Surface Integral

7 Key excerpts on "Surface Integral"

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  eBook - PDF

  Multivariable Calculus, Metric Edition

  • James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Finally, we compute, by definition, yy S F  dS as the sum of the surface integrals of F over the pieces S 1 and S 2 : y S y F  dS - y S1 y F  dS 1 y S2 y F  dS -  2 1 0 -  2 ■ Although we motivated the Surface Integral of a vector field using the example of fluid flow, this concept also arises in other physical situations. For instance, if E is an electric field (see Example 16.1.5), then the Surface Integral y S y E  dS is called the electric flux of E through the surface S. One of the important laws of electro- statics is Gauss’s Law, which says that the net charge enclosed by a closed surface S is 11 Q - « 0 y S y E  dS S¡ y z x S™ FIGURE 13 Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 1192 CHAPTER 16 Vector Calculus where « 0 is a constant (called the permittivity of free space) that depends on the units used. (In the SI system, « 0 < 8.8542 3 10 212 C 2 yN  m 2 .) Therefore, if the vector field F in Example 4 represents an electric field, we can conclude that the charge enclosed by S is Q - 4 3 « 0 . Another application of Surface Integrals occurs in the study of heat flow. Suppose the temperature at a point s x, y, zd in a body is us x, y, zd. Then the heat flow is defined as the vector field F - 2K =u where K is an experimentally determined constant called the conductivity of the sub- stance. The rate of heat flow across the surface S in the body is then given by the Surface Integral y S y F  dS - 2K y S y =u  dS EXAMPLE 6 The temperature u in a metal ball is proportional to the square of the distance from the center of the ball.

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  eBook - PDF

  Calculus, Metric Edition

  • James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Finally, we compute, by definition, yy S F  dS as the sum of the surface integrals of F over the pieces S 1 and S 2 : y S y F  dS - y S1 y F  dS 1 y S2 y F  dS -  2 1 0 -  2 ■ Although we motivated the Surface Integral of a vector field using the example of fluid flow, this concept also arises in other physical situations. For instance, if E is an electric field (see Example 16.1.5), then the Surface Integral y S y E  dS is called the electric flux of E through the surface S. One of the important laws of electro- statics is Gauss’s Law, which says that the net charge enclosed by a closed surface S is 11 Q - « 0 y S y E  dS S¡ y z x S™ FIGURE 13 Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 1230 CHAPTER 16 Vector Calculus where « 0 is a constant (called the permittivity of free space) that depends on the units used. (In the SI system, « 0 < 8.8542 3 10 212 C 2 yN  m 2 .) Therefore, if the vector field F in Example 4 represents an electric field, we can conclude that the charge enclosed by S is Q - 4 3 « 0 . Another application of Surface Integrals occurs in the study of heat flow. Suppose the temperature at a point s x, y, zd in a body is us x, y, zd. Then the heat flow is defined as the vector field F - 2K =u where K is an experimentally determined constant called the conductivity of the sub- stance. The rate of heat flow across the surface S in the body is then given by the Surface Integral y S y F  dS - 2K y S y =u  dS EXAMPLE 6 The temperature u in a metal ball is proportional to the square of the distance from the center of the ball.

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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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  Calculus

  Multivariable

  • William G. McCallum, Deborah Hughes-Hallett, Andrew M. Gleason, David O. Lomen, David Lovelock, Jeff Tecosky-Feldman, Thomas W. Tucker, Daniel E. Flath, Joseph Thrash, Karen R. Rhea, Andrew Pasquale, Sheldon P. Gordon, Douglas Quinney, Patti Frazer Lock(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  1026 Chapter Nineteen FLUX INTEGRALS AND DIVERGENCE 19.1 THE IDEA OF A FLUX INTEGRAL Flow Through a Surface Imagine water flowing through a fishing net stretched across a stream. Suppose we want to measure the flow rate of water through the net, that is, the volume of fluid that passes through the surface per unit time. Example 1 A flat square surface of area A, in m 2 , is immersed in a fluid. The fluid flows with constant velocity v , in m/sec, perpendicular to the square. Write an expression for the rate of flow in m 3 /sec. v ✛ A Figure 19.1: Fluid flowing perpendicular to a surface Solution In one second a given particle of water moves a distance of ‖v ‖ in the direction perpendicular to the square. Thus, the entire body of water moving through the square in one second is a box of length ‖v ‖ and cross-sectional area A. So the box has volume ‖v ‖A m 3 , and Flow rate = ‖v ‖A m 3 /sec. This flow rate is called the flux of the fluid through the surface. We can also compute the flux of vector fields, such as electric and magnetic fields, where no flow is actually taking place. If the vector field is constant and perpendicular to the surface, and if the surface is flat, as in Example 1, the flux is obtained by multiplying the speed by the area. Next we find the flux of a constant vector field through a flat surface that is not perpendicular to the vector field, using a dot product. In general, we break a surface into small pieces which are approximately flat and where the vector field is approximately constant, leading to a flux integral. Orientation of a Surface Before computing the flux of a vector field through a surface, we need to decide which direction of flow through the surface is the positive direction; this is described as choosing an orientation. 1 At each point on a smooth surface there are two unit normals, one in each direction. Choosing an orientation means picking one of these normals at every point of the surface in a contin- uous way.

