Curl of a Vector Field

12 Key excerpts on "Curl of a Vector Field"

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  Linear Operators and Transforms in Mathematics (Concepts & Applications)

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter 2 Curl (Mathematics) In vector calculus, the curl (or rotor ) is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl is represented by a vector. The attributes of this vector (length and direction) characterize the rotation at that point. The direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl is the magnitude of rotation. If the vector field represents the flow velocity of a moving fluid, then the curl is the circulation density of the fluid. A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the Curl of a Vector Field to the line integral of the vector field around the boundary curve. The alternative terminology rotor or rotational and alternative notations rot F and ∇ × F are often used (the former especially in many European countries, the latter using the del operator and the cross product) for curl and curl F . Unlike the gradient and divergence, curl does not generalize as simply to other dimensions; some generalizations are possible, but only in three dimensions is the geometrically defined Curl of a Vector Field again a vector field. This is a similar pheno-menon as in the 3 dimensional cross product, and the connection is reflected in the notation ∇ × for the curl. The name curl was first suggested by James Clerk Maxwell in 1871. Definition The Curl of a Vector Field F , denoted curl F or ∇ × F , at a point is defined in terms of its projection onto various lines through the point.

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  Linear Operators in Calculus

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter 2 Curl (Mathematics) In vector calculus, the curl (or rotor ) is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl is represented by a vector. The attributes of this vector (length and direction) characterize the rotation at that point. The direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl is the magnitude of rotation. If the vector field represents the flow velocity of a moving fluid, then the curl is the circulation density of the fluid. A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the Curl of a Vector Field to the line integral of the vector field around the boundary curve. The alternative terminology rotor or rotational and alternative notations rot F and ∇ × F are often used (the former especially in many European countries, the latter using the del operator and the cross product) for curl and curl F . Unlike the gradient and divergence, curl does not generalize as simply to other dimensions; some generalizations are possible, but only in three dimensions is the geometrically defined Curl of a Vector Field again a vector field. This is a similar phenomenon as in the 3 dimensional cross product, and the connection is reflected in the notation ∇ × for the curl. The name curl was first suggested by James Clerk Maxwell in 1871. Definition The Curl of a Vector Field F , denoted curl F or ∇ × F , at a point is defined in terms of its projection onto various lines through the point. If is any unit vector, the projection of the curl of F onto is defined to be the limiting value of a closed line integral in a plane

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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)

  Contents 20.1 The Curl of a Vector Field . . . . . . . . . . . . . . . . 1000 Circulation Density . . . . . . . . . . . . . . . . . . . . . . 1000 Definition of the Curl . . . . . . . . . . . . . . . . . . . . 1001 Why Do the Two Definitions of Curl Give the Same Result? . . . . . . . . . . . . . . . . . . . 1004 Curl-Free Vector Fields . . . . . . . . . . . . . . . . . . . 1004 20.2 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . 1008 The Boundary of a Surface . . . . . . . . . . . . . . . . 1008 Calculating the Circulation from the Circulation Density . . . . . . . . . . . . . . . . . . . . . 1008 Curl-Free Vector Fields . . . . . . . . . . . . . . . . . . . 1010 Curl Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011 20.3 The Three Fundamental Theorems . . . . . . . . . 1015 The Gradient and the Curl . . . . . . . . . . . . . . . . . 1015 The Curl and the Divergence . . . . . . . . . . . . . . 1016 Chapter Twenty THE CURL AND STOKES’ THEOREM 1000 Chapter 20 THE CURL AND STOKES’ THEOREM 20.1 THE Curl of a Vector Field The divergence is a scalar derivative which measures the outflow of a vector field per unit volume. Now we introduce a vector derivative, the curl, which measures the circulation of a vector field. Imagine holding the paddle-wheel in Figure 20.1 in the flow shown by Figure 20.2. The speed at which the paddle-wheel spins measures the strength of circulation. Notice that the angular velocity depends on the direction in which the stick is pointing. If the stick is pointing horizontally the paddle- wheel does not spin; if the stick is vertical, the paddle wheel spins. Figure 20.1: A device for measuring circulation Figure 20.2: A vector field (in the planes  = 1,  = 2,  = 3) with circulation about the -axis Circulation Density We measure the strength of the circulation using a closed curve.

