Begin2.DVI

The integral relation (8.51) follows from the divergence theorem. In the diver-

gence theorem, substitute F
=H
×C ,
where C
is an arbitrary constant vector. By

using the vector relations

∇·(H
 ×C
) = C
·(∇× H
)−H
 ·(∇× C
)

div F
=∇·F
=∇·(H
×C
) = C
·(∇× H
) and

F
·ˆen= (H
×C
)·ˆen=H
·(C
׈en) = C
·(ˆen×H
) (triple scalar product),

(8 .54)

the divergence theorem can be written as

∫∫∫

V

div (H
×C
)dV =

∫∫∫

V

C
·(∇× H
)dV =∫∫

S

(H
 ×C
)·ˆendS =

∫∫

S

C
·(ˆen×H
)dS (8 .55)

Since C
is a constant vector one may write

C
·

∫∫

V

∇× H dV
 =C
·

∫∫

S

ˆen×H dS.
 (8 .56)

For arbitrary C
this relation implies

∫∫∫

V

∇× H dV
 =

∫∫

S

ˆen×H dS.
 (8 .57)

In this integral replace H
by F
(H
is arbitrary) to obtain the relation (8.51).

The integral (8.52) also is a special case of the divergence theorem. If in the

divergence theorem one makes the substitution F
=φC ,
where φis a scalar function

of position and C
is an arbitrary constant vector, there results

∫∫

V

div F dV
=

∫∫∫

V

∇(φC
)dV =

∫∫∫

V

C
·∇ φ dV =

∫∫

S

C φ d
 
S. (8 .58)

where the vector identity ∇(φC
) = (∇φ)·C
+φ(∇× C
)has been employed. The relation

given by equation (8.58), for an arbitrary constant vector C
, produces the integral

relation (8.52).

The integral (8.53) is a special case of Stokes theorem. If in Stokes theorem one

substitutes F
=φC ,
where C
is a constant vector, there results

∫∫

S

( curl F
)·dS
=

∫∫

S

∇× (φC
)·ˆendS =

∫∫

S

(∇φ×C
)·ˆendS

=

∫∫

S

(ˆen×∇ φ)·C dS
 =

∫

C

©C φ d
r.

(8 .59)