Vector differentiation gradient

Differential operations with scalars and vectors. The gradient or grad of a scalar field is... 

Gradient operator, vector differential operator, also known as... 

To reduce the material balance conditions (11,1) to differential equations for the composition and pressure, flux relations must be used to relate the vectors to the gradients of the composition and pressure... 

Two approaches to this equation have been employed. (/) The scalar product is formed between the differential vector equation of motion and the vector velocity and the resulting equation is integrated (1). This is the most rigorous approach and for laminar flow yields an expHcit equation for AF in terms of the velocity gradients within the system. (2) The overall energy balance is manipulated by asserting that the local irreversible dissipation of energy is measured by the difference ... 

The constant K, which maintains the equaUty, has been termed the hydraulic conductivity, permeabiUty, or simply conductivity. The permeabiUty is generally accepted to be a constant for a saturated soil, except for very small gradients (2—4). Here represents the hydraulic head at location whereas A/is the hydraulic length between points 1 and 2. is an area perpendicular to the discharge vector. In differential form... 

Assume that a differentiable scalar field depends on certain independent variables as S = S(xi, x2, X3). The gradient of the scalar field produces a vector, described below in different coordinate systems. 

Let us express Eqs. (58)-(61) in terms of potentials, rather than fields. Toward that end, let us invoke a general result from field theory (see, e.g., Kellogg [88], p. 76) Any vector field F(r,w), sufficiently differentiable, is the sum of a gradient and a curl. Then, fields P, N are given by... 

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... 

At the time of this writing, the only implementation of Cl second derivatives is that of Fox etal. (1983). The superiority of this method to the numerical differentiation of the Cl gradient is not clear to the present author. Cl second derivatives require the perturbed Cl vectors. The determination of the latter should cost about as much as the Cl energy calculation. Therefore, Cl gradient calculations should be roughly equivalent to a Cl second derivative calculation. Another difficulty in the Cl second derivative algorithm is the need to store the derived Cl coefficients. Hopefully, these difficulties will be eliminated in the future. Cl third derivatives are attractive in principle but are not yet quite practicable. 

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