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  Calculus

  Multivariable

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

    (Publisher)

  The coordinate definition of curl gives curl   = ( (0)  −   )   + (   − (0)  )   + (   −   )   =  0 . (b) This vector field appears to be rotating around the -axis. By the right-hand rule, the circulation density around   is negative, so we expect the -component of the curl to point down. The coordinate definition gives curl   = ( (0)  − (−)  )   + (   − (0)  )   + ( (−)  −   )   = −2  . (c) At first glance, you might expect this vector field to have zero curl, as all the vectors are parallel to the -axis. However, if you find the circulation around the curve  in Figure 20.6, the sides contribute nothing (they are perpendicular to the vector field), the bottom contributes a negative quantity (the curve is in the opposite direction to the vector field), and the top contributes a larger positive quantity (the curve is in the same direction as the vector field and the magnitude of the vector field is larger at the top than at the bottom). Thus, the circulation around  is positive and hence we expect the curl to be nonzero and point up. The coordinate definition gives curl   = ( (0)  − (0)  )   + ( (−( + 1))  − (0)  )   + ( (0)  − (−( + 1))  )   =  . Another way to see that the curl is nonzero in this case is to imagine the vector field representing the velocity of moving water. A boat sitting in the water tends to rotate, as the water moves faster on one side than the other. 20.1 THE Curl of a Vector Field 1059    Figure 20.6: Rectangular curve in -plane        Figure 20.7: Rotating flywheel Example 3 A flywheel is rotating with angular velocity   and the velocity of a point  with position vector   is given by   =   ×   .

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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)

  Contents 20.1 The Curl of a Vector Field . . . . . . . . . . . . . . . . 1000 Circulation Density . . . . . . . . . . . . . . . . . . . . . . 1000 Definition of the Curl . . . . . . . . . . . . . . . . . . . . 1001 Why Do the Two Definitions of Curl Give the Same Result? . . . . . . . . . . . . . . . . . . . 1004 Curl-Free Vector Fields . . . . . . . . . . . . . . . . . . . 1004 20.2 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . 1008 The Boundary of a Surface . . . . . . . . . . . . . . . . 1008 Calculating the Circulation from the Circulation Density . . . . . . . . . . . . . . . . . . . . . 1008 Curl-Free Vector Fields . . . . . . . . . . . . . . . . . . . 1010 Curl Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011 20.3 The Three Fundamental Theorems . . . . . . . . . 1015 The Gradient and the Curl . . . . . . . . . . . . . . . . . 1015 The Curl and the Divergence . . . . . . . . . . . . . . 1016 Chapter Twenty THE CURL AND STOKES’ THEOREM 1000 Chapter 20 THE CURL AND STOKES’ THEOREM 20.1 THE Curl of a Vector Field The divergence is a scalar derivative which measures the outflow of a vector field per unit volume. Now we introduce a vector derivative, the curl, which measures the circulation of a vector field. Imagine holding the paddle-wheel in Figure 20.1 in the flow shown by Figure 20.2. The speed at which the paddle-wheel spins measures the strength of circulation. Notice that the angular velocity depends on the direction in which the stick is pointing. If the stick is pointing horizontally the paddle- wheel does not spin; if the stick is vertical, the paddle wheel spins. Figure 20.1: A device for measuring circulation Figure 20.2: A vector field (in the planes  = 1,  = 2,  = 3) with circulation about the -axis Circulation Density We measure the strength of the circulation using a closed curve.

