Del operator

A third combination which involves del operating twice on a vector is... 

The cross product of two dels operating on a scalar function 0 yields... 

In (1.25), the terms inside the brackets can be reformulated by use of vector and tensor notations. By comparing the terms inside the brackets with the mathematical definitions of the nabla or del operator, the vector product between this nabla operator and the mass flux vector we recognize that these... 

Ra = homogeneous reactions involving A within the control volume V = del operator. 

We can define derivatives corresponding to successive application of the del operator. The first such operator is the divergence of the gradient. If / is a scalar function, the divergence of the gradient of / is given in Cartesian coordinates by... 

Two other possibilities for successive operation of the del operator are the curl of the gradient and the gradient of the divergence. The curl of the gradient of any differentiable scalar function always vanishes. 

This identity is verified by employing summation notation for v and the del operator V in rectangular coordinates, because unit vectors Sy, and are not functions of position. This strategy applies to all vector-tensor identities because all unit vectors in rectangular coordinates can be moved to the left of... 

Add the tensorial rank of each cofactor and subtract 2 for the dot operation. Scalars are zeroth-rank tensors and vectors are first-rank tensors. Since the del operator is a vector and convective momentum flux is a second-rank tensor, V pvv is a vector. The fcth component of V pvv is... 

Even though this vector-tensor identity was verified using summation notation in rectangular coordinates, it is valid in any coordinate system. It is extremely tedious to verify vector-tensor identities that involve the del operator in curvilinear coordinate systems because the unit vectors exhibit spatial dependence. Now it is possible to combine terms in the equation of motion due to the accumulation rate process and convective momentum flux. Equations (8-24) and (8-25) yield ... 

The terms and symbols used in this equation have the following meaning. Cj is the molar concentration of species j (kmol/m fluid), so that dCj/dt is the non-steady-state term expressing accumulation or depletion. V is the nabla or del operator. In a rectangular coordinate system, x, y, z with unit vectors Sy, and 8, the gradient of a scalar function / is represented by V/ and the divergence... 

V = gradient del operator fj, = permeability J = current density at any point... 

A function of the position vector x is called a field. We can have a scalar field, a vector field or a tensor field. Derivatives with respect to position vectors are performed using the vector differential operator V, know as the del operator. It is written as 0/6xi in the Cartesian tensor notation. The operator can be treated as a vector but it cannot stand alone. It must operate on a scalar, vector or a tensor. 

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