Math 21a Supplement on Integration and ElectroMagnetism
The Math 21a Supplement on Electricity and Magnetism presented the equations that
describe all known electric and magnetic phenomena.
In particular, recall that Maxwell’s equations
describe electric fields by a vector valued function of space and time,
E
(t, x), while magnetic fields
are described by a similar object,
B
(t, x).
Then, Maxwell postulates that only such pairs (
E
,
B
)
which obey the constraints
•
div
E
=
ρ
,
•
div
B
= 0,
•
∂
∂
t
E
= curl
B

j
,
•
∂
∂
t
B
=  curl
E
,
(1)
appear in nature.
Here, the function
ρ
and the vector valued function
j
(both functions of time and
space) are determined by the distribution and velocities of the various charged materials or particles
present.
The pair (
ρ
,
j
) must also satisfy a constraint, which is
∂
∂
t
ρ
+ div
j
= 0 .
(2)
The purpose of this supplement is illustrate various important ramifications of (1) and (2)
which follow from applications of either Stokes’ Theorem or the Divergence Theorem.
a)
A short review
Before getting to the purpose at hand, I will remind you of two definitions.
For the first,
suppose that
v
is a vector valued function on
R
3
.
(In the examples below the vector valued
functions under consideration will be
E
,
B
and
j
, thus they may also depend on time.)
Now,
suppose that S is a surface in
R
3
with a chosen normal direction,
n
.
For example, S could be the
whole surface of a sphere and
n
the outward pointing normal, or S could be something much more
intricate.
With regard to the normal vector
n
, keep in mind that most surfaces have two possible
normal directions which point opposite.
The first definition to review is that of the
flux
of
v
through S in the direction of the normal
vector
n
.
In particular, remember that this flux is defined to be the value of the surface integral
S
∫
v
•
n
dA .
(3)
Why call (3) ‘flux’?
Here is one reason:
The value of (3) is evidently proportional to the average
amount that
v
points outward from S along the normal
n
.
Thus, if
v
is the velocity vector of a
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moving fluid at each point, then the positivity of (3) indicates that there is a net flow of fluid
through S in the direction
n
.
On the other hand , if (3) is negative, then the net flow of fluid
through S is in the direction of 
n
.
So, when
v
is the velocity of a fluid, the integral in (3) acts just
like flux should act.
When
v
represents some other physical quantity, the corresponding integral
in (3) is still called a flux integral.
The second definition in this review requires the specification of a closed path
γ
in
R
3
.
That
is,
γ
is some (possibly very twisty) path whose endpoint coincides with its starting point.
Having
chosen a direction around
γ
, we introduced the path integral
γ
∫
v
•d
x
.
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 Fall '11
 GregoryJ.Galloway
 Equations, Multivariable Calculus, Flux, right hand, qV

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