Line Integral

6 Key excerpts on "Line Integral"

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  Anton's Calculus

  Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  58. Writing Discuss the similarities and differences between the definition of a definite integral over an interval (Defini- tion 5.5.1) and the definition of the Line Integral with respect to arc length along a curve (Definition 15.2.1). 59. Writing Describe the different types of Line Integrals, and discuss how they are related. QUICK CHECK ANSWERS 15.2 1. 2 √ 2 2. 3 √ 2 2 3. 4 4. 0 15.3 INDEPENDENCE OF PATH; CONSERVATIVE VECTOR FIELDS In this section we will show that for certain kinds of vector fields F the Line Integral of F along a curve depends only on the endpoints of the curve and not on the curve itself. V ector fields with this property, which include gravitational and electrostatic fields, are of special importance in physics and engineering. WORK INTEGRALS We saw in the last section that if F is a force field in 2-space or 3-space, then the work performed by the field on a particle moving along a parametric curve C from an initial point P to a final point Q is given by the integral ∫ C F ⋅ dr or equivalently ∫ C F ⋅ T ds 996 Chapter 15 / Topics in Vector Calculus Accordingly, we call an integral of this type a work integral. Recall that a work integral can also be expressed in scalar form as ∫ C F ⋅ dr = ∫ C f (x, y) dx + g(x, y) dy 2-space (1) ∫ C F ⋅ dr = ∫ C f (x, y, z) dx + g(x, y, z) dy + h(x, y, z) dz 3-space (2) where f , g, and h are the component functions of F. INDEPENDENCE OF PATH The parametric curve C in a work integral is called the path of integration. One of the important problems in applications is to determine how the path of integration affects the work performed by a force field on a particle that moves from a fixed point P to a fixed point Q. We will show shortly that if the force field F is conservative (i.e., is the gradient of some potential function ), then the work that the field performs on a particle that moves from P to Q does not depend on the particular path C that the particle follows.

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  Calculus

  Concepts and Contexts, Enhanced Edition

  • James Stewart(Author)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 13.2 Line IntegralS 921 This integral is often abbreviated as and occurs in other areas of physics as well. Therefore we make the following definition for the Line Integral of any continuous vector field. Definition Let be a continuous vector field defined on a smooth curve given by a vector function , . Then the Line Integral of along C is When using Definition 13, remember that is just an abbreviation for , so we evaluate simply by putting , , and in the expression for . Notice also that we can formally write . Find the work done by the force field in moving a particle along the quarter-circle , . SOLUTION Since and , we have and Therefore the work done is Note: Even though and integrals with respect to arc length are unchanged when orientation is reversed, it is still true that because the unit tangent vector is replaced by its negative when is replaced by Line Integral of a vector field Evaluate , where and is the twisted cubic given by SOLUTION We have F r t t 3 i t 5 j t 4 k r t i 2 t j 3 t 2 k r t t i t 2 j t 3 k 0 t 1 z t 3 y t 2 x t C F x , y , z xy i y z j z x k x C F d r EXAMPLE 8 C . C T y C F d r y C F d r x C F d r x C F T ds 2 cos 3 t 3 0 2 2 3 y C F d r y 2 0 F r t r t dt y 2 0 2 cos 2 t sin t dt r t sin t i cos t j F r t cos 2 t i cos t sin t j y sin t x cos t 0 t 2 r t cos t i sin t j F x , y x 2 i xy j EXAMPLE 7 d r r t dt F x , y , z z z t y y t x x t F r t F x t , y t , z t F r t y C F d r y b a F r t r t dt y C F T ds F a t b r t C F 13 x C F d r Figure 12 shows the force field and the curve in Example 7. The work done is negative because the field impedes movement along the curve. FIGURE 12 0 1 1 y x Figure 13 shows the twisted cubic in Example 8 and some typical vectors acting at three points on .

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  Calculus

  One and Several Variables

  • Saturnino L. Salas, Garret J. Etgen, Einar Hille(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  CHAPTER 18 Line IntegralS AND SURFACE INTEGRALS In this chapter we will study integration over curves and integration over surfaces. At the heart of this subject lie three great integration theorems: Green’s theorem, Gauss’s theorem (commonly known as the divergence theorem), and Stokes’s theorem. All three theorems are ultimately based on The Fundamental Theorem of Integral Calculus, and all can be cast in the same general form: an integral over a set S = a related integral over the boundary of S. A word about terminology. Suppose that S is some subset of the plane or of three- dimensional space. A function that assigns a scalar to each point of S (say, the tem- perature at that point or the mass density at that point) is known in science as a scalar field. A function that assigns a vector to each point of S (say, the wind velocity at that point or the gradient of a function f at that point) is called a vector field. We will be using this “field” language throughout. ■ 18.1 Line IntegralS We are led to the definition of Line Integral by the notion of work. The Work Done by a Varying Force over a Curved Path The work done by a constant force F on an object that moves along a straight line is, by definition, the component of F in the direction of the displacement multiplied by the length of the displacement vector r (Project 13.3): W = (comp d F)r. We can write this more briefly as a dot product: (18.1.1) W = F · r 938 18.1 Line IntegralS ■ 939 This elementary notion of work is useful, but it is not sufficient. Consider, for example, an object that moves through a magnetic field or a gravitational field. The path of the motion is usually not a straight line but a curve, and the force, rather than remaining constant, tends to vary from point to point. What we want now is a notion of work that applies to this more general situation. Let’s suppose that an object moves along a curve C : r(u ) = x (u ) i + y (u ) j + z (u ) k, u ∈ [a , b] subject to continuous force F.

