Derivatives vector functions

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.)... 

In Section XIV.A.2, we intend to obtain the vector function x q, 0) for a given distribution of conical intersections. Thus, first we have to derive an expression for a conical intersection removed from the origin, namely, assumed to be located at some point, ( /),0jo), in the plane. 

If V is the volume bounded by a closed surface S and A is a vector function of position with continuous derivatives, then... 

To derive the boundary conditions, we introduce some vector function X(p) ... 

Clearly grad is a vector function whose (x, y, z) components are the first partial derivatives of . The gradient of a vector function is undefined. Consider an infinitesimal vector displacement such that... 

The first of these expresses the condition that the centripetal constraint force does no work, because the velocity is perpendicular to the radius. The second states that the radial component of the acceleration is directed inwards and equal to the square of the speed. This relation can be used to calculate the constraint force by taking the scalar product of the equation of motion (2) with the vector function r[t], and using the constraint and its time derivatives to obtain... 

From the various versions of this method we will choose only one. Let V < 0 and, only at the rest point under study c, V - 0. Then let Vhave its minimum, V(c) = at the point c and for some e > Vmin the set specified by the inequality V(c0) < e is finite. Therefor any initial conditions c0 from this set the solution of eqn. (73) is c(t, k, c0) - c at t - oo. V(c) is called a Lyapunov function. The arbitrary function whose derivative is negative because of the system is called a Chetaev or sometimes a dissipative function. Physical examples are free energy, negative entropy, mechanical energy in systems with friction, etc. Studies of the dissipative functions can often provide useful information about a given system. A modern representation for the second Lyapunov method, including a method of Lyapunov vector functions, can be found in ref. 20. 

Note that Lgh(x) is itself a scalar function of x and that Lgk(x) IR —> IR. Consequently, we can calculate its directional derivative along the vector function g, as... 

In fact, two types of vector function can be derived from the spherical harmonics, and these are defined by... 

In order to make these qualitative statements more precise, we will derive all the electromagnetic field components of a fundamental Gaussian beam from two vector functions written in cylindrical coordinates as... 

In this formula, I is the nuclear quantum number, r(Bj) the first derivative lineshape function, B the resonance position and P the transition probability. 0 and i are the Euler angles expressing the orientation of the magnetic field vector B with respect to the principal axes of the tensors. Integration is needed since in powder samples, the crystallites take all possible orientations with respect to the magnetic field. Since the principal tensor axes and the crystal axes are assumed to be coincident, integration can be restricted to one octant of the unit sphere. 

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

There are two principal vector derivatives of vector functions. The divergence of F is defined in Cartesian coordinates by... 

Show that for a potential function, the curl is a zero vector. A DERIVE session is given in Table 1.3. Further, we can express homogeneity property of the energy U S, V, n) as a vector function with the set of extensive variables as the scalar product of the vector e with the gradient of U(e)... 

Note that the entities H, F, g, and their partial derivatives are functions of y and u. Such an entity, say, H, is denoted by H when evaluated at the optimum, i.e., for the vector of optimal controls u and the corresponding vector of optimal states y. 

Two choices are possible for defining the first derivative of a vector function (i.e. each point in space is associated with a vector). The divergence is denoted with V and produces a scalar. 

We now turn to the molecular derivation of the equation of change for angular momentum The quantity [rf x pj ] is the angular momentum of a bead with respect to some arbitrarily chosen fixed reference frame. The beads are regarded as point particles, and hence possess no intrinsic angular momentum. Consequently, to obtain Eq. (9.1) from the statistical mechanical approach, we consider the following vector function B in the phase spaces = E K x p ] 5 (r - r) (9.4)... 

With these definitions of vectors y and derivative vectors, the first and second derivative vectors can be written in terms of the function vector y using matrix notation... 

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