Second partial derivative

Harmonic Functions Both the real and the imaginary )arts of any analytic function/= u + iij satisfy Laplaces equation d /dx + d /dy = 0. A function which possesses continuous second partial derivatives and satisfies Laplace s equation is called a harmonic function. 

If ti satisfies necessaiy conditions [Eq. (3-80)], the second term disappears in this last line. Sufficient conditions for the point to be a local minimum are that the matrix of second partial derivatives F is positive definite. This matrix is symmetric, so all of its eigenvalues are real to be positive definite, they must all be greater than zero. 

Using the commutation of second partial derivative together with (A.3) and (A.4)... 

If V is a function of more than one variable, then more complex criteria for determining maxima and minima are obtained. Generally, but not always, the second partial derivatives of the function with respect to all its variables are sufficient to determine the character of a stationary value of V. For such functions, the theory of quadratic forms as described by Langhaar [B-1] should be examined. 

Following the procedures of Chapter 3, Section 4 and taking into account that the second partial derivative d ul di" is continuous on the line of discontinuity k = of the functions k[x, t) and f x,t), we deduce through such an analysis that... 

The Laplacian is constructed from second partial derivatives, so it is essentially a measure of the curvature of the function in three dimensions (Chapter 6). The Laplacian of any scalar field shows where the field is locally concentrated or depleted. The Laplacian has a negative value wherever the scalar field is locally concentrated and a positive value where it is locally depleted. The Laplacian of the electron density, p, shows where the electron density is locally concentrated or depleted. To understand this, we first look carefully at a onedimensional function and its first and second derivatives. 

It can be shown from a Taylor series expansion that if/(x) has continuous second partial derivatives, /(x) is concave if and only if its Hessian matrix is negative-semidefinite. For/(x) to be strictly concave, H must be negative-definite. For /(x) to be convex H(x) must be positive-semidefinite and for/(x) to be strictly convex, H(x) must be positive-definite. 

Newton s method makes use of the second-order (quadratic) approximation of fix) at x and thus employs second-order information about fix), that is, information obtained from the second partial derivatives of fix) with respect to the independent variables. Thus, it is possible to take into account the curvature of fix) at x and identify better search directions than can be obtained via the gradient method. Examine Figure 6.9b. 

It requires both first and second partial derivatives, which may not be practical to obtain. 

The Kuhn-Tucker necessary conditions are satisfied at any local minimum or maximum and at saddle points. If (x, A, u ) is a Kuhn-Tucker point for the problem (8.25)-(8.26), and the second-order sufficiency conditions are satisfied at that point, optimality is guaranteed. The second order optimality conditions involve the matrix of second partial derivatives with respect to x (the Hessian matrix of the... 

Here, the coefficients b represent first and second partial derivatives of the rate expression f(xx, x2 K) or functions thereof. 

In order to stably levitate an object, the net force on it must be zero, and the forces on the body, if it is perturbed, must act to return it to its original position. The object must be at a local potential minimum that is, the second derivatives with respect to all spatial coordinates of the potential must be positive. This may seem, at first sight, to be trivial to arrange. However, any system whose potential is a solution to Laplace s equation is automatically unstable A statement in words of Laplace s equation is that the sum of the second partial derivatives of the potential is zero, and so not all can be simultaneously positive. This has long been known for electrostatic potentials, having been stated by Earnshaw(n) Millikan s scheme for suspending charged particles is thus only neutrally stable, since the fields within a Millikan capacitor provide no lateral constraint. 

The F matrix and -q vector in equation (4) are defined as the second partial derivatives of the Lagrangian, taken at the zero perturbation strength ... 

There is no natural way to generalize the one-dimensional bisection method to solve this multidimensional problem. But it is possible to generalize Newton s method to this situation. The one-dimensional Newton method was derived using a Taylor expansion, and the multidimensional problem can be approached in the same way. The result involves a 3/V x 3/V matrix of derivatives, J, with elements 7y = dg, /dxj. Note that the elements of this matrix are the second partial derivatives of the function we are really interested in, E(x). Newton s method defines a series of iterates by... 

Due to the simple structure of the function f, its first and second partial derivatives are easy to compute... 

The following proposition justifies the rehance on spherical harmonics in spherically symmetric problems involving the Laplacian. To state it succinctly, we introduce the vector space C2 C 2(R3) continuous functions whose first and second partial derivatives are all continuous. 

It suffices to show that K Pl (V- -) = 0. So suppose that f e K fi (V" "), i.e., suppose that f and its first and second partial derivatives are continuous, that Df = 0 and that f is orthogonal to every solution obtained by separation of variables. We will show that f = 0. 

Here the first equality follows from the fact that f e C. The technical continuity condition on f and its first and second partial derivatives allows us to exchange the derivative and the integral sign (disguised as a complex scalar product). See, for example, [Bart, Theorem 31.7]. The third equality follows from the Hermitian symmetry of It follows that is an element... 

If we concentrate on a thin slice of the concentration profiles, then the rate of change of concentration in time in that slice is given by the net rate of diffusion in and the chemical rate of removal by reaction. The former is given by a term involving the second partial derivative with respect to space and the diffusion coefficient, the latter by the term k ab2. Thus, for the reactant A, we have the governing equation... 

The use of the conditions expressed by Equations (5.97>-(5.102) and the fact that each second partial derivative in Equation (5.105) is identical to the same derivative for the double-prime part shows that 82E = d2E", and therefore Equation (5.108) can be written as... 

The transitions between phases discussed in Section 10.1 are classed as first-order transitions. Ehrenfest [25] pointed out the possibility of higher-order transitions, so that second-order transitions would be those transitions for which both the Gibbs energy and its first partial derivatives would be continuous at a transition point, but the second partial derivatives would be discontinuous. Under such conditions the entropy and volume would be continuous. However, the heat capacity at constant pressure, the coefficient of expansion, and the coefficient of compressibility would be discontinuous. If we consider two systems, on either side of the transition point but infinitesimally close to it, then the molar entropies of the two systems must be equal. Also, the change of the molar entropies must be the same for a change of temperature or pressure. If we designate the two systems by a prime and a double prime, we have... 

The first partial derivative in this equation is equal to — Sm(H + ), where Sm(H + ) is the molar entropy of the hydrogen ion. To evaluate the second partial derivative in equation 4.1-10, we need to recall that the chemical potential of species is given by... 

The Schrodinger equation is a second-order partial differential equation, involving a relation between the independent variables x, y, z and their second partial derivatives. This kind of equation can be solved only in some very simple cases (for example, a particle in a box). Now, chemical problems are N-body problems the motion of any electron will depend on those of the other N — 1 particles of the system, because all the electrons and all the nuclei are mutually interacting. Even in classical mechanics, these problems must be solved numerically. 

There are four second partial derivatives of the function z(x, y). These are... 

This useful equation is known as the cross-derivative rule. There are nine second partial derivatives of a function h(x, y, z) of three variables. Calculating derivatives for a few of these functions can convince the reader that the cross-derivative rule also holds for such functions. Thus, using the notation of Eq. (14),... 

Problem definition requires specification of the initial state of the system and boundary conditions, which are mathematical constraints describing the physical situation at the boundaries. These may be thermal energy, momentum, or other types of restrictions at the geometric boundaries. The system is determined when one boundary condition is known for each first partial derivative, two boundary conditions for each second partial derivative, and so on. In a plate heated from ambient temperature to 1200°F, the temperature distribution in the plate is determined by the heat equation 8T/dt = a V2T. The initial condition is T = 60°F at / = 0, all over the plate. The boundary conditions indicate how heat is applied to the plate at the various edges y = 0, 0[Pg.86]

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