8Some applications of integral theorems
IA Vector Calculus
8.3 Conservation laws
Definition
(Conservation equation)
.
Suppose we are interested in a quantity
Q
. Let
ρ
(
r, t
) be the amount of stuff per unit volume and
j
(
r, t
) be the flow rate
of the quantity (eg if Q is charge, j is the current density).
The conservation equation is
∂ρ
∂t
+ ∇ · j = 0.
This is stronger than the claim that the total amount of
Q
in the universe is
fixed. It says that
Q
cannot just disappear here and appear elsewhere. It must
continuously flow out.
In particular, let
V
be a fixed time-independent volume with boundary
S = ∂V . Then
Q(t) =
Z
V
ρ(r, t) dV
Then the rate of change of amount of Q in V is
dQ
dt
=
Z
V
∂ρ
∂t
dV = −
Z
V
∇ · j dV = −
Z
S
j · ds.
by divergence theorem. So this states that the rate of change of the quantity
Q
in
V
is the flux of the stuff flowing out of the surface. ie
Q
cannot just disappear
but must smoothly flow out.
In particular, if
V
is the whole universe (ie
R
3
), and
j →
0 sufficiently rapidly
as
|r| → ∞
, then we calculate the total amount of
Q
in the universe by taking
V
to be a solid sphere of radius
R
, and take the limit as
R → ∞
. Then the surface
integral → 0, and the equation states that
dQ
dt
= 0,
Example.
If
ρ
(
r, t
) is the charge density (i.e.
ρδV
is the amount of charge in
a small volume
δV
), then
Q
(
t
) is the total charge in
V
.
j
(
r, t
) is the electric
current density. So j · dS is the charge flowing through δS per unit time.
Example.
Let
j
=
ρu
with
u
being the velocity field. Then (
ρu δt
)
·δS
is equal
to the mass of fluid crossing δS in time δt. So
dQ
dt
= −
Z
S
j · dS
does indeed imply the conservation of mass. The conservation equation in this
case is
∂ρ
∂t
+ ∇ · (ρu) = 0
For the case where
ρ
is constant and uniform (i.e. independent of
r
and
t
), we
get that ∇ · u = 0. We say that the fluid is incompressible.