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  Vector Calculus

  • William Cox(Author)
  • 1998(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  So the total work done by F in moving along the curve from t = a to t = b is b W = fF(f(t» . r'(z) dt a Note that in general the work done will depend on the curve traced between a and b. Vectorially, the integral expression for W is a scalar quantity. It is called the tangential Line Integral of the vectorfield F along the curve C. In fluid dynamics and electromagnetism we often have occasion to consider the 'flux' of a vector field quantity across a surface. This leads us to another important type of integral of a vector field. Thus, suppose F is a vector field, and let S be a surface with outward unit normal represented by n. Then F .n is the component of Integration in Vector Fields 187 F normal to the surface at a point with normal D. We say that thefiux ofF across an element of area do of the surface S, with normal D, is F . D do. The flux of F across the surface S is then given by the surface integral Having defined line and surface integrals we will be in a position to discuss three great theorems of integration in vector calculus -Green's theorem, Gauss's diver-gence theorem and Stokes's theorem. These theorems look complicated at first sight, but actually they all have a very simple foundation. They essentially relate an n-tuple integration to an (n -1 )-tuple integration, by the general process an integral over a region R = a related integral over the boundary of R For example, we might have surface integral over R = Line Integral around the boundary of R (double) (single) Another viewpoint, related to this, is that they come about by a sort of generalized integration by parts. Note that in this chapter I am going to use R to denote a general region, rather than the g used in Chapter 7. I used g in Chapter 7 because I wanted to reserve R for rectangular regions in the discussion of multiple integrals -there is no need for the distinction now.

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  Concise Vector Analysis

  The Commonwealth and International Library of Science, Technology, Engineering and Liberal Studies: Mathematics Division

  • C. J. Eliezer, W. J. Langford, E. A. Maxwell, I. N. Sneddon(Authors)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  Chapter 4 Vector Calculus 4.1. Line Integrals SUPPOSE C is an arc of a curve with terminal points A and Β (Fig. 1). Let φ(Ρ) be a scalar point function. The integral of φ over the curve C may be defined in the following ways. The arc is divided Pm Β Fig. 1 into m parts by m + 1 points Pq, P ^, ... where A = Po, Β = P^. Let A-y^ where q = 1, 2, ... , m be the length of the arc Λ -1 let Xq be point on the curve within the arc P^_ ^ P^. The expression (1) is formed for each interval As^, and the diSerent expressions as q takes the values 1, 2, ... , m are summed, thus yielding φ{Χ ,)Α8, + φ(Χ2 )Α8. + ... + 0 ( ^ J A 5 , . (2) If as m CO, and each Δ^^ 0, the expression (2) tends to a finite hmit which is independent of the mode of division of C into 58 VECTOR CALCULUS 59 its elements Δ.$·^, the value of this Umit is called the integral of φ over C and is written φ ds. (3) Such an integral over a curve C is called a Line Integral or a curvilinear integral and the curve C is called the path of integration. It will be noted that this is a simple generahzation of the Riemann integral of a function of a single variable (4) where/ (JC ) is a function of the variable x, and the hmits of integra-tion are a and b, that is, the path of integration is along the X -axis from the point χ = ato the point χ = 6. The existence of the Umit of the sum in expression (2) may be shown, in respect of the functions which we usually use in Apphed Mathematics, by the methods used in respect of the integral (4) and will be assumed in the rest of this book. The functions we deal with are assumed to be single valued, continuous and dif-ferentiable. In practice the evaluation of the integral would not be through the direct use of the hmiting process. Like integral (4), it is determined by utihzing its connection with the indefinite integral. The integrands in fine integrals may be scalars or vectors.

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  Calculus

  Multivariable

  • Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  994 Chapter 18 Line IntegralS Solution We want to find a function  (, ) for which   =  and   = . One possibility for  is  (, ) =  2 2 +  2 2 . You can check that grad  =    +    . Now we can use the Fundamental Theorem to compute the Line Integral. Since   = grad  we have   1   ⋅   =   1 grad  ⋅   =  (0, 2) −  (1, 0) = 3 2 . Notice that the calculation looks exactly the same for  2 . Since the value of the integral depends only on the values of  at the endpoints, it is the same no matter what path we choose. Path-Independent, or Conservative, Vector Fields In the previous example, the Line Integral was independent of the path taken between the two (fixed) endpoints. We give vector fields whose Line Integrals have this property a special name. A vector field   is said to be path-independent, or conservative, if for any two points  and , the Line Integral ∫    ⋅   has the same value along any piecewise smooth path  from  to  lying in the domain of   . If, on the other hand, the Line Integral ∫    ⋅   does depend on the path  joining  to , then   is said to be a path-dependent vector field. Now suppose that   is any continuous gradient field, so   = grad  . If  is a path from  to , the Fundamental Theorem for Line Integrals tells us that     ⋅   =  () −  ( ). Since the right-hand side of this equation does not depend on the path, but only on the endpoints of the path, the vector field   is path-independent. Thus, we have the following important result: If   is a continuous gradient vector field, then   is path- independent. Why Do We Care About Path-Independent, or Conservative, Vector Fields? Many of the fundamental vector fields of nature are path-independent—forexample, the gravitational field and the electric field of particles at rest.

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