Derivatives, partial

Partial Derivatives Are the Slopes of the Tangents to Multivariate Functions 

The partial derivative is the slope of one particular tangent to the curt e at a given point. 

Derivatives of functions of several variables are partial derivatives, taken with respect to only one variable, with all the other variables held constant. For example, the partial derivatives of /(x,y) are defined by 

Just like ordinary derh ativcs, partial derivatives arc functions of the independent ariables. The can be further differentiated to yield second- and higher-order dcri atives. The following example shows how these rules are applied to the ideal gas law. We explore the physics of this law later. For now our purpose is Just to illustrate partial derivatives. 

If you know the value of fix) at some point x = a, you can use the Taylor series expansion Equation (4.22) to compute fix) near that point  

The Ihree partial derivatives of the equation of a conic section are  

Process engineering and design using Visual Basic 

When a variable such as energy U T, V, Nk) is a function of many variables y, T and N, its partial derivative with respect to each variable is defined by holding all other variables constant. Thus, if U T, V, N) = (5/2)NRT — aN /V then the partial derivatives are 

The subscripts indicate the variables that are held constant during the differentiation. In cases where the variables being held constant are understood, the subscripts are often dropped. The change in V, i.e. the differential dU, due to changes in N, V and T is given by 

For functions of many variables, there is a second derivative corresponding to every pair of variables d U/dTdV, d U/dNdV, d U/dT etc. For the cross derivatives such as d U/dT dV that are derivatives with respect to two different variables, the order of differentiation does not matter. That is 

The same is valid for all higher derivatives such as d U/dT dV, i.e. the order of differentiation does not matter. 

Most physical systems are characterized by more than two quantitative variables. Experience has shown that it is not always possible to change such quantities at will, but that specification of some of them will determine values for others. Functional relations involving three or more variables lead us to branches of calculus that make use of partial derivatives and multiple integrals. 

We have already snuck in the concept of partial differentiation in several instances by evaluating the derivative with regard to x of a function of the form f(x, y,..while treating y. as if they were constant quantities. The correct notation for such operations makes use of the curly dee symbol 9, for example. 

To begin with, we will consider functions of just two independent variables, sucb as z = f(x, y). Generalization to more than two variables is usually 

The subscript y or x denotes the variable that is held constant. If there is no ambiguity, tbe subscript can be omitted, as in Eq. (10.1). Some alternative notations for (dz/dx)y are dz/dx, df/dx, Zx A, and /0 )(x, y). As shown in Fig. 10.1, a partial derivative such as (dz/dx)y can be interpreted geometrically as the instantaneous slope at the point (x, y) of the curve formed by the intersection of the surface z = z(x, y) and the plane y = constant, and analogously for (dz/dy)x- Partial derivatives can be evaluated by the same rules as for ordinary differentiation, treating all but one variable as constants. 

Products of partial derivatives can be manipulated in the same way as products of ordinary derivatives provided that the same variables are held constant. 

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The procedure would then require calculation of (2m+2) partial derivatives per iteration, requiring 2m+2 evaluations of the thermodynamic functions per iteration. Since the computation effort is essentially proportional to the number of evaluations, this form of iteration is excessively expensive, even if it converges rapidly. Fortunately, simpler forms exist that are almost always much more efficient in application. 

DET Determinant of Jacobian partial derivative matrix whose... 

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

DGT Partial derivative of the enthalpy balance equation (7-14) with respect to the temperature. 

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

The expression hU/i n J.s>, n signifies, by common convention, the partial derivative of U with respect to the number of moles n- of a particular species, holdmg. S, V and the number of moles n.j of all other species (/ )... 

Equation (A2.1.26) is equivalent to equation (A2.1.25) and serves to identify T, p, and p. as appropriate partial derivatives of tire energy U, a result that also follows directly from equation (A2.1.23) and the fact that dt/ is an exact differential. 

All of these quantities are state fiinctions, i.e. the differentials are exact, so each of the coefficients is a partial derivative. For example, from equation (A2.1.35) p = —while from equation (A2.1.36)... 

We define the field intensity tensor Fi,c as a function of a so far undetermined vector operator X = Xj, and of the partial derivatives dt... 

Throughout, the space coordinates and other vectorial quantities are written either in vector fomi x, or with Latin indices k— 1,2,3) the time it) coordinate is Ap = ct. A four vector will have Greek lettered indices, such as Xv (v = 0,1,2,3) or the partial derivatives 0v- m is the electronic mass, and e the charge. 

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... 

The equality also holds if we take the partial derivative of both sides of Eq. (8.16) with respect to p. 

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

Partial Derivative The abbreviation z =f x, y) means that is a function of the two variables x and y. The derivative of z with respect to X, treating y as a constant, is called the partial derivative with respecd to x and is usually denoted as dz/dx or of x, y)/dx or simply/. Partial differentiation, hke full differentiation, is quite simple to apply. Conversely, the solution of partial differential equations is appreciably more difficult than that of differential equations. 

Also divide both numerator and denominator of a partial derivative by dw while bolding a variable y constant to get... 

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. 

For partial derivatives an indication of which variable is being transformed avoids confusion. Thus, if... 

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. 

If the changes are indeed small, then the partial derivatives are constant among all the samples. Then the expected value of the change, E dY), is zero. The variances are given by the following equation... 

In all these equations the partial derivatives are taken with composition held constant. 

Replacement of each of the four partial derivatives through the appropriate Maxwell relation gives finely... 

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 minimum value of both F and G is found when the partial derivatives of F with respecl to are set equal to zero ... 

The second class of multivariable optimization techniques in principle requires the use of partial derivatives, although finite difference formulas can be substituted for derivatives such techniques are called indirect methods and include the following classes ... 

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