Begin2.DVI

Chapter7

Vector Calculus I

One aspect of vector calculus can be described as taking many of the concepts

from scalar calculus, generalizing these concepts and representing them in a vector

format. These alternative vector representations have many applications in repre-

senting two-dimensional and three-dimensional physical problems. Let us begin by

examining the representation of curves using vectors.

Curves

Atwo-dimensional curve can be defined

(i) Explicitly y=f(x)

(ii) Implicitly F(x, y ) = 0

(iii) Parametrically x=x(t), y =y(t)

(iv ) As a vector
r =
r (t) = x(t)eˆ 1 +y(t)ˆe 2 or
r =
r (x) = xˆe 1 +f(x)ˆe 2

Athree-dimensional curve can be defined

(i) Parametrically x=x(t), y =y(t), z =z(t)

(ii) As a vector
r =
r (t) = x(t)ˆe 1 +y(t)eˆ 2 +z(t)eˆ 3

(iii) A curve in space is sometimes defined as the intersection of two surfaces

F(x, y, z ) = 0 and G(x, y, z ) = 0 and in this special case the curve

is defined by a set of (x, y, z)values which are common to both surfaces.

It is assumed that the functions used to define these curves are continuous single-

valued functions which are everywhere differentiable. Also note that the parametric

and vector representations of a curve are not unique.

In two-dimensions a parametric curve {x(t), y(t)}, for a≤t≤bhas end points

(x(a), y (a)) and (x(b), y (b)).A curve is called a closed curve if its end points coincide

and x(a) = x(b)and y(a) = y(b). If (x 0 , y 0 )is a point on the given curve, which is not an

end point, such that there exists more than one value of the parameter tsuch that

(x(t), y (t)) = (x 0 , y 0 ), then the point (x 0 , y 0 )is called a multiple point or a point where

the curve crosses itself. A curve is called a simple closed curve if it has no multiple

points and the end points coincide. Simple closed curves are defined by one-to-one

mappings. The above definitions of end points, closed curve, simple closed curve and

multiple points apply to parametric curves {x(t), y (t), z (t)}in three-dimensions and to

n-dimensional parametric curves defined by {x 1 (t), x 2 (t),... , x n(t)}as the parameter t

ranges from ato b.