Page 431 - Advanced engineering mathematics

12.9 Stokes’s Theorem 411

For the converse, it is enough to show that, if F has curl zero, then F · d R is independent of
C
path, since then we can define a potential function by choosing P 0 and setting
(x,y,z)
ϕ(x, y, z) = F · dR.
P 0
To show this independence of path, let C and K be paths in D from P 0 to P 1 . Form a closed path

L =C (−K). Since D is simply connected, there is a piecewise smooth surface in D having
C as boundary. By Stokes’s theorem,

F · dR = F · dR − F · dR
L C K

(∇× F) · ndσ = 0.
=

12.9.2 Maxwell’s Equations
The theorems of Gauss and Stokes are used in the analysis of vector fields. We will illustrate this
with electric and magnetic fields and Maxwell’s equations. To begin, we will use the following
standard notation and relationships:

E= electric intensity = permitivity of the medium
J= current density σ = conductivity
μ= permeability D = E = electric flux density
Q = charge density B = μH = magnetic flux density
H= magnetic intensity

q = QdV = total charge in a region V
V

ϕ = B · ndσ = magnetic flux across

i = J · ndσ = flux of current across .

In these, flux is computed using an outer unit normal to the closed surface .
We also have the following relationships, which have been observed and verified experimen-
tally.

∂ϕ
Faraday’s law E · dR =− .
C ∂t
Here C is any piecewise smooth closed curve in the medium. We may think of this as saying that
the rate of change of the magnetic flux across is the negative of the measure of the tangential
component of the electric intensity around any closed curve bounding .

Ampère’s law H · dR = i.

This says that the measure of the tangential component of the magnetic intensity about C is the
current flowing through any surface bounded by C.

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October 14, 2010 14:53 THM/NEIL Page-411 27410_12_ch12_p367-424