Page 224 - Engineering Electromagnetics, 8th Edition

206 ENGINEERING ELECTROMAGNETICS

Because this is true for any volume, it is true for the differential volume dν,
Tdν = 0
and therefore

T = 0
or

∇ · ∇× A ≡ 0 (31)

6
Equation (31) is a useful identity of vector calculus. Of course, it may also be
proven easily by direct expansion in rectangular coordinates.
Let us apply the identity to the non-time-varying magnetic field for which
∇× H = J
This shows quickly that

∇ · J = 0
which is the same result we obtained earlier in the chapter by using the continuity
equation.
Before introducing several new magnetic field quantities in the following section,
we may review our accomplishments at this point. We initially accepted the Biot-
Savart law as an experimental result,

IdL × a R
H =
4πR 2
and tentatively accepted Amp`ere’s circuital law, subject to later proof,

H · dL = I

From Amp`ere’s circuital law the definition of curl led to the point form of this same
law,
∇× H = J

We now see that Stokes’ theorem enables us to obtain the integral form of Amp`ere’s
circuital law from the point form.

D7.6. Evaluate both sides of Stokes’ theorem for the field H = 6xya x −
2
3y a y A/m and the rectangular path around the region, 2 ≤ x ≤ 5, −1 ≤ y ≤
1, z = 0. Let the positive direction of dS be a z .
Ans. −126 A; −126 A

6 This and other vector identities are tabulated in Appendix A.3.