Let C be a smooth curve contained in ℝ^{p}. A curve C is an “oriented curve” if the only parameterizations considered are those which lie in exactly one of the two equivalence classes, each of which is called an “orientation”. In simple language, orientation specifies a direction over which motion along the curve is to take place. Thus, it specifies the order in which the points of C are encountered. The pair of concepts consisting of the set of points making up the curve along with a direction of motion along the curve is called an oriented curve.
Next the concept of work done by a force field F on an object as it moves along the curve 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 straight line was discussed but here the object moves over a curve. In order to define what is meant by the work, consider the following picture.
In this picture, the work done by a constant force F on an object which moves from the point x

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

or in other words,

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

This motivates the following definition of work.
Definition 23.1.2 Let F

where the function x is one of the allowed parameterizations of C in the given orientation of C. In other words, there is an interval
Theorem 23.1.3 The symbol ∫ _{C}F ⋅ dR, is well defined in the sense that every parametrization in the given orientation of C gives the same value for ∫ _{C}F ⋅ dR.
Proof: Suppose g :


Regardless the physical interpretation of F, this is called the line integral. When F is interpreted as a force, the line integral measures the extent to which the motion over the curve in the indicated direction is aided by the force. If the net effect of the force on the object is to impede rather than to aid the motion, this will show up as the work being negative.
Does the concept of work as defined here coincide with the earlier concept of work when the object moves over a straight line when acted on by a constant force? If it doesn’t, then the above is not a good definition because it will contradict earlier and more basic constructions. Math is not like religion which often abounds in apparent contradictions.
Let p and q be two points in ℝ^{n} and suppose F is a constant force acting on an object which moves from p to q along the straight line joining these points. Then the work done is F ⋅

Therefore, the new definition adds to but does not contradict the old one. Therefore, it is not unreasonable to use this as the definition.
Example 23.1.4 Suppose for t ∈
To find this line integral use the above definition and write

In evaluating this replace the x in the formula for F with t, the y in the formula for F with cos

Example 23.1.5 Let C denote the oriented curve obtained by r

Find ∫ _{C}F⋅dR.
You use the definition.


Suppose you have a curve specified by r
Proposition 23.1.6 Suppose C is an oriented smooth curve parameterized by r
Prove: Let ϕ

Now define R

and so
The arc length parameter is just like any other parameter, in so far as considerations of line integrals are concerned, because it was shown above that line integrals are independent of parametrization. However, when things are defined in terms of the arc length parametrization, it is clear they depend only on geometric properties of the curve itself and for this reason, the arc length parametrization is important in differential geometry.
Definition 23.1.7 As to piecewise smooth curves, recall these are just smooth curves joined together at a succession of points p_{1},p_{2},