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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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  Calculus: Single and Multivariable

  • Deborah Hughes-Hallett, William G. McCallum, Andrew M. Gleason, Eric Connally, Daniel E. Flath, Selin Kalaycioglu, Brigitte Lahme, Patti Frazer Lock, David O. Lomen, David Lovelock, Guadalupe I. Lozano, Jerry Morris, David Mumford, Brad G. Osgood, Cody L. Patterson, Douglas Quinney, Karen R. Rhea, Ayse Arzu Sahin, Ad(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  In practice this problem seldom arises; however, one way to avoid it is to define flux integrals by the method used to compute them shown in Section 21.3. Flux and Fluid Flow If   is the velocity vector field of a fluid, we have Rate fluid flows through surface  = Flux of   through  =     ⋅    The rate of fluid flow is measured in units of volume per unit time. Example 3 Find the flux of the vector field   (, , ) shown in Figure 19.8 through the square  of side 2 shown in Figure 19.9, oriented in the   direction, where   (, , ) = −   +     2 +  2 . Figure 19.8: The vector field   in planes  = 0,  = 1,  = 2, where   (, , ) = −   +     2 +  2    3 1 2    Figure 19.9: Flux of   through the square  of side 2 in -plane and oriented in   direction     2 1 3 Δ   ✛ ✛ ✛ Δ ✛ ✛ Δ Figure 19.10: A small patch of surface with area ‖Δ   ‖ = ΔΔ Solution Consider a small rectangular patch with area vector Δ   in  , with sides Δ and Δ so that ‖Δ   ‖ = Δ Δ. Since Δ   points in the   direction, we have Δ   =   ΔΔ. (See Figure 19.10.) At the point (, 0, ) in  , substituting  = 0 into   gives   (, 0, ) = (1∕)   . Thus, we have Flux through small patch ≈   ⋅ Δ   = ( 1    ) ⋅ (   ΔΔ) = 1  Δ Δ. 966 Chapter 19 FLUX INTEGRALS AND DIVERGENCE Therefore, Flux through surface =     ⋅    = lim ‖Δ   ‖→0    ⋅ Δ   = lim Δ → 0 Δ → 0  1  Δ Δ. This last expression is a Riemann sum for the double integral ∫  1  , where  is the square 1 ≤  ≤ 3, 0 ≤  ≤ 2. Thus, Flux through surface =     ⋅    =   1   =  2 0  3 1 1    = 2 ln 3. The result is positive since the vector field is passing through the surface in the positive direction.

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  Mathematical Methods and Physical Insights

  An Integrated Approach

  • Alec J. Schramm(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  Taking advantage of symmetry, use Faraday’s law to find the induced electric field (magnitude and direction) inside and outside the solenoid of Problem 14.30 with time-dependent current I (t). 15 The Theorems of Gauss and Stokes The complementary relationship between differentiation and integration is laid down by the funda- mental theorem of calculus,  b a df dx dx = f (b) − f (a). (15.1) Along a one-dimensional path C threading through three-dimensional space from  r 1 to  r 2 , the theorem becomes  C  ∇f · d   = f ( r 2 ) − f ( r 1 ). (15.2) The fundamental theorem simply states that the net change of a continuous function between  r 1 and  r 2 is given by the cumulative effect of infinitesimal changes in f over a curve C connecting the points. We now want to expand the theorem’s purview beyond one-dimensional integrals. In our discussions of derivatives and integrals of vector fields, we focused particular attention on four measures: the divergence and the curl, the flux and the circulation. As the names suggest, the divergence and flux give measures of the surging of a vector field, the curl and circulation the swirl of a vector field. But though related, they are not identical: divergence is not flux, nor is curl circulation. The divergence and curl are themselves fields (scalar and vector, respectively), assigning definite values to every point in space — the expansion rate of sawdust or the spin rate of a propeller. They are local measures. By contrast, the flux and circulation are both (scalar) quantities defined over extended regions — a surface S and a curve C, respectively. The correct relationship entails the accumulation (integration) over extended regions of local, infinitesimal changes (derivatives) — in essence, the fundamental theorem of calculus. 15.1 The Divergence Theorem dx dy dz z x y Figure 15.1 Infinitesimal rectangular block.

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