Page 242 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 10] LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS 233

ð
Theorem 2. Anecessary and sufficient condition that the line integral, A dr be independent of path
is that r A ¼ 0. C
þ
Theorem 3. If r A ¼ 0, then the line integral of A over an allowable closed path is 0, i.e., A dr ¼ 0.
If C is a plane curve, then Theorem 3 follows immediately from Green’s theorem, since in the plane
case r A reduces to

@A 1 @A 2
@y ¼ @x

EXAMPLE. Newton’s second law for forces is F ¼ dðmvÞ , where m is the mass of an object and v is its velocity.
dt
When F has the representation F ¼ r ,itissaid to be conservative. The previous theorems tell us that the
integrals of conservative fields of force are independent of path. Furthermore, showing that r F ¼ 0 is the
preferred way of showing that F is conservative, since it involves differentiation, while demonstrating that exists
such that F ¼ r requires integration.

SURFACE INTEGRALS
Our previous double integrals have been related to a very special surface, the plane. Now we
consider other surfaces, yet, the approach is quite similar. Surfaces can be viewed intrinsically, i.e., as
non-Euclidean spaces; however, we do not do that. Rather, the surface is thought of as embedded in a
three-dimensional Euclidean space and expressed through a two-parameter vector representation:

x ¼ rðv 1 ; v 2 Þ
While the purpose of the vector representation is to be general (that is, interpretable through any
allowable three-space coordinate system), it is convenient to initially think in terms of rectangular
Cartesian coordinates; therefore, assume
r ¼ xi þ yj þ zk
and that there is a parametric representation

x ¼ rðv 1 ; v 2 Þ; y ¼ rðv 1 ; v 2 Þ; z ¼ rðv 1 ; v 2 Þ ð11Þ
The functions are assumed to be continuously differentiable.
The parameter curves v 2 ¼ const and v 1 ¼ const establish a coordinate system on the surface (just as
y ¼ const, and x ¼ const form such a system in the plane). The key to establishing the surface integral
of a function is the differential element of surface area. (For the plane that element is dA ¼ dx; dy.)
At any point, P,of the surface
@r @r
dv 2
dx ¼ dv 1 þ
@v 1 @v 2
spans the tangent plane to the surface. In particular, the directions of the coordinate curves v 2 ¼ const
@r @r
dv 2 , respectively (see Fig. 10-3).
and v 1 ¼ const are designated by dx 1 ¼ dv 1 and dx 2 ¼
@v 1 @v 2
The cross product
@r @r
dv 1 dv 2
dx 1 xdx 2 ¼
@v 1 @v 2

@r
@r
is normal to the tangent plane at P,and its magnitude is the area of a differential coordinate
@v 1 @v 2
parallelogram.