Divergence of a Vector Field

Note that J(r, i) and v(r, f) are vector fields, while p(r, t) is a scalar field. 

FIGURE 11.12 Flux of a vector field J out of an element of volume Aywith surface 

Divide each side by AV and let the element of volume be shrunken to a point. In the limit Ax, Ay, Az 0, the terms containing Jx reduce to a partial 

The right-hand side has the stucture of a scalar product of V with J  

The divergence of the current density at a point x, y, z represents the net outward flux of electric charge from that point. Since electric charge is conserved, the flow of charge from every point must be balanced by a reduction of the charge density p(r, t) in the vicinity of that point. This leads to the equation of continuity 

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The divergence of a vector field u, written div u, is given in Cartesian coordinates by... 

Another type of derivative, the divergence of a vector field, is defined in Section 1.3.5. 

When contraction is performed once (summation over repeated indices), the divergence is obtained instead of the gradient. The divergence of a vector field v is a scalar... 

The divergence of a vector field u(x, y,z) is a scalar field which for each point of the vector field gives the inflow or outflow per unit volume in the immediate vicinity of this point ... 

In defining the divergence of a vector field A(r, t), we have transformed an integration over sxuface area into an integration over volume, as shown... 

The differential operators encountered often in the description of the physical properties of solids are the gradient of a scalar field V,4)(r), the divergence of a vector field Vr F(r), the curl of a vector field x F(r), and the laplacian of a scalar field V vector field is simply the vector addition of the laplacian of its components, V F = + V F z). These operators in three dimensions are... 

Multiplication sign, used for Scalar Product (or Dot Product) of two vectors. Divergence of a Vector Field, and other... 

Alternatively, one may attempt to estimate the integral over the derivative of the displacement field that entered in the expression for the coupling constant g= pc Jy cPr du/2. Since da is the divergence of a vector, the integral is reduced to that over a surface within the droplet s boundary ... 

The divergence operator is the three-dimensional analogue of the differential du of the scalar function u x) of one variable. The analogue of the derivative is the net outflow integral that describes the flux of a vector field across a surface S... 

On the other hand, it is well known that there is a relationship between Lyapunov exponents and the divergence of the vector field deduced from the differential equations describing a dynamical system. This relation provides a test on the numerical values obtained from the simulation algorithm. This relationship is, according to the definition of Lyapunov exponents ... 

Following the Morbidelli and Varma criterion, several other methods have been proposed in recent years in order to characterize the highly sensitive behavior of a batch reactor when it reaches the runaway boundaries. Among the most successful approaches, the evidence of a volume expansion in the phase space of the system has been widely exploited to characterize runaway conditions. For example, Strozzi and Zaldivar [9] defined the sensitivity as a function of the sum of the time-dependent Lyapunov exponents of the system and the runaway boundaries as the conditions that maximize or minimize this Lyapunov sensitivity. This has put the basis for the development of a new class of runaway criteria referred to as divergence-based approaches [5,10,18]. These methods usually identify runaway with the occurrence of a positive divergence of the vector field associated with the mathematical model of the reactor. 

Solenoidal field Field whose flux lines close and whose flux is divergenceless. The divergence of the cinl of ary vector field is zero. Ar r field that is the cinl of a vector field is solenoidal. 

Gauss s theorem equates the flux of a vector field through a closed surface with the divergence of that same held throughout its volume. This result will be useful in the following chapters. 

Several other functimis of the derivatives of vectors occur frequently in discussions of electromagnetic fields. The divergence of a vector A, written divA, is defined as... 

Taking the curl of both sides of the third and fourth Maxwell equations and using the identity that relates the curl of the curl of a vector field to its divergence and its laplacian, Eq. (G.21), we find... 

The divergence div u and gradient Vm of a vector field u(x) are respectively a scalar and a second-order tensor field, defined by ... 

Proof. It suflBces to consider the case vector field at O is negative, it is also negative in a small neighborhood of O. This implies that the flow near O contracts areas. The latter immediately implies that the Poincare map between any two cross-sections is indeed a contraction. 

In the (a, 6)-parameter plane, find the transition boundary saddle-focus for the origin, and equations for its linear stable and unstable subspaces. Detect the curves in the parameter plane that correspond to the vanishing of the saddle value a of the equilibrium state at the origin. Find where the divergence of the vector field at the saddle-focus vanishes. Plot the curves found in the (a 6)-plane. ... 

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