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Unformatted text preview: Chapter 9. Line Integrals 9.1 Introduction 9.1.1 Work Done I (i) Let F be a constant force acting on a particle in the displacement direction as shown in figure (i) above. Suppose the distance moved by the particle is s . The work done is given by W = k F k × s. (ii) Let F be a constant force acting on a particle in the direction which form an angle θ against 2 MA1505 Chapter 9. Line Integrals the displacement direction (see figure (ii) above). Suppose the distance moved by the particle is s . The work done is given by W = k F k cos θ × s = ( F · T ) × s = F · s T where T is the unit vector in the displacement direction. 9.1.2 Work Done II Let F ( x,y,z ) be a variable force acting on a particle which moves along the curve C with vector equation r ( t ) = x ( t ) i + y ( t ) j + z ( t ) k as shown in the figure below. Suppose the particle moves from point P to point Q . What is the work done? Solution: To solve this problem, we divide the 2 3 MA1505 Chapter 9. Line Integrals curve C into n segments. If a segment i is small enough, it can be treated as a straight line segment and the force within which can be assumed to be con stant F i . Then the work done for such a segment is approximately given by W i ≈ F i · Δ r i where r i = s T i and T i is the unit tangent vector along this segment. 3 4 MA1505 Chapter 9. Line Integrals So the total work done is approximately W total ≈ n X 1 F i · Δ r i . As n → ∞ , we write this as Z C F · d r which gives the actual total work done. 9.1.3 Vector Fields The vector function F is called in general a vector field and the above integral is called the line integral of F along the curve C . We shall see in section 9.3.7 how to evaluate this type of integral. 4 5 MA1505 Chapter 9. Line Integrals 9.2 Vector Fields 9.2.1 Vector field (two variables) Let R be a region in xyplane. A vector field on R is a vector function F that assigns to each point ( x,y ) in R a twodimensional vector F ( x,y ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . y ........................................................................................................................................ . . . . . . . . . . x .................................. . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • .................. . . . . . . . . . • . . . . . . . . . ....
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 Spring '11
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