Page 238 - Schaum's Outline of Theory and Problems of Advanced Calculus

Line Integrals, Surface

Integrals, and Integral

Theorems

Construction of mathematical models of physical phenomena requires functional domains of greater
complexity than the previously employed line segments and plane regions. This section makes progress
in meeting that need by enriching integral theory with the introduction of segments of curves and
portions of surfaces as domains. Thus, single integrals as functions defined on curve segments take
on new meaning and are then called line integrals. Stokes’s theorem exhibits a striking relation between
the line integral of a function on a closed curve and the double integral of the surface portion that is
enclosed. The divergence theorem relates the triple integral of a function on a three-dimensional region
of space to its double integral on the bounding surface. The elegant language of vectors best describes
these concepts; therefore, it would be useful to reread the introduction to Chapter 7, where the impor-
tance of vectors is emphasized. (The integral theorems also are expressed in coordinate form.)

LINE INTEGRALS
The objective of this section is to geometrically view the domain of a vector or scalar function as a
segment of a curve. Since the curve is defined on an interval of real numbers, it is possible to refer the
function to this primitive domain, but to do so would suppress much geometric insight.
A curve, C,in three-dimensional space may be represented by parametric equations:
x ¼ f 1 ðtÞ; y ¼ f 2 ðtÞ; z ¼ f 3 ðtÞ; a @ t @ b ð1Þ

or in vector notation:
x ¼ rðtÞ ð2Þ
where
rðtÞ¼ xi þ yj þ zk
(see Fig. 10-1).
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