14.4. LINE INTEGRALS 265

14.4.2 Line Integrals And WorkLet C be a smooth curve contained in Rp. A curve C is an “oriented curve” if the onlyparameterizations considered are those which lie in exactly one of the two equivalenceclasses, each of which is called an “orientation”. In simple language, orientation specifiesa direction over which motion along the curve is to take place. Thus, it specifies the order inwhich the points of C are encountered. The pair of concepts consisting of the set of pointsmaking up the curve along with a direction of motion along the curve is called an orientedcurve.

Definition 14.4.8 Suppose F (x) ∈ Rp is given for each x ∈ C where C is a smooth ori-ented curve and suppose x→ F (x) is continuous. The mapping x→ F (x) is called avector field. In the case that F (x) is a force, it is called a force field.

Next the concept of work done by a force field F on an object as it moves along thecurve C, in the direction determined by the given orientation of the curve will be defined.This is new. Earlier the work done by a force which acts on an object moving in a straightline was discussed but here the object moves over a curve. In order to define what is meantby the work, consider the following picture.

x(t)

F (x(t))

x(t +h)In this picture, the work done by a constant force F on an object which moves from the

point x(t) to the point x(t +h) along the straight line shown would equal F ·(x(t +h)−x(t)).It is reasonable to assume this would be a good approximation to the work done in movingalong the curve joining x(t) and x(t +h) provided h is small enough. Also, provided h issmall,

x(t +h)−x(t)≈ x′ (t)h

where the wriggly equal sign indicates the two quantities are close. In the notation ofLeibniz, one writes dt for h and

dW = F (x(t)) ·x′ (t)dt

or in other words,dWdt

= F (x(t)) ·x′ (t) .

Defining the total work done by the force at t = 0, corresponding to the first endpoint ofthe curve, to equal zero, the work would satisfy the following initial value problem.

dWdt

= F (x(t)) ·x′ (t) , W (a) = 0.

This motivates the following definition of work.