Scalarizing function

This wave equation is tire basis of all wave optics and defines tire fimdamental stmcture of electromagnetic tlieory witli tire scalar function U representing any of tire components of tire vector functions E and H. (Note tliat equation (C2.15.5) can be easily derived by taking tire curl of equation (C2.15.1) and equation (C2.15.2) and substituting relations (C2.15.3) and (C2.15.4) into tire results.)... 

Equality Constrained Problems—Lagrange Multipliers Form a scalar function, called the Lagrange func tion, by adding each of the equality constraints multiplied by an arbitrary iTuiltipher to the objective func tion. 

This implies that, at constant k, the line integral of the differential form s de, parametrized by time t, taken over the closed curve h) zero. This is the integrability condition for the existence of a scalar function tj/ e) such that s = d j//de (see, e.g., Courant and John [13], Vol. 2, 1.10). This holds for an elastic closed cycle at any constant values of the internal state variables k. Therefore, in general, there exists a function ij/... 

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... 

When is a one component scalar function, one can take the square root of Eq. (9-237) and one thus obtains the relativistic equation describing a spin 0 particle discussed in Section 9.4. This procedure, however, does not work for a spin particle since we know that in the present situation the amplitude must be a multicomponent object, because in the nonrelativistic limit the amplitude must go over into the 2-component nonrelativistic wave function describing a spin particle. Dirac, therefore, argued that the square root operator in the present case must involve something operating on these components. 

Thus, we have expressed the vector field g in terms of a scalar function, U p), by a relatively simple operation. 

What type of objective function should we minimize This is the question that we are always faced with before we can even start the search for the parameter values. In general, the unknown parameter vector k is found by minimizing a scalar function often referred to as the objective function. We shall denote this function as S(k) to indicate the dependence on the chosen parameters. 

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... 

The term scalar field is used to describe a region of space in which a scalar function is associated with each point. If there is a vector quantity specified at each point, the points and vectors constitute a vector field. 

Suppose that 4> x,y,z) is a scalar point function, that is, a scalar function that is uniquely defined in a given region. Under a change of coordinate system to, say, x y z, it will take on another form, although its value at any point remains the same. Applying the chain rule (Section 2.12),... 

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

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... 

Unlike grad (j) div a is seen to be a scalar function. The Laplacian function in terms of this notation is written... 

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. 

It is a property of the second derivative of a scalar function such as p that it determines where the function is locally concentrated and locally depleted, in the absence of corresponding maxima and minima in the function itself. Consider a function f(x) with both maxima and minima. At a maximum in/(%) the curvature is negative, d2f(x)/dx2 < 0, and the value of f(x) is greater than the average of its values at x+dx and x-dx. The reverse is true at a minimum in f(x) where d2f(x)/dx2>0 and the value of f(x) is less than the average of its values at x+dx and x-dx. It is... 

The final matrix characteristic covered here involves differentiation of function of a vector with respect to a vector. Suppose/(x) is a scalar function of n variables (xx, x2, xn). The first partial derivative of/(x) with respect to x is... 

For a scalar function, the matrix of second derivatives, called the Hessian matrix, is... 

We start by considering an arbitrary measurable10 one-point11 scalar function of the random fields U and 0 Q U, 0). Note that, based on this definition, Q is also a random field parameterized by x and t. For each realization of a turbulent flow, Q will be different, and we can define its expected value using the probability distribution for the ensemble of realizations.12 Nevertheless, the expected value of the convected derivative of Q can be expressed in terms of partial derivatives of the one-point joint velocity, composition PDF 13... 

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

The variation d/(x) of an n-multivariate scalar function /(x) along a displacement direction x can be written as... 

Using the guessed value of F and the resulting controller settings, a multivariable Nyquist plot of the scalar function = 1 -i- Det [/ +... 

In the absence of friction, there are two forces acting on the mass m whose position vector at time t is denoted by the vector r[r] measured relative to the support point, which is the origin of a set of Cartesian axes with three-component k in the upward vertical direction. The first is the force of gravity on the mass, which acts downwards with a value —mgk. The second is the centripetal force, unknown for the moment, which is directed along the support towards the universal point. We denote this force by — Tr t, where Tis a scalar function of time to be found. The Newtonian equations of motion can then be written as... 

For / = 0, Eq. (7.1) simply represents the integral over the charge distribution, which is the total charge—a scalar function described as the monopole. The higher moments are, in ascending order of / the dipole, a vector the quadrupole, a... 

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

Suppose that, given a scalar function xp and an arbitrary constant vector c, we construct a vector function M ... 

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