Divergence of a Vector Field

11 Key excerpts on "Divergence of a Vector Field"

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  eBook - PDF

  Calculus

  Multivariable

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

    (Publisher)

  . . . . . . . . . . . . . . . . . . . 1033 19.3 The Divergence of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039 Definition of Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039 Why Do the Two Definitions of Divergence Give the Same Result? . . . . . . . 1041 Divergence-Free Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1042 Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043 19.4 The Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048 The Boundary of a Solid Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048 Calculating the Flux from the Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . 1048 The Divergence Theorem and Divergence-Free Vector Fields . . . . . . . . . . . 1050 Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051 1018 Chapter 19 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 , in m 2 , is immersed in a fluid. The fluid flows with constant velocity   , in m/sec, perpendicular to the square. Write an expression for the rate of flow in m 3 /sec.   ✛  Figure 19.1: Fluid flowing perpendicular to a surface Solution In one second a given particle of water moves a distance of ‖   ‖ 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 ‖   ‖ and cross-sectional area .

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  Calculus

  Multivariable

  • William G. McCallum, Deborah Hughes-Hallett, Daniel E. Flath, Andrew M. Gleason, Selin Kalaycioglu, Brigitte Lahme, Patti Frazer Lock, Guadalupe I. Lozano, Jerry Morris, David Mumford, Brad G. Osgood, Cody L. Patterson, Douglas Quinney, Ayse Arzu Sahin, Adam H. Spiegler, Jeff Tecosky-Feldman, Thomas W. Tucker, Aaron(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  We say that the origin is a source. Figure 19.30 suggests flow into the origin; in this case we say that the origin is a sink. In this section we use the flux out of a closed surface surrounding a point to measure the outflow per unit volume there, also called the divergence, or flux density.   Figure 19.29: Vector field showing a source   Figure 19.30: Vector field showing a sink Definition of Divergence To measure the outflow per unit volume of a vector field at a point, we calculate the flux out of a small sphere centered at the point, divide by the volume enclosed by the sphere, then take the limit of this flux-to-volume ratio as the sphere contracts around the point. Geometric Definition of Divergence The divergence, or flux density, of a smooth vector field   , written div   , is a scalar-valued function defined by div   (, , ) = lim Volume→0 ∫    ⋅    Volume of  . Here  is a sphere centered at (, , ), oriented outward, that contracts down to (, , ) in the limit. The limit can be computed using other shapes as well, such as the cubes in Example 2. 3 Although not all vector fields represent physically realistic fluid flows, it is useful to think of them in this way. 19.3 THE Divergence of a Vector Field 983 In Cartesian coordinates, the divergence can also be calculated using the following formula. We show these definitions are equivalent later in the section. Cartesian Coordinate Definition of Divergence If   =  1   +  2   +  3   , then div   =  1  +  2  +  3  . The dot product formula gives an easy way to remember the Cartesian coordinate definition, and suggests another common notation for div   , namely ∇ ⋅   . Using ∇ =     +     +     , we can write div   = ∇⋅   = (     +     +     ) ⋅ ( 1   +  2   +  3   ) =  1  +  2  +  3  . Example 1 Calculate the divergence of   (  ) =   at the origin (a) Using the geometric definition.

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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)

