Vector differentiation divergence

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 ... 

The differential operator divergence allows the passage from vectorial to scalar fields. For a vector [Pg.810]

In general, as seen in Section 2.8.4, the vector of surface forces (per unit volume) on a differential element can be represented as the divergence of the tensor stress field... 

Since there is not a continuously differentiable relationship between the inlet and outlet flows, the Gauss divergence theorem (i.e., the V- operation) has no practical application. Recall that, by definition, the surface unit vector n is directed outward. The sign of the mass-fraction difference in Eq. 16.68 is set by recognizing that the inlet flow velocity is opposite the direction of n, and vice versa for the exit. The overall mass-continuity equation,... 

Show that the volume integral of the divergence of a continuously differentiable second-order tensorial field in a finite region D is equivalent to the integral over the surface contour S enclosing that volume multiplied by the oriented normal vector n at each point of the surface. 

The machinery of geometric algebra makes it possible to differentiate and integrate functions of vector variables in a coordinate-free manner. The conventionally separated concepts of the gradient, divergence, and curl are... 

It is advantageous to replace Eq. (11) by an equivalent expression with lower differentiability requirements on the basis functions, in which the boundary conditions are automatically satisfied. The appropriate integration by parts formula is the Surface Divergence Theorem (SDT Weatherburn 1927), which is an integral relation for a piecewise-differentiable vector-value function F defined on a surface ... 

The scalar product v da gives the volume dV, which is multiplied by the local concentration ci to find differential flow J, da, which is the amount of the substance passing an area at any angle with the velocity vector V,-. For a volume enclosed by a surface area a, the total amount of species i leaving that volume is /gJi da. The divergence of the flow J, is... 

The mathematical operator consisting of the dot product, or scalar product, of the differential vector V with another vector is called the divergence operator. For instance, the charge concentration p in a volume is linked to the electric displacement D through one of Maxwell s relationships ... 

In other words, one can substitute any physically meaningful vector, whether q, q, or any other, for mathematical vector v in equation (5) so long as the differentiability requirements are satisfied. In doing so, equation (5) immediately reveals an unarguable relationship between such a physically meaningful vector and its divergence. 

The divergence of a vector function can similarly be expressed in orthogonal coordinates. If F is a vector function, it must be expressed in terms of the unit vectors of the coordinate system in which we are to differentiate. 

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... 

S. The Gauss law (1.212) relates the surface-integrated field on the boundary of a volume to the total net charge inside the volume. Using the divergence theorem fg B nda — - Bdr for any vector field B leads to the differential form of the... 

Figure 15 displays the above velocity field a(x) by showing its streamlines and the corresponding plots of the vector field. It can be shown by differentiation, that this velocity field a(x) is divergence-free, as required by equation (28). Note, that it is not uncommon to have orders of magnitude differences in the... 

Remember that in differential form, the divergence of a vector D in a point r is the limit to which the left-hand side of this equation tends under a contraction S (and V) to a point ... 

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