Gradient of a function

The property of the method of steepest ascent lies in the fact that movement along the gradient of a function must be preceded by a local description of the response surface by means of full or fractional factorial experiments [49]. It has been demonstrated that by processing FUFE or FRFE experimental outcomes we may obtain a mathematical model of a research subject in the form of a linear regression ... 

Gradient Extremal (GE), 338 Gradient norm minimization, 333 Gradient of a function, 238 Greens function, 257 GROMOS force field, 40 Gross atomic charge, 218... 

Gradient dynamical system. The vector field of a gradient dynamical system is the gradient of a function called potential function, i.e. X(m) = VV( x ca ), where x implicitly denotes the set of the q variables of R4 defining the point m of the manifold M and where ca stands for the control space parameters. 

The restriction in Eq. (25) to variations for which /A(r) dr = 0 is computationally inconvenient. To avoid this, we introduce the notion of a functional derivative, p = Po Just as the gradient of a function at a point is defined as that vector, V/(x) x =, which maps small changes in x about x0 to the resulting changes in the value of the function according to... 

EXAMPLE 7.27 Find the expression for the gradient of a function of spherical polar coordinates / =... 

The gradient of a function of three variables is defined by (5.31). The gradient V/ of a function /( i, 2> >of variables is defined as the n-dimensional vector whose components are the first partial derivatives off ... 

A function with phase /i(r) corresponds to the vector potential Ai(r), while the function with the phase Xi (r) + X (i") corresponds to A2 (r), where Vx = A2 (r) — Ai (r). Therefore, these two vector potentials differ by the gradient of a function, and this is allowed [according to Eq. (G.IO)] without having to modify any physical phenomena. 

Note that the physically equivalent vector potentials also may contain a vector field component A (the same in both cases), which is not a gradient (of any function). For example, it contains a vortexlike field, which is not equivalent to any gradient field. The curl of such a field is nonzero, while the curl of a gradient of any function does equal zero. Thus, the A itself may contain an unknown admixture of the gradient of a function. Hence, any experimental observation is determined solely by the non-gradient component of the field. For example, the magnetic field H2 for the vector potential A2 is... 

Thus, in order to be a holonomic constraint every row of G p) must be a gradient of a function. A necessary condition for a vector to be a gradient is that its derivative is a symmetric matrix, e.g. [OR70]. Thus, the z-th constraint is nonholonomic, iff... 

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