Chain rule

The chain rule is useful for composite functions, functions of veiriables that are themselves functions of other variables. Suppose that the variables x, y, and z in a function f(x,y,z) are each a function of another variable u, that is a = x(u),y = y(u), and z = z(u). Then f x,y,z) is a composite function / = f(x(u),y(u),z(u)). The total derivative df in terms of the variables ix,y,z) is 

Substitution of Equations (5.50) into Equation (5.49) produces the chain rule for partial differentiation  

We will show in Chapter 8 that the state of equilibrium in many laboratory experiments is defined by the total differential dG = 0, where the Gibbs free energy G(T, p) is a function of temperature T and pressure p. However, suppose that you want to know the relationship between the measurable quantities T and p at equilibrium when dG = 0. This is the basis for phase equilibria. How is the relationship of T and p defined at constant G We U return to the ph sics later. For now let s look at the mathematics. 

Let s look at a general case in which a function f(x,y,z) is constrained to be constant and therefore the total differential equals zero  

The variables x, y, and z are not independent of each other, owing to the constraint that df = 0. How are they related Let s look at the relations between pairs of variables. If both / and z are held constant, df and dz equal zero, and you have 

On the other hand, for an isochoric change dT = 0. This means that TdS = CvdT. It does not matter whether the heat capacity is a function of entropy and volume C = C (5, V, n) or a function of temperature and volume C = C (r, V, n). By the use of the chain rule, we can get 

In contrast, since the temperature is a function of both S and V, T S, V), the derivative with respect to the volume is more comphcated, namely 

Observe the shorthand notation fx and fy. We can also build further derivatives of higher order, dP/dy = f y and dQ/dx = fy. Since f y = fyx, 

This is the law of Schwarz, which will be discussed in detail in a separate section. Thermodynamics makes extensive use of the law of Schwarz. 

n) is the total differential of first order. The temperature, the pressure, and the chemical potential are to be treated as the partial derivatives of the energy 

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Reference [73] presents the first line-integral study between two excited states, namely, between the second and the third states in this series of states. Here, like before, the calculations are done for a fixed value of ri (results are reported for ri = 1.251 A) but in contrast to the previous study the origin of the system of coordinates is located at the point of this particulai conical intersection, that is, the (2,3) conical intersection. Accordingly, the two polar coordinates (adiabatic coupling term i.e. X(p (— C,2 c>(,2/ )) again employing chain rules for the transformation... 

In conjunction with the use of isoparametric elements it is necessary to express the derivatives of nodal functions in terms of local coordinates. This is a straightforward procedure for elements with C continuity and can be described as follows Using the chain rule for differentiation of functions of multiple variables, the derivative of a function in terms of local variables ij) can be expressed as... 

In many applications, derivative operators need to be expressed in spherical coordinates. In converting from cartesian to spherical coordinate derivatives, the chain rule is employed as follows ... 

Very often in experimental sciences and engineering functions and their derivatives are available only through their numerical values. In particular, through measurements we may know the values of a function and