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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)

  Chapter Twenty THE CURL AND STOKES’ THEOREM Contents 20.1 The Curl of a Vector Field . . . . . . . . . . 1070 Circulation Density . . . . . . . . . . . . . 1070 Definition of the Curl . . . . . . . . . . . . 1071 Why Do the Two Definitions of Curl Give the Same Result? . . . . . . . . . . . . 1073 Curl-Free Vector Fields . . . . . . . . . . . 1074 20.2 Stokes’ Theorem . . . . . . . . . . . . . . 1078 The Boundary of a Surface . . . . . . . . . 1078 Calculating the Circulation from the Circulation Density . . . . . . . . . 1078 Curl Free Vector Fields . . . . . . . . . . . 1080 Curl Fields . . . . . . . . . . . . . . . . . 1081 20.3 The Three Fundamental Theorems . . . . . 1084 The Gradient and the Curl . . . . . . . . . . 1085 The Curl and the Divergence . . . . . . . . 1086 REVIEW PROBLEMS . . . . . . . . . . . 1090 PROJECTS . . . . . . . . . . . . . . . . . 1094 1070 Chapter Twenty THE CURL AND STOKES’ THEOREM 20.1 THE Curl of a Vector Field The divergence is a scalar derivative which measures the outflow of a vector field per unit volume. Now we introduce a vector derivative, the curl, which measures the circulation of a vector field. Imagine holding the paddle-wheel in Figure 20.1 in the flow shown by Figure 20.2. The speed at which the paddle-wheel spins measures the strength of circulation. Notice that the angular velocity depends on the direction in which the stick is pointing. If the stick is pointing horizontally the paddle-wheel does not spin; if the stick is vertical, the paddle wheel spins. Figure 20.1: A device for measuring circulation x y z Figure 20.2: A vector field with circulation about the z-axis Circulation Density We measure the strength of the circulation using a closed curve. Suppose C is a circle with centre P = (x, y, z ) in the plane perpendicular to n , traversed in the direction determined from n by the right-hand rule.

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  Mathematical Methods and Physical Insights

  An Integrated Approach

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

    (Publisher)

  That sort of continuous transformation resembles a rotation — and in fact,  ∇ ×  F evaluated at a point P is a measure of the swirl of the vector field at P. Imagine water rotating with constant angular velocity  ω. The velocity of the water at  r is  v =  ω × r ; a simple calculation reveals  ∇ ×  v = 2  ω. (12.66) So we can understand the curl as twice 5 the angular speed ω of the water, oriented along the axis of rotation — something easily measured by placing a propeller into the water and watching it spin. The flow-of-water metaphor allows us to interpret 1 2 (  ∇ ×  F) · ˆ n as the angular speed of a small propeller placed at a point P with its rotation axis along ˆ n. The direction of the curl, then, is along the axis around which the observed ω is greatest. Fields with vanishing curl are said to be irrotational. 5 Twice?! See BTW 12.1. 150 12 GRAD, DIV, AND CURL Example 12.16 Swirl I: xˆ ı vs. yˆ ı Look back at the fields in Figure 12.5. It’s not hard to see that a small propeller placed at any point in xˆ ı would not spin. And to be sure,  ∇ × (xˆ ı ) = 0 — as is true of any field of the form f (x j ) ˆ e j . This may seem obvious, if only because parallel streamlines seem incongruous with any sense of swirl. On the other hand, despite its parallel streamlines, yˆ ı clearly has a uniform, non-zero curl pointing into the page (that is, a clockwise rotation),  ∇ × (yˆ ı ) = − ˆ k. Example 12.17 Swirl II: Significant Signs With a curl of 2 ˆ k,  F(x, y) = (x − y)ˆ ı + (x + y) ˆ j is a velocity field with unit angular speed ω = 1. Indeed, in Figure 12.8 the counter-clockwise swirl is immediately obvious. As is the behavior of a small propeller at the origin. Further from the origin propeller behavior may be less clear — but as we saw in the previous example, even parallel streamlines can have a curl. Figure 12.8 What’s in a sign? But what a difference a sign makes! Consider the effect of the substitution y → −y.

