Partial derivative of function

A A data list in Fig. 9b fu Partial derivative of function / with... 

Main objective in this paper is the revision and application of parametric methods in imcertainty propagation. However, before applying these methods we must ensure that output vector really follows a normal multivariate distribution. If no information is available about the joint density function in the output, the widespread procedure to ensure this assumption is by means of a multinormal contrast of goodness of fit. Fortunately, following theorem ensures the as3nnptotic j oint normal distribution o f the output vector, when this normality is fulfilled in the input vector, under some weakly conditions about the differentiability of functions in the simulator, providing also the mean vector and covariance matrix of the output vector as functions of their equivalents parameters in the input and the partial derivates of functions in the simulator. 

J matrix of partial derivatives of function with respect to parameters (Chapter 2) Jacobian matrix ... 

DFA Partial derivative of the Rachford-Rice objective function (7-13) with respect to the vapor-feed ratio. 

Change in extract-feed ratio from one iteration to the next. Partial derivative of Rachford-Rice objective function with respect to extract-feed ratio. 

P, Jy, and J , are the components of the total orbital angular momentum J of the nuclei in the IX frame. The Euler angles a%, b, cx appear only in the P, P and P angular momentum operators. Since the results of their operation on Wigner rotation functions are known, we do not need then explicit expressions in temis of the partial derivatives of those Euler angles. 

Cartesian coordinates, the vector x will have 3N components and x t corresponds to the current configuration of fhe system. SC (xj.) is a 3N x 1 matrix (i.e. a vector), each element of which is the partial derivative of f with respect to the appropriate coordinate, d"Vjdxi. We will also write the gradient at the point k as gj.. Each element (i,j) of fhe matrix " "(xj.) is the partial second derivative of the energy function with respect to the two coordinates r and Xj, JdXidXj. is thus of dimension 3N x 3N and is... 

Function F is identical with G because the summation term is zero. However, the partial derivatives of F and G with respect to are different, because function F incorporates the constraints of the material balances. 

The partial derivative of the material balance function with regard to concentration can be measured because f... 

Note that if Bn is zero, then T13 and T23 are also zero, so Equation (5.81) reduces to the specially orthotropic plate solution. Equation (5.65), if D11 =D22- Because Tn, T12, and T22 are functions of both m and n, no simple conclusion can be drawn about the value of n at buckling as could be done for specially orthotropic laminated plates where n was determined to be one. Instead, Equation (5.81) is a complicated function of both m and n. At this point, recall the discussion in Section 3.5.3 about the difference between finding a minimum of a function of discrete variables versus a function of continuous variables. We have already seen that plates buckle with a small number of buckles. Consequently, the lowest buckling load must be found in Equation (5.81) by a searching procedure due to Jones involving integer values of m and n [5-20] and not by equating to zero the first partial derivatives of N with respect to m and n. 

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. 

Referring to the earlier treatment of linear least-squares regression, we saw that the key step in obtaining the normal equations was to take the partial derivatives of the objective function with respect to each parameter, setting these equal to zero. The general form of this operation is... 

Thus, z and pc can be approximated by finding the intersection of the functions Y N + 1) and Y N) the iV —) 00 result is obtained by extrapolation. The other critical exponents may be obtained through scaling of the corresponding partial derivatives of t and the usual scaling relations. 

All the coefficients will, in general, be functions of both independent variables, and since we know that the heat absorbed depends on the path of change, it follows that the coefficients are not, in general, partial derivatives of a function of the two independent variables, for SQ would then be a perfect differential (cf. H. M., 115). 

Vector VS(k) contains the first partial derivatives of the objective function S(k) with respect to k and it is often called the gradient vector. For simplicity, we denoted it as g(k) in this chapter. 

Thus far in this chapter, functions of only a single variable have been considered. However, a function may depend on several independent variables. For example, z — f(x,y), where x and y are independent variables. If one of these variables, say y, is held constant the function depends only on x. Then, the derivative can be found by application of the methods developed in this chapter. In this case the derivative is called the partial derivative of z with respect to jc, which is represented by dz/dx or Bf/Bx. The partial derivative with respect to y is analogous. The same principle can be applied to implicit functions of several independent variables by the method developed in Section 2.5. Clearly, the notion of partial derivatives can be extended to functions of any number of independent variables. However, it must be remembered that when differentiating with respect to a given independent variable, all others are held constant. 

In general, the matrix, known as the Jacobian, contains entries for the partial derivative of each residual function Rj with respect to each unknown variable x,-. For a system of n equations in n unknowns, the Jacobian is an n x n matrix with n2 entries ... 

To do so, we calculate the Jacobian matrix, which is composed of the partial derivatives of the residual functions with respect to the unknown variables. Differentiating the mass action equations for aqueous species Aj (Eqn. 4.2), we note that,... 

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... 

For conservative systems, it is possible to define another quantity, the potential energy V, which is a function of the coordinates x,yi,Zj of all particles [i = 1 — n). The force components acting on each particle are equal to the negative partial derivatives of the potential energy with respect to the coordinates... 

---