Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR INTEGRATION AND INTEGRAL THEOREMS
Green's theorem
Green's theorem is an important corollary of the divergence theorem, and it
has many applications in many branches of physics. Recall that the divergence
theorem Eq. (1.78) states that
Z
I
rAdV ˆ A da:
V
Let A ˆ B, where is a scalar function and B a vector function, then rA
becomes
rA ˆr… B† ˆ rB ‡ B r :
Substituting these into the divergence theorem, we have
I
Z
B da ˆ … rB ‡ B r †dV:
S
V
S
…1:84†
If B represents an irrotational vector ®eld, we can express it as a gradient of a
scalar function, say, ':
Then Eq. (1.84) becomes
I
Z
B da ˆ
Now
S
V
B r':
‰ r…r'†‡…r'†…r †ŠdV: …1:85†
B da ˆ…r'†^nda:
The quantity …r'†^n represents the rate of change of in the direction of the
outward normal; it is called the normal derivative and is written as
…r'†^n @'=@n:
Substituting this and the identity r…r'† ˆr 2 ' into Eq. (1.85), we have
I
@'
S @n
ZV
da ˆ ‰ r 2 ' ‡r' r ŠdV: …1:86†
Eq. (1.86) is known as Green's theorem in the ®rst form.
Now let us interchange ' and , then Eq. (1.86) becomes
I
' @
S @n
ZV
da ˆ ‰'r 2 ‡r' r ŠdV:
Subtracting this from Eq. (1.85):
I
@'
@n ' @
@n
S
Z
da ˆ
V
43
r 2 ' 'r 2
dV: …1:87†