  We say that the origin is a source. Figure 19.30 suggests flow into the origin; in this case we say that the origin is a sink. In this section we use the flux out of a closed surface surrounding a point to measure the outflow per unit volume there, also called the divergence, or flux density.   Figure 19.29: Vector field showing a source   Figure 19.30: Vector field showing a sink Definition of Divergence To measure the outflow per unit volume of a vector field at a point, we calculate the flux out of a small sphere centered at the point, divide by the volume enclosed by the sphere, then take the limit of this flux-to-volume ratio as the sphere contracts around the point. Geometric Definition of Divergence The divergence, or flux density, of a smooth vector field   , written div   , is a scalar-valued function defined by div   (, , ) = lim Volume→0 ∫    ⋅    Volume of  . Here  is a sphere centered at (, , ), oriented outward, that contracts down to (, , ) in the limit. The limit can be computed using other shapes as well, such as the cubes in Example 2. 3 Although not all vector fields represent physically realistic fluid flows, it is useful to think of them in this way. 19.3 THE Divergence of a Vector Field 983 In Cartesian coordinates, the divergence can also be calculated using the following formula. We show these definitions are equivalent later in the section. Cartesian Coordinate Definition of Divergence If   =  1   +  2   +  3   , then div   =  1  +  2  +  3  . The dot product formula gives an easy way to remember the Cartesian coordinate definition, and suggests another common notation for div   , namely ∇ ⋅   . Using ∇ =     +     +     , we can write div   = ∇⋅   = (     +     +     ) ⋅ ( 1   +  2   +  3   ) =  1  +  2  +  3  . Example 1 Calculate the divergence of   (  ) =   at the origin (a) Using the geometric definition.

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  Calculus

  Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  These names originate in the study of fluid flow, in which case the divergence relates to the way in which fluid flows toward or away from a point and the curl relates to the rotational properties of the fluid at a point. We will investigate the physical interpretations of these operations in more detail later, but for now we will focus only on their computation. 15.1.4 DEFINITION If F(x, y, z) = f (x, y, z)i + g(x, y, z) j + h(x, y, z)k, then we de- fine the divergence of F, written div F, to be the function given by div F = ∂ f ∂ x + ∂ g ∂ y + ∂ h ∂ z (7) 15.1 Vector Fields 975 15.1.5 DEFINITION If F(x, y, z) = f (x, y, z)i + g(x, y, z) j + h(x, y, z)k, then we de- fine the curl of F, written curl F, to be the vector field given by curl F =  ∂ h ∂ y − ∂ g ∂ z  i +  ∂ f ∂ z − ∂ h ∂ x  j +  ∂ g ∂ x − ∂ f ∂ y  k (8) REMARK Observe that div F and curl F depend on the point at which they are computed, and hence are more properly written as div F(x, y, z) and curl F(x, y, z). However, even though these functions are expressed in terms of x, y, and z, it can be proved that their values at a fixed point depend only on the point and not on the coordinate system selected. This is important in applications, since it allows physicists and engineers to compute the curl and divergence in any convenient coordinate system. Before proceeding to some examples, we note that div F has scalar values, whereas curl F has vector values (i.e., curl F is itself a vector field). Moreover, for computational purposes it is useful to note that the formula for the curl can be expressed in the determinant form curl F =         i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z f g h         (9) You should verify that Formula (8) results if the determinant is computed by interpreting a “product” such as (∂ / ∂ x)(g) to mean ∂ g / ∂ x.

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  Vectors in Physics and Engineering

  • Alan Durrant(Author)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  5.3.3 Divergence and physical law

  In many applications the flux of a vector field across a surface describes the rate of flow of mass, energy or some other entity across the surface. For example, the pattern of heat flow in a conducting material is described by a vector field h (Wm –2 ) defined in Section 5.2.3 . The flux of h across a surface is the rate at which heat flows across the surface (in watts) and so divh (Wm –3 ) is the rate of outward flow of heat per unit of enclosed volume, evaluated as a limit at a point (as in Eq (5.25 )). When conditions are steady (i.e. temperatures are independent of time) and when there are no sources of heat (such as electrical heating) inside the material, the spatial pattern of heat flow obeys the physical law

  div  =  0

  ( steady conditions; no sources )

  (5.32)

  This equation states that the net outflow of heat per unit volume evaluated at any point is zero. In simple terms this means that the heat flowing into any small region of the material is balanced by an equal amount of heat flowing out of it.

