Let be an open, two-sided surface bounded by a closed non-intersecting curve (simple closed curve). Consider a directed line normal to as positive if it is on one side of , and negative if it is on the other side of . The choice of which side is positive is arbitrary but should be decided upon in advance. Call the direction or sense of positive if an observer, walking on the boundary of with his head pointing in the direction of the positive normal, has the surface on his left. Then if are single-valued, continuous, and have continuous first partial derivatives in a region of space including , we have
In vector form with and , this is simply expressed as
In words this theorem, called Stoke’s theorem, states that the line integral of the tangential component of a vector taken around a simple closed curve is equal to the surface integral of the normal component of the curl of taken over any surface having as a boundary. Note that if, as a special case in (39), we obtain the result (28).
If the equation of a curve C in the plane is given as , the line integral (14) is evaluated by placing in the integrand to obtain the definite integral
which is then evaluated in the usual manner.
Similarly if C is given as , then and the line integral becomes
If C is given in parametric form , the line integral becomes
where and denote the values of corresponding to points and respectively.
Combination of the above methods may be used in the evaluation.
Similar methods are used for evaluating line integrals along space curve.
It is often convenient to express a line integral in vector form as an aid in physical or geometric understanding as well as for brevity of notation. For example, we can express the line integral (15) in the form
where and . The line integral (14) is a special case of this with .
If at each point (x, y, z) we associate a force F acting on an object (i.e. if a force field is defined), then
represents physically the total work done in moving the object along the curve C.
Let C be a curve in the xy plane which connects points and , (see Fig. 6-2). Let and be single-valued functions defined at all points of C. Subdivide C into n parts by choosing n – 1 points on it given by . Call and and suppose that points are chosen so that they are situated on C between points and . Form the sum
The limit of this sum as in such a way that all quantities approaches zero, if such limit exists, is called a line integral along C and is denoted by
The limit does exist if P and Q are continuous (or piecewise continuous) at all points of C. The value of the integral depends in general on P, Q, the particular curve C, and on the limits and .
In an exactly analogous manner one may define a line integral along a curve C in three dimensional space as
where , and are functions of , and .
Other types of line integrals, depending on particular curves, can be defined. For example, if denotes the arc length along curve C in the above figure between points and , then
is called the line integral of along curve C. Extensions to three (or higher) dimensions are possible.