Divergence Theorem

3 Key excerpts on "Divergence Theorem"

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  eBook - ePub

  Transmission Lines and Wave Propagation

  • Philip C. Magnusson, Andreas Weisshaar, Vijai K. Tripathi, Gerald C. Alexander(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  Laplacian of a vector field , have been defined.

  (1) The Divergence Function

  As an example, consider a small enclosing surface in the electric-flux density D field, in a region of finite charge density. If the volume enclosed is small enough, the charge density will be essentially uniform throughout that volume, and the volume integral becomes simply the product of the local charge density and the volume, which may be indicated by Δv ?. Volume may then be divided into the surface integral to yield an expression for average charge density in terms of the surface integral. Let the enclosing surface be shrunk around some discrete point. The given ratio then approaches, as a limit, the charge density at that point. This mathematical result is defined as the divergence of D , and is written div D or ▿ · D . It is a scalar function of location in the D field:

  ∇ ⋅ D  =  [

  ∮ D ⋅ d s

  Δ v

  ]

  lim Δ v → 0

  ⁢ ( A.8 )

  It may be noted that the inverted delta symbol introduced above is customarily called "del" when it is designated separately. In applications in rectangular coordinates the symbol del resembles a vector, but it does not transform as a vector into other coordinate systems.2 The paragraph preceding Eq. A-8 gives the definition of divergence, and accordingly it is preferable that the expression "▿ · D " be read "divergence of D " rather than "del dot D ." The same principle applies to other derivative functions with symbols which incorporate del.

  (2) The Curl Function

  Ampére's law for static fields, Eq. 10-2 , provides a good example from which to develop this concept. Substitution of Eq. A-7 for i yields the following in terms of the current-density vector field J

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  eBook - ePub

  From Vehicles to Grid to Electric Vehicles to Green Grid

  Many a Little Makes a Miracle

  • Fuhuo Li, Shigeru Kanemitsu;Jianjie Zhang(Authors)
  • 2019(Publication Date)
  • WSPC

    (Publisher)

  or more concretely,

  Proof.

  The name comes from the fact that the left-hand side of (3.23) is equal to the integral of divergence defined by (3.11).

  Theorem 3.5. (Stokes’ theorem) Let be a domain, its boundary and let ω be a C1 class differential form of degree 1 in 3 variables:

  Then we have

  or more concretely,

  3.5Divergence

  Let v be the vector field of velocity of the fluid flowing in a domain Ω ⊂ 3 and let X ⊂ Ω be a bounded closed domain with its boundary ∂X forming a surface. Let ρ denote the density (distribution) function of the fluid in Ω. Then f = ρv is the vector field describing the flow of mass distribution of the fluid.

  At a point X ∋ P : x = x(u, v), the normal vector n is given by

  and we write

  for the component of f in the direction of n. Hence fn expresses the ratio of mass flowing out at P.

  Therefore its integral over all ∂X

  expresses the totality of masses flowing out of X in a unit time interval.

  Lemma 3.1. Let be a continuous vector field in Ω ⊂ 3 , ω = Pdydz + Qdzdx + Rdxdy a differential form of degree 2, S ⊂ Ω a smooth surface given by S : x = x(u, v), (u, v) ∈ 2 . Then we have

  where the right-hand side is the surface integral defined by

  Proof. By (3.28) and (3.29), we have

  Substituting this, we have which amounts to (3.31).

  Suppose there is no source or sink in X. Then since Q(∂X) in (3.30) indicates the rate of decrease of all masses in X, we have

  By Lemma 3.1 and Theorem 3.4 , the left-hand side of (3.35) is ∫∫∫X div F:

  Hence

  If is continuous, then we may change the integration and differentiation on the right of (3.35) and we obtain

  Since (3.38) holds for any bounded domain X, we must have

  which is called the equation of continuity of fluid. This will be deduced again in Example 3.1

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  eBook - ePub

  Pocket Book of Electrical Engineering Formulas

  • Richard C. Dorf, Ronald J. Tallarida(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  and z, (d i v e r g e n c e) div V = ▽ · V = δ a δ x + δ b δ y + δ c δ z (c u r l) curl V = ▽ × V = | i j k δ δ x δ δ y δ δ z a b c | Also, div grad g = ▽ 2 g = δ 2 g δ x 2 +[--=PLGO-SEPARATOR=-. -]δ 2 g δ y 2 + δ 2 g δ z 2 and curl grad g = 0; div curl V = 0; curl curl V = grad div V − (i ߜ 2 a + j ߜ 2 b + k ߜ 2 c). 3.  Divergence Theorem (Gauss) Given a vector function F with continuous partial derivatives in a region R bounded by a closed surface S, then ∭ R d i v F d V = ∬ S n · F d S, where n is the (sectionally continuous) unit normal. to S. 4.  Stokes’ Theorem Given a vector function with continuous gradient over a surface S that consists of portions that are piecewise smooth and bounded by regular closed curves such as C, then ∬ S n · curl F d S = ∮ C F · d r 5.  Planar Motion in Polar Coordinates Motion in a plane may be expressed with regard to polar coordinates (r, θ). Denoting the position vector by r and its magnitude by r, we have r = r R (θ), where R is the unit vector. Also, d R / dθ = P, a unit vector perpendicular to R. The velocity and acceleration are then v = d r d t R + r d θ d t P ; a = [ d 2 r d t 2 − r (d θ d t) 2 ] R + [ r d 2 θ d t 2 + 2 d r d t d θ d t ] P. Note that the component of acceleration in the P direction (transverse component) may also be written 1 r d d t (r 2 d θ d t) so that in purely radial motion it is zero. and r 2 d θ d t = C (constant) which means that the position vector sweeps out area at a constant rate (see Area in Polar Coordinates, Section 7.4).

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