Page 387 - Advanced engineering mathematics

LINE INTEGRALS GREEN’S THEOREM
AN EXTENSION OF GREEN’S THEOREM
CHAPTER 12 INDEPENDENCE OF PATH AND
POTENTIAL THEORY SURFACE
Vector Integral

Calculus

The primary objects of vector integral calculus are line and surface integrals and relationships
between them involving the vector differential operators gradient, divergence and curl.

12.1 Line Integrals

For line integrals we need some preliminary observations about curves. Suppose a curve C
has parametric equations
x = x(t), y = y(t), z = z(t) for a ≤ t ≤ b.
These are the coordinate functions of C. It is convenient to think of t as time and C as the
trajectory of an object, which at time t is at C(t) = (x(t), y(t), z(t)). C has an orientation,
since the object starts at the initial point (x(a), y(a), z(a)) at time t = a and ends at the
terminal point (x(b), y(b), z(b)) at time t =b. We often indicate this orientation by putting
arrows along the graph.

We call C:

• continuous if each coordinate function is continuous;
• differentiable if each coordinate function is differentiable;
• closed if the initial and terminal points coincide: (x(a), y(a), z(a)) = (x(b), y(b), z(b));
• simple if a < t 1 < t 2 < b implies that
(x(t 1 ), y)t 1 ), z(t 1 )) = (x(t 2 ), y(t 2 ), z(t 2 ));
and
• smooth if the coordinate functions have continuous derivatives which are never all zero
for the same value of t.

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October 14, 2010 14:53 THM/NEIL Page-367 27410_12_ch12_p367-424