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  Mathematical Methods for Mathematicians, Physical Scientists and Engineers

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

    (Publisher)

  Sec. 10.3] The Curl of a Vector Field 273 η CD Fig. 10.3 Specify an axis at some point Ρ (Fig. 10.3), by choosing a unit vector ή along it. Let 8S be a small surface element, which becomes plane as 8S—>0 with its normal lying along ή. The boundary of bS is a simple closed curve, and an element of this is dr. Then, the circulation vector at P, denoted by curl ν is defined to have its component in direction ή given by the integral being around the bounding curve of 5S. Thus, (10.1) defines the 'circulation per unit area' about the axis ή, and for well-behaved vector fields this limit does exist. The direction of integration is, however, undefined, but it is customary to choose ή and dr to obey the right-hand rule as shown in the diagram. Also, since the right-hand side of (10.1) is a scalar, ή · curl ν is a scalar product. The easiest way of showing that the limit (10.1) exists for well-behaved fields ν is to evaluate the components of curl ν in particular coordinate sys-tems. First, consider Cartesian coordinates x, y, z. Here the ζ component (curl v) 2 , is given by taking ή along the ζ axis. Choose the surface 6S to be a small rectangle PjP2p3P4 perpendicular to ή as shown in Fig. 10.4, the in-tegration round the boundary being in the sense indicated by the arrows. ή · curl ν = lim tfvdr (10.1) ζ ή χ y Fig. 10.4

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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)

  Thus t s y, zd - hszd and f z s x, y, zd - 3xy 2 z 2 1 h9 szd Then (7) gives h9 szd - 0. Therefore f s x, y, zd - xy 2 z 3 1 K ■ The reason for the name curl is that the curl vector is associated with rotations. One connection is explained in Exercise 39. Another occurs when F represents the velocity field in fluid flow (see Example 16.1.3). In Section 16.8 we show that particles near s x, y, zd in the fluid tend to rotate about the axis that points in the direction of curl Fs x, y, zd, following the right-hand rule, and the length of this curl vector is a mea- sure of how quickly the particles move around the axis (see Figure 1). If curl F - 0 at a point P, then the fluid is free from rotations at P and F is called irrotational at P. In this case, a tiny paddle wheel moves with the fluid but doesn’t rotate about its axis. If curl F ± 0, the paddle wheel rotates about its axis. As an illustration, each vector field F in Figure 2 represents the velocity field of a fluid. In Figure 2(a), curl F ± 0 at most points, including P 1 and P 2 . A tiny paddle wheel placed at P 1 would rotate counterclockwise about its axis (the fluid near P 1 flows roughly in the same direction but with greater velocity on one side of the point than on the other), so the curl vector at P 1 points in the direction of k. Similarly, a paddle wheel at P 2 would rotate clockwise and the curl vector there points in the direction of 2k. In Figure 2(b), curl F - 0 everywhere. A paddle wheel placed at P moves with the fluid but doesn’t rotate about its axis. (x, y, z) curl F(x, y, z ) FIGURE 1 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.

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  Mathematical Methods in Engineering and Physics

  • Gary N. Felder, Kenny M. Felder(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  Similar arguments show that the front and back faces contribute f y ∕y and the top and bottom ones contribute f z ∕z. Put it all together and you get Equation 8.6.1. Curl The curl can be defined similarly. Instead of a surface we draw a closed loop around the point, and take the line integral of the vector field around that loop. (We’ve been bandying the term “circulation” about when talking about the curl; the circulation of a vector field around a closed loop means the line integral of that field around that loop.) The line integral matches the curl just as well as the surface integral matches the divergence, because it measures how much the field is circling around the point, ignoring the components that radiate in or out. But remember that curl is a vector! If our loop lies flat in the xy 6 We are following the standard convention in fluid flow where “flux” refers to the component of the vector field pointingthroughthesurfaceand“flow”referstothatcomponentmultipliedbythearea,i.e.thevalueofthesurface integral. 418 Chapter 8 Vector Calculus (horizontal) plane, then it will measure the z-component of the curl. (Remember the right- hand rule.) So we need three different loops to get the three different components of the curl. Other than that, the outline is similar to what we did for the divergence. Just as before, we shrink our loop down to focus on the region immediately around the point. Just as before, a smaller loop will lead to a smaller line integral, so we look at the circulation per unit area enclosed to reach a definition: The z-component of the Curl of a Vector Field  v at a point P is the circulation per unit area of  v around a loop surrounding P, in a plane perpendicular to  k, in the limit where the area of that loop shrinks to zero.

