Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR AND TENSOR ANALYSIS
name is associated with this theorem because of his extensive work on general
problems of double and triple integrals.
If a continuous, di€erentiable vector ®eld A is de®ned in a simply connected
region of volume V bounded by a closed surface S, then the theorem states that
Z
I
rAdV ˆ A da;
…1:78†
V
where dV ˆ dx 1 dx 2 dx 3 . A simple connected region V has the property that every
simple closed curve within it can be continuously shrunk to a point without
leaving the region. To prove this, we ®rst write
Z
V
Z
rAdV ˆ
S
X 3
V iˆ1
@A i
@x i
dV;
then integrate the right hand side with respect to x 1 while keeping x 2 x 3 constant,
thus summing up the contribution from a rod of cross section dx 2 dx 3 (Fig. 1.20).
The rod intersects the surface S at the points P and Q and thus de®nes two
elements of area da P and da Q :
Z
V
@A 1
@x 1
dV ˆ
I
S
dx 2 dx 3
Z Q
P
I
@A 1
dx
@x 1 ˆ
1 S
Z Q
dx 2 dx 3 dA 1 ;
where we have used the relation dA 1 ˆ…@A 1 =@x 1 †dx 1 along the rod. The last
integration on the right hand side can be performed at once and we have
Z
I
@A 1
dV ˆ ‰A
@x 1 …Q†A 1 …P†Šdx 2 dx 3 ;
1
V
S
where A 1 …Q† denotes the value of A 1 evaluated at the coordinates of the point Q,
and similarly for A 1 …P†.
The component of the surface element da which lies in the x 1 -direction is
da 1 ˆ dx 2 dx 3 at the point Q, and da 1 ˆdx 2 dx 3 at the point P. The minus sign
P
Figure 1.20. A square tube of cross section dx 2 dx 3 .
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