Conservative Vector Field

5 Key excerpts on "Conservative Vector Field"

• 

  eBook - PDF

  Mathematical Methods in Engineering and Physics

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

    (Publisher)

  8.11.2 Explanation: Conservative Vector Fields In introductory mechanics you may have learned that “conservative” forces (such as grav- ity) conserve mechanical energy, while non-conservative forces (such as friction) do not. In Section 8.5 we defined a “conservative” vector field as the gradient of a scalar field. In this section we will elaborate on the definition of “conservative” as a mathematical property of a vector field and show how it relates to the physical definition of a conservative force. Four Definitions of a Conservative Field We begin by giving four different definitions, the first two of which should look familiar. (We number these definitions, not because they come in any order, but so we can refer back to them later by number.) Conservative Fields 1. A Conservative Vector Field is the gradient of some scalar field:  v =  ∇F .Toputthesamething a different way, a Conservative Vector Field has a potential function. (In math texts F would generally be the potential of  v, but here we follow most physics texts and say F is minus the potential of  v.) 2. The line integrals of a Conservative Vector Field are “path independent.” That is, ∫ C  v ⋅ d  s depends only on the starting and ending points, not on the specific curve drawn between them. 3. The line integral of a Conservative Vector Field around a closed path (one where the starting and ending points are the same) is always zero: ∮  v ⋅ d  s = 0. 4. A Conservative Vector Field is “irrotational,” meaning it has no curl:  ∇ ×  v =  0. (This is not a perfect definition, as discussed below.) We are not saying “If you want to show that a particular field is conservative, you have to show that it meets all of these criteria.” Rather, we are saying that any one of these statements is a sufficient definition—with an exception for Definition 4—because each one implies the others.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Calculus

  Late Transcendental

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

    (Publisher)

  Conversely, a vector field satisfying these conditions on a suitably restricted region is conservative on that region if f , g, and h are continuous and have con- tinuous first partial derivatives in the region. Some problems involving Formulas (18) and (19) are given in the review exercises at the end of this chapter. CONSERVATION OF ENERGY If F(x, y, z) is a conservative force field with a potential function φ(x, y, z), then we call V (x, y, z) = −φ(x, y, z) the potential energy of the field at the point (x, y, z). Thus, it follows from the 3-space version of Theorem 15.3.1 that the work W done by F on a particle that moves along any path C from a point (x 0 , y 0 , z 0 ) to a point (x 1 , y 1 , z 1 ) is related to the potential energy by the equation W =  C F · dr = φ(x 1 , y 1 , z 1 ) − φ(x 0 , y 0 , z 0 ) = −[V (x 1 , y 1 , z 1 ) − V (x 0 , y 0 , z 0 )] (20) That is, the work done by the field is the negative of the change in potential energy. In particular, it follows from the 3-space analog of Theorem 15.3.2 that if a particle traverses a piecewise smooth closed path in a Conservative Vector Field, then the work done by the field is zero, and there is no change in potential energy. To take this a step further, suppose that a particle of mass m moves along any piecewise smooth curve (not necessarily closed) in a conservative force field F, starting at (x 0 , y 0 , z 0 ) with speed v i and ending at (x 1 , y 1 , z 1 ) with speed v f . If F is the only force acting on the particle, then an argument similar to the derivation of Equation (6) in Section 5.6 shows that the work done on the particle by F is equal to the change in kinetic energy 1 2 mv 2 f − 1 2 mv 2 i of the particle.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Calculus

  Multivariable

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  Conversely, a vector field satisfying these conditions on a suitably restricted region is conservative on that region if f , g, and h are continuous and have continu- ous first partial derivatives in the region. Some problems involving Formulas (18) and (19) are given in the review exercises at the end of this chapter. Conservation of Energy If F(x, y, z) is a conservative force field with a potential function φ(x, y, z), then we call V (x, y, z) = −φ(x, y, z) the potential energy of the field at the point (x, y, z). Thus, it follows from the 3-space version of Theorem 15.3.1 that the work W done by F on a particle that moves along any path C from a point (x 0 , y 0 , z 0 ) to a point (x 1 , y 1 , z 1 ) is related to the potential energy by the equation W =  C F · dr = φ(x 1 , y 1 , z 1 ) − φ(x 0 , y 0 , z 0 ) = −[V (x 1 , y 1 , z 1 ) − V (x 0 , y 0 , z 0 )] (20) That is, the work done by the field is the negative of the change in potential energy. In particular, it follows from the 3-space analogue of Theorem 15.3.2 that if a particle traverses a piecewise smooth closed path in a Conservative Vector Field, then the work done by the field is zero, and there is no change in potential energy. To take this a step further, suppose that a particle of mass m moves along any piecewise smooth curve (not necessarily closed) in a conservative force field F, starting at (x 0 , y 0 , z 0 ) with speed v i and ending at (x 1 , y 1 , z 1 ) with speed v f . If F is the only force acting on the particle, then an argument similar to the derivation of Equation (6) in Section 6.6 shows that the work done on the particle by F is equal to the change in kinetic energy 1 2 mv 2 f − 1 2 mv 2 i of the particle.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Calculus

