York, 1986.
V
S
Continuity equation
Consider a ¯uid of density …r† which moves with velocity v(r) in a certain region.
If there are no sources or sinks, the following continuity equation must be satis-
®ed:
where j is the current
@…r†=@t ‡rj…r† ˆ0;
j…r† ˆ…r†v…r†
…1:79†
…1:79a†
and Eq. (1.79) is called the continuity equation for a conserved current.
To derive this important equation, let us consider an arbitrary surface S enclosing
a volume V of the ¯uid. At any time the mass of ¯uid within V is M ˆ
RV
dV
and the time rate of mass increase (due to mass ¯owing into V )is
@M
@t ˆ @ Z Z
@
dV ˆ
@t
@t dV;
V
39
V