Page 239 - Schaum's Outline of Theory and Problems of Advanced Calculus

230 LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS [CHAP. 10

Fig. 10-1

For this discussion it is assumed that r is continuously differentiable. While (as we are doing) it is
convenient to refer the Euclidean space to a rectangular Cartesian coordinate system, it is not necessary.
(For example, cylindrical and spherical coordinates sometimes are more useful.) In fact, one of the
objectives of the vector language is to free us from any particular frame of reference. Then, a vector
A½xðtÞ; yðtÞ; zðtފ or a scalar, ,is pictured on the domain C, which according to the parametric repre-
sentation, is referred to the real number interval a @ t @ b.
The Integral

ð
A dr ð3Þ
C
of a vector field A defined on a curve segment C is called a line integral. The integrand has the
representation

A 1 dx þ A 2 dy þ A 3 dz
obtained by expanding the dot product.
The scalar and vector integrals
ð n
X
ðtÞ dt ¼ lim ð k ; k ; k Þ t k ð4Þ
C n!1 k¼1
n
ð
X
AðtÞdt ¼ lim Að k ; k ; k Þ tÞ k ð5Þ
C n!1 k¼1
can be interpreted as line integrals; however, they do not play a major role [except for the fact that the
scalar integral (3) takes the form (4)].
The following three basic ways are used to evaluate the line integral (3):

1. The parametric equations are used to express the integrand through the parameter t. Then

ð ð
t 2 dr
dt
A dr ¼ A
C t 1 dt
2. If the curve C is a plane curve (for example, in the xy plane) and has one of the representations
y ¼ f ðxÞ or x ¼ gðyÞ, then the two integrals that arise are evaluated with respect to x or y,
whichever is more convenient.