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  Tensor and Vector Analysis

  With Applications to Differential Geometry

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

    (Publisher)

  i to be

  Notice that the curl of λ j can be expressed as

  which is a useful form for calculation. Note that the Curl of a Vector Field F is a vector field. It is commonly denoted by Δ × F. The curl in general coordinates has components . It should be noted that if then λ i,j = λ j,i with the consequence that the curl of a gradient vector is identically zero. Any vector for which the curl is a zero vector is called irrotational

  The curl will be discussed further after introduction of Stokes’ theorem in Chapter 11 . A physical interpretation which reveals the reason for the designation “curl” of a vector will now be explained. The coordinates are orthogonal cartesian.

  The velocity vector V = υ i ei of a point P in a body rotating with constant angular velocity was shown in Section 7–5 to be Ω × R, where Ω is the constant angular velocity vector of the rotation, and R is the position vector to P . Write R = x i ei and Ω = ωi ei , where ω i are constants. Then

  Now the curl of V is given by

  By use of (7 –12) , further reduction of (12) leads to

  From (13) it follows that curl V, which means that the angular velocity vector of a uniformly rotating body is one-half the curl of the velocity vector of any point in the body. If one prefers, he may calculate Δ × V in (13) from the determinant form in (10) with λ i replaced by υ i . The computation in (13) was chosen in order to give the reader more facility with the notation employed.

  10–4. Physical Components. Care must be exercised in applying formula (6) and the formula for the curl, that is, . It was seen in Section 6–4 that if a vector in any coordinate system has components λ i , then the scalar projections of the vector on the tangents to the coordinate curves are . The three quantities may be defined as physical components of vector because they are the components of interest in physical applications. Note that the λ i might even be angles which would not behave as displacements. The components

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  Available until 16 Feb |Learn more

  Advanced Engineering Mathematics with MATLAB

  • Dean G. Duffy(Author)
  • 2021(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  R=--]) 3 / 2 − 3 x 2 + 3 y 2 + 3 z 2 (x 2 + y 2 + z 2) 5 / 2 = 0. (4.2.12) Another important vector function involving the vector field v is the curl of v, written curl(v) or rot(v) in some older textbooks. In fluid flow problems it is proportional to the instantaneous angular velocity of a fluid element. In rectangular coordinates, c u r l (v) = ∇ × v = (w y − v z) i + (u z − w x) j + (v x − u y) k, (4.2.13) where v =u i + v j + w k as before. However, it is best remembered in the mnemonic form_ ∇ × F = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z u v w | = (w y − v z) i + (u z − w x) j + (v x − u y) k. (4.2.14) If the Curl of a Vector Field is zero everywhere within a region, then the field is irrotational. Figure 4.2.2 illustrates graphically some vector fields that do and do not possess divergence and curl. Let the vectors that are illustrated represent the motion of fluid particles. In the case of divergence only, fluid is streaming from the point at which the density is falling. Alternatively, the point could be a source. In the case where there is only curl, the fluid rotates about the point and the fluid is incompressible. Finally, the point that possesses both divergence and curl is a compressible fluid with rotation. Figure 4.2.2: Examples of vector fields with and without divergence and curl. Some useful computational formulas exist for both the divergence and curl. operations: ∇ × (F + G) = ∇ × F + ∇ × G, (4.2.15) ∇ × ∇ φ = 0, (4.2.16) ∇ · ∇ × F = 0, (4.2.17) ∇ × (φ F) = φ ∇ × F + ∇ φ × F, (4.2.18) ∇ (F · G) = (F · ∇) G + (G · ∇) F + F × (∇ × G)[--=PLGO-SEPARATOR=--. ]+ G × (∇ × F), (4.2.19) ∇ × (F × G) = (G · ∇) F − (F · ∇) G + F (∇ · G) − G (∇ · F), (4.2.20) ∇ × (∇ × F) = ∇ (∇ · F) − (∇ · ∇) F, (4.2.21) and ∇ · (F × G) = G · ∇[--=PLGO-SEP. ARATOR=--]× F − F · ∇ × G. (4.2.22) In this book, the operation ∇ F is undefined. • Example 4.2.3 If F = xz 3 i − 2 x 2 yz j + 2 yz 4 k, compute the curl of F and verify that ∇·∇× F = 0. From the definition of

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