Introduction to vector and tensor analysis

point in opposite direction). Consequently, the only contributions comes from
the surface S of V .
a(r) · dS ≈
a(r) · dSi
(2.59)
S
i
B(r0,i)
Since each term on the rhs. of Eq. (2.58) equals a corresponding term on the
rhs. of Eq. (2.59) the Gauss teorem is demonstrated.
Gauss theorem is often used in conjunction with following mathematical
theorem
d
dt
V
φ(r, t)dV =
V
∂
φ(r, t)dV,
∂t
where t is time and φ is a time dependent scalar field (The theorem works in
arbitrary spatial dimension). The two theorems are central in deriving partial
differential equations for dynamical systems, in particular the so called continuity
equations, linking a flow field to the time changes of scalar field advected
by the flow. For instance if ρ(r, t) is a density and v(r, t), is the velocity field
of a fluid then the vector j(r) = ρ(r)v(r) gives the density current. It can then
by shown (try it) that under mass conservation then
∂ρ
+ ∇ · j = 0.
∂t
The divergence of a vector field therefore has the physical meaning of giving the
net “outflux” of a scalar advected by the field within an infinitesimal volume.
Stokes theorem
Stokes theorem states that if S is the “curl analogue” of the divergence theorem
and relates the integral of the curl of a vector field over an open surface S to the
line integral of the vector field around the perimeter C bounding the surface.
∇ × a (r) · dS = a(r) · dr (2.60)
S
Following the same lines as for the derivation of the divergence theorem the
surface S can be divided into many small areas Si with boundaries Ci and unit
normals ni. For each small area one can show that
(∇ × a) · niSi ≈ a · dr.
Summing over i one finds that on the rhs. all parts of all interior boundaries
that are not part of C are included twice, being traversed in opposite directions
on each occasion and thus cancelling each other. Only contributions from line
elements that are also part of C survive. If each Si is allows to to tend to zero,
Stokes theorem, Eq. (2.60), is obtained.
25
C
Ci