  Early Transcendentals

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

    (Publisher)

  Conversely, a vector field satisfying these conditions on a suitably restricted region is conservative on that region if f , g, and h are continuous and have con- tinuous first partial derivatives in the region. Some problems involving Formulas (18) and (19) are given in the review exercises at the end of this chapter. CONSERVATION OF ENERGY If F(x, y, z) is a conservative force field with a potential function φ(x, y, z), then we call V (x, y, z) = −φ(x, y, z) the potential energy of the field at the point (x, y, z). Thus, it follows from the 3-space version of Theorem 15.3.1 that the work W done by F on a particle that moves along any path C from a point (x 0 , y 0 , z 0 ) to a point (x 1 , y 1 , z 1 ) is related to the potential energy by the equation W =  C F · dr = φ(x 1 , y 1 , z 1 ) − φ(x 0 , y 0 , z 0 ) = −[V (x 1 , y 1 , z 1 ) − V (x 0 , y 0 , z 0 )] (20) That is, the work done by the field is the negative of the change in potential energy. In particular, it follows from the 3-space analog of Theorem 15.3.2 that if a particle traverses a piecewise smooth closed path in a Conservative Vector Field, then the work done by the field is zero, and there is no change in potential energy. To take this a step further, suppose that a particle of mass m moves along any piecewise smooth curve (not necessarily closed) in a conservative force field F, starting at (x 0 , y 0 , z 0 ) with speed v i and ending at (x 1 , y 1 , z 1 ) with speed v f . If F is the only force acting on the particle, then an argument similar to the derivation of Equation (6) in Section 6.6 shows that the work done on the particle by F is equal to the change in kinetic energy 1 2 mv 2 f − 1 2 mv 2 i of the particle.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Anton's Calculus

  Early Transcendentals

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

    (Publisher)

  To take this a step further, suppose that a particle of mass m moves along any piecewise smooth curve (not necessarily closed) in a conservative force field F, starting at (x 0 , y 0 , z 0 ) with speed v i and ending at (x 1 , y 1 , z 1 ) with speed v f . If F is the only force acting on the particle, then an argument similar to the deriva- tion of Equation (6) in Section 6.6 shows that the work done on the particle by F is equal to the change in kinetic energy 1 2 mv 2 f − 1 2 mv 2 i of the particle. (An argument for smooth curves appears in the Making Connections exercises.) If we let V i denote the potential energy at the starting point and V f the potential energy at the final point, then it follows from (20) that 1 2 mv 2 f − 1 2 mv 2 i = −[V f − V i ] which we can rewrite as 1 2 mv 2 f + V f = 1 2 mv 2 i + V i This equation states that the total energy of the particle (kinetic energy + potential energy) does not change as the particle moves along a path in a Conservative Vector Field. This result, called the conservation of energy principle, explains the origin of the term “Conservative Vector Field.” QUICK CHECK EXERCISES 15.3 (See page 1005 for answers.) 1. If C is a piecewise smooth curve from (1, 2, 3) to (4, 5, 6), then ∫ C dx + 2 dy + 3 dz = 2. If C is the portion of the circle x 2 + y 2 = 1 where 0 ≤ x, oriented counterclockwise, and f (x, y) = ye x , then ∫ C f ⋅ dr = 3. A potential function for the vector field F(x, y, z) = yzi + (xz + z)j + (xy + y + 1)k is (x, y, z) = . 4. If a, b, and c are nonzero real numbers such that the vector field x 5 y a i + x b y c j is a Conservative Vector Field, then a = , b = , c = 1004 Chapter 15 / Topics in Vector Calculus EXERCISE SET 15.3 C CAS 1–6 Determine whether F is a Conservative Vector Field. If so, find a potential function for it. 1. F(x, y) = 2xi + 2y j 2. F(x, y) = 3y 2 i + 6xy j 3. F(x, y) = x 2 yi + 6xy 2 j 4. F(x, y) = e x cos yi − e x sin y j 5. F(x, y) = (cos y + y cos x)i + (sin x + x sin y) j 6.

  Sign up to read

  Learn more about book