Advanced Calculus fi..

Chapter 5 Vector Integral Calculus 273in terms of the parameter x. If the direction of C is that of increasing x, (5.4) and(5.5) becomeThe ordinary definite integral fab y dx, where y = g(x), is a special case of (5.6). 1Similarly, if C is represented in the formand the direction of C is that of increasing y, then. In most applications the line integrals appear as a combination,which is abbreviated as follows:the brackets being used only when necessary.In the formulas thus far the direction of C has been that of increasing parameter.If the opposite direction is chosen, upper and lower limits are reversed on all integrals.Thus (5.4) becomesThe line integral is therefore multiplied by - 1. Often it is convenient to specify thepath by its equations in some form and to indicate the direction by using the initialand terminal points as lower and upper limits:(x?. v?)LBpdX+edy or / P ~ X Q + ~ Y .C(XI.",)