  When there are sources of heat in the conducting material the heat flow is governed by

  divh = S

  ( steady conditions with heat soures )

  (5.33)

  where S is a scalar source field representing the rate at which heat is generated per unit volume of material (Wm –3 ). Eq (5.33 ) governs the steady pattern of heat flow in the bar of an electric fire in which heat is generated electrically, or in a nuclear reactor fuel rod in which heat is generated by nuclear fission.

  It is often important to consider heat flow under non-steady conditions. Suppose an electric bar heater is switched off at time t = 0. Then for t

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  Tensor and Vector Analysis

  With Applications to Differential Geometry

  • C. E. Springer(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  25) as

  from which it follows, by use of (9 –64), that

  and consequently

  where, as usual, the repeated i in the right-hand member indicates a sum. Contraction of the tensor in (2) , and use of the result in (5) , produces

  which provides a useful formula, in general coordinates, for the divergence of the vector with components λ i . Observe that a vector field was obtained from a scalar field by differentiating the scalar field, whereas the Divergence of a Vector Field produces a scalar field. The abbreviations div F and ∇·F are commonly employed for the divergence of a vector F.

  A physical interpretation of div F will appear after discussion of the Gauss divergence theorem in Section 11–8 . However, in order to give the reader some feeling for the concept of divergence as a measure of change of density in a fluid at a point, the following example is presented.

  Let V = υ i ei be the velocity vector of a particle at P (x i ) in a fluid moving through space, where x i are orthogonal cartesian coordinates. Consider an infinitesimal parallelopiped (Fig. 21 ) of the volume determined by the increments Δx i at P (x i ). The net increase in the amount of fluid in this box will be calculated by considering the amount flowing into each face and the outflow through the opposite face in time Δt . The inflow through the face perpendicular to the x 1 -direction and through P is given

  Fig.21

  by the product of the velocity component υ 1 times the area Δx 2 Δx 3 of the face, times the density of the fluid, that is, by ρυ 1 Δx 2 Δx 3 Δt in time Δt . By Taylor’s theorem, the value of ρυ 1 at the point x 1 + Δx 1 is given by . Hence, the net increase (or decrease) in the mass of volume due to flow in the x 1

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  Calculus: Early Transcendentals, 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)

  SOLUTION By the definition of divergence (Equation 9 or 10), we have div F - =  F - - -x s x zd 1 - -y s xy zd 1 - -z s2y 2 d - z 1 x z ■ If F is a vector field on R 3 , then curl F is also a vector field on R 3 . As such, we can compute its divergence. The next theorem shows that the result is 0. 11 Theorem If F - P i 1 Q j 1 R k is a vector field on R 3 and P, Q, and R have continuous second-order partial derivatives, then div curl F - 0 In Section 16.8 we give a more detailed explanation of curl and its interpretation (as a consequence of Stokes' Theorem). 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. 1166 CHAPTER 16 Vector Calculus PROOF Using the definitions of divergence and curl, we have div curl F - =  s= 3 Fd - - -x S -R -y 2 -Q -z D 1 - -y S -P -z 2 -R -x D 1 - -z S -Q -x 2 -P -y D - - 2 R -x -y 2 - 2 Q -x -z 1 - 2 P -y -z 2 - 2 R -y -x 1 - 2 Q -z -x 2 - 2 P -z -y - 0 because the terms cancel in pairs by Clairaut’s Theorem. ■ EXAMPLE 5 Show that the vector field Fs x, y, zd - x z i 1 xy z j 2 y 2 k can’t be written as the curl of another vector field, that is, F ± curl G for any vector field G. SOLUTION In Example 4 we showed that div F - z 1 x z and therefore div F ± 0. If it were true that F - curl G, then Theorem 11 would give div F - div curl G - 0 which contradicts div F ± 0. Therefore F is not the curl of another vector field. ■ Again, the reason for the name divergence can be understood in the context of fluid flow.

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  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)

  SOLUTION By the definition of divergence (Equation 9 or 10), we have div F - =  F - - -x s x zd 1 - -y s xy zd 1 - -z s2y 2 d - z 1 x z ■ If F is a vector field on R 3 , then curl F is also a vector field on R 3 . As such, we can compute its divergence. The next theorem shows that the result is 0. 11 Theorem If F - P i 1 Q j 1 R k is a vector field on R 3 and P, Q, and R have continuous second-order partial derivatives, then div curl F - 0 In Section 16.8 we give a more detailed explanation of curl and its interpretation (as a consequence of Stokes' Theorem). 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. 1204 CHAPTER 16 Vector Calculus PROOF Using the definitions of divergence and curl, we have div curl F - =  s= 3 Fd - - -x S -R -y 2 -Q -z D 1 - -y S -P -z 2 -R -x D 1 - -z S -Q -x 2 -P -y D - - 2 R -x -y 2 - 2 Q -x -z 1 - 2 P -y -z 2 - 2 R -y -x 1 - 2 Q -z -x 2 - 2 P -z -y - 0 because the terms cancel in pairs by Clairaut’s Theorem. ■ EXAMPLE 5 Show that the vector field Fs x, y, zd - x z i 1 xy z j 2 y 2 k can’t be written as the curl of another vector field, that is, F ± curl G for any vector field G. SOLUTION In Example 4 we showed that div F - z 1 x z and therefore div F ± 0. If it were true that F - curl G, then Theorem 11 would give div F - div curl G - 0 which contradicts div F ± 0. Therefore F is not the curl of another vector field. ■ Again, the reason for the name divergence can be understood in the context of fluid flow.

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  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)

  Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 16.9 The Divergence Theorem 1205 radius a, where a is chosen to be small enough so that S 1 is contained within S. Let E be the region that lies between S 1 and S. Then Equation 7 gives 8 y y E y div E dV - 2 y S1 y E  dS 1 y S y E  dS You can verify that div E - 0. (See Exercise 25.) Therefore from (8) we have y S y E  dS - y S1 y E  dS The point of this calculation is that we can compute the surface integral over S 1 because S 1 is a sphere. The normal vector at x is xy | x | . Therefore E  n - «Q | x | 3 x  S x | x | D - «Q | x | 4 x  x - «Q | x | 2 - «Q a 2 since the equation of S 1 is | x | - a. Thus we have y S y E  dS - y S1 y E  n dS - «Q a 2 y S1 y dS - «Q a 2 AsS 1 d - «Q a 2 4a 2 - 4«Q This shows that the electric flux of E is 4«Q through any closed surface S that contains the origin. [This is a special case of Gauss’s Law (Equation 16.7.11) for a single charge. The relationship between « and « 0 is « - 1 ys4 « 0 d.] ■ Another application of the Divergence Theorem occurs in fluid flow. Let vs x, y, zd be the velocity field of a fluid with constant density . Then F - v is the rate of flow per unit area. If P 0 s x 0 , y 0 , z 0 d is a point in the fluid and B a is a ball with center P 0 and very small radius a, then div FsPd < div FsP 0 d for all points P in B a since div F is con- tinuous. We approximate the flux over the boundary sphere S a as follows: y Sa y F  dS - y y Ba y div F dV < y y Ba y div FsP 0 d dV - div FsP 0 dVsB a d This approximation becomes better as a l 0 and suggests that 9 div FsP 0 d - lim a l 0 1 VsB a d y Sa y F  dS Equation 9 says that div FsP 0 d is the net rate of outward flux per unit volume at P 0 . (This is the reason for the name divergence.) If div FsPd . 0, the net flow is outward near P and P is called a source.

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  Mathematical Methods for Mathematicians, Physical Scientists and Engineers

  • Jeremy Dunning-Davies(Author)
  • 2003(Publication Date)
  • Woodhead Publishing

    (Publisher)

  Only the component of ν along ή contributes to the flow since any compo-nent normal to ή does not flow through dS (Fig. 10.6). The rate of flow may be written (vn)dS. This is negative if the normal com-ponent of ν is in the opposite direction to ή. More generally, the flux of any vector field, through an element dS, is defined as (v-n)dS, which is clearly a scalar but depends on both the magnitude of dS and the direction of its normal ή. 278 Vector Analysis [Ch. 10 Fig. 10.6 Now consider a finite region V of 3-space bounded by a simple closed surface S. Then, the flux of ν out of the region V is given by summing the above expression over all small elements 5S of the surface S. In the limit with every 5S—»0, the outward flux becomes the surface integral / / ( v n ) d S , the nor-mal ή pointing out of V at every point of S. Earlier, it was found convenient to define curl ν to measure the 'circulation of ν at a point'. It is convenient also to define a scalar field to measure the 'outflow at a point'. This scalar field is called the divergence of v, is denoted by div v, and is defined by div ν = lim — 6V-*0 δν (vn)dS, (10.4) Fig. 10.7 Sec. 10.4] The Divergence of a Vector Field 279 An expression for div ν may be found, fairly easily, in terms of arbitrary orthogonal coordinates u u u 2 , u 3 . The volume bV in (10.4) may be chosen to be a nearly rectangular solid with opposite corners at points A(u x ,u 2 ,U3) and C ' ( « i + δ^!, u 2 + hu 2 , u a + bu 3 ) as shown in Fig. 10.7. The edges AB, AD, AA' are along the orthogonal coordinate axes and so are of lengths hfiui, h 2 bu 2 , h 3 hu 3 respectively. Now to calculate the contribution to //(v-ft)dS from the shaded faces ABCD and A'B'C'D'. Suppose ν has components υ χ , υ 2 , v 3 along the coordinate axes, then v n = -v 3 on ABCD + v 3 on A'B'C'D' .

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  Mathematical Methods and Physical Insights

  An Integrated Approach

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

    (Publisher)

  In fact, from (15.46) we have  ∇ ·  A  =  ∇ ·  A + ∇ 2 λ. (15.48) So an astute choice of λ can give  A  virtually any divergence we’d like! For instance, back in Example 14.28 we found a vector potential  A 2 = −xy 2 zˆ ı − x 2 yz ˆ j for which  B =  ∇ ×  A 2 = x 2 yˆ ı − xy 2 ˆ j . But what if we’d prefer to work with a vector potential whose divergence vanishes? In Problem 15.23 you’re asked to show that the choice λ = 1 2 x 2 y 2 z yields a field  A  ≡  A 2 +  ∇λ = 1 2 x 2 y 2 ˆ k, (15.49) which is both divergenceless and whose curl is  B. (This is  A 1 of Example 14.28.) 210 15 THE THEOREMS OF GAUSS AND STOKES 15.3 The Fundamental Theorem of Calculus — Revisited The theorems of Gauss and Stokes are both incarnations of the fundamental theorem of calculus. In each, the cumulation of infinitesimal changes within the region of integration equals the net change of the field over the boundary of the region. For instance, in (15.2), f (P 2 ) − f (P 1 ) =  C  ∇f · d  , (15.50) we have the accumulation of  ∇f · d   and the zero-dimensional boundary (endpoints) P 1 and P 2 of the curve C. Similarly (15.7),  S  E · d a =  V  ∇ ·  E dτ, (15.51) declares the accumulation of  ∇ ·  E dτ equal to the evaluation of  E over the two-dimensional boundary S of the volume V . Likewise in (15.32),  C  B · d   =  S (  ∇ ×  B) · d a, (15.52) we have (  ∇ ×  B) · d a and the one-dimensional boundary C of the surface S. Note that in each case, the orientations are self-consistent. In (15.32), the direction around C and the direction of the normal ˆ n on the open surface S are connected by the right-hand rule; in (15.7), the volume dictates an outward-pointing ˆ n on the closed boundary S; and the minus sign in (15.2) reflects the direction of integration along C — that is, from P 1 to P 2 .

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