16 3 The Fundamental Theorem for Line Integrals Copyright Cengage Learning All rights reserved The Fundamental Theorem for Line Integrals Recall that Part 2 of the Fundamental Theorem of Calculus can be written as where F is continuous on a b We also called Equation 1 the Net Change Theorem The integral of a rate of change is the net change 2 The Fundamental Theorem for Line Integrals If we think of the gradient vector f of a function f of two or three variables as a sort of derivative of f then the following theorem can be regarded as a version of the Fundamental Theorem for line integrals 3 Example 1 Find the work done by the gravitational field in moving a particle with mass m from the point 3 4 12 to the point 2 2 0 along a piecewise smooth curve C Solution We know that F is a conservative vector field and in fact F f where 4 Example 1 Solution cont d Therefore by Theorem 2 the work done is W C F dr C f dr f 2 2 0 f 3 4 12 5 Independence of Path 6 Independence of Path Suppose C1 and C2 are two piecewise smooth curves which are called paths that have the same initial point A and terminal point B We know that in general C1 F dr C2 F dr But one implication of Theorem 2 is that C1 f dr C2 f dr whenever f is continuous In other words the line integral of a conservative vector field depends only on the initial point and terminal point of a curve 7 Independence of Path In general if F is a continuous vector field with domain D we say that the line integral C F dr is independent of path if C1 F dr C2 F dr for any two paths C1 and C2 in D that have the same initial and terminal points With this terminology we can say that line integrals of conservative vector fields are independent of path A curve is called closed if its terminal point coincides with its initial point that is r b r a See Figure 2 A closed curve Figure 2 8 Independence of Path If C F dr is independent of path in D and C is any closed path in D we can choose any two points A and B on C and regard C as being composed of the path C1 from A to B followed by the path C2 from B to A See Figure 3 Figure 3 9 Independence of Path Then C F dr C1 F dr C2 F dr C1 F dr C2 F dr 0 since C1 and C2 have the same initial and terminal points Conversely if it is true that C F dr 0 whenever C is a closed path in D then we demonstrate independence of path as follows Take any two paths C1 and C2 from A to B in D and define C to be the curve consisting of C1 followed by C2 10 Independence of Path Then 0 C F dr C1 F dr C2 F dr C1 F dr C2 F dr and so C1 F dr C2 F dr Thus we have proved the following theorem Since we know that the line integral of any conservative vector field F is independent of path it follows that C F dr 0 for any closed path 11 Independence of Path The physical interpretation is that the work done by a conservative force field as it moves an object around a closed path is 0 The following theorem says that the only vector fields that are independent of path are conservative It is stated and proved for plane curves but there is a similar version for space curves 12 Independence of Path We assume that D is open which means that for every point P in D there is a disk with center P that lies entirely in D So D doesn t contain any of its boundary points In addition we assume that D is connected this means that any two points in D can be joined by a path that lies in D The question remains How is it possible to determine whether or not a vector field F is conservative Suppose it is known that F P i Q j is conservative where P and Q have continuous first order partial derivatives Then there is a function f such that F f that is and 13 Independence of Path Then there is a function f such that F f that is and Therefore by Clairaut s Theorem The converse of Theorem 5 is true only for a special type of region 14 Independence of Path To explain this we first need the concept of a simple curve which is a curve that doesn t intersect itself anywhere between its endpoints See Figure 6 r a r b for a simple closed curve but r t1 r t2 when a t1 t2 b Types of curves Figure 6 15 Independence of Path In Theorem 4 we needed an open connected region For the next theorem we need a stronger condition A simply connected region in the plane is a connected region D such that every simple closed curve in D encloses only points that are in D Notice from Figure 7 that intuitively speaking a simply connected region contains no hole and can t consist of two separate pieces Figure 7 16 Independence of Path In terms of simply connected regions we can now state a partial converse to Theorem 5 that gives a convenient method for verifying that a vector field on is conservative 17 Example 2 Determine whether or not the vector field F x y x y i x 2 j is conservative Solution Let P x y x y and Q x y x 2 Then Since P y Q x F is not conservative by Theorem 5 18 Conservation of Energy 19 Conservation of Energy Let s apply the ideas of this chapter to a continuous force field F that moves an object along a path C given by r t a t b where r a A is the initial point and r b B is the terminal point of C According to Newton s Second Law of Motion the force F r t at a point on C is related to the acceleration a t r t by the equation F r t m r t So the work done by the force on the object is 20 Conservation of Energy By formula u t v t u t v t u t v t Fundamental Theorem of Calculus Therefore where v r is the velocity 21 Conservation of Energy The quantity that is half the mass times the square of the speed is called the kinetic energy of the object Therefore we can rewrite Equation 15 as W K B K A which says that the work done by the force field along C is equal to the change in kinetic energy at the endpoints of …
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