Scalar function, gradient

The vector potential is not uniquely defined since the gradient of any scalar function may be added (the curl of a derivative is always zero). It is convention to select it as... 

Nearly two years ago, studying electrodynamics in curved space-time I found1 that Maxwell s equations impose on space-time a restriction which can be formulated by saying that a certain vector q determined by the curvature field must be the gradient of a scalar function, or... 

Hence curl grad = 0 for all (j>. Again, conversely may be inferred that if 6 is a vector function with identically zero curl, then 6 must be a gradient of some scalar function. Vector functions with identically zero curl are said to be irrotational. 

The gradient vector grad f(x) of the scalar function f(x) is an n-vector defined as... 

Hence u may be written as the gradient of some scalar function, i.e.,... 

The vector potential A is dehned only up to the gradient of a scalar function this additional degree of freedom can be used to turn A into a solenoidal potential... 

The result is two coupled Maxwell-Heaviside equations. Jackson shows that potentials A and in these two equations are arbitrary (i.e., yield the same force fields) [19,20]7 in a specific sense, since the A vector can be replaced with A = A + VA, where is a scalar function and VA is its gradient. The field is given by = V x A, so that the new B field becomes... 

The quantity A appears in these equations and is the vector potential of electromagnetic theory. In a very elementary discussion of the static electric field we are introduced to the theory of Coulomb. It is demonstrated that the electric field can be written as the gradient of a scalar potential E = —Vc)>, constant term to this potential leaves the electric field invariant. Where you choose to set the potential to zero is purely arbitrary. In order to describe a time-varying electric field a time dependent vector potential must be introduced A. If one takes any scalar function % and uses it in the substitutions... 

Consider first a general gauge transformation of the vector potential. For any scalar function / (r), the curl of the gradient vanishes identically ... 

Notation is a scalar function a,/3 are vector functions V is the gradient operator Jv d3r is an integration over the volume of the system Js d2r is an integration over the boundary of the system J, dr is an integration over a path i n is the outer unit normal to the boundary. 

Consider the relation dL - R(r) dr If dL is an exact differential prove that dL is the differential of a scalar function F(r) satisfying the relation R(r) - V F(r) and show that the necessary and sufficient condition for existence of a gradient of a scalar function is that it be irrotational V x R - 0. Show that this latter requirement coincides with the necessary and sufficient conditions rendering dL an exact differential. 

Suppose the system can be written in the form x = -VV, for some continuously differentiable, single-valued scalar function V(x). Such a system is calied a gradient system with potential function V. 

The final form of this equation, often used in the literature, is obtained by introducing a transformation of the potential energy term, In general, a force field is called conservative if it is the gradient of a scalar function. 

Inviscid, incompressible and irrotational flow fields are governed by Laplace s eqnation and are called potential flows, as the velocity in such flows can be expressed as the gradient of a scalar function called the velocity potential. 

The gradient of the phase indicator function, being a generalized scalar function [54], can be expressed as ... 

If we take the divergence of the gradient of the scalar function ip, as is done for the pressure field formulating an equation for the pressure, we obtain ... 

To solve (4-292), (4-293), (4—296), (4-299), and (4-300), we note that the continuity equation for an inviscid, irrotational flow is satisfied by a velocity field that can be expressed as the gradient of a scalar function, namely,... 

In effect, (7-29) represents a decomposition of the general vector field a into an irrotational part, associated with V, and a solenoidal (or divergence-free) part, represented by V A 0AVX). It should be noted that general proofs exist that show not only that (7-29) can represent any arbitrary vector field a but also that an arbitrary, irrotational vector field can be represented in terms of the gradient of a single scalar function

solenoidal vector field can be represented in the form of the second term of (7-29). Because... 

The gradient operator is a vector derivative operator that produces a vector when applied to a scalar function. 

The gradient of / is sometimes denoted by grad / instead of V/. The direction of the gradient of a scalar function is the direction in which the function is increasing most rapidly, and its magnitude is the rate of change of the function in that direction. 

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. 

Remember now that a gradient of any scalar function added to the vector potential satisfies the Coulomb gauge because... 

The distributive law holds for finding a gradient. Thus if S and T are scalar functions... 

Basically, the gradient applies to a scalar function in three variables, say /(x, y, z). The V operator in cartesian coordinates is a differential vector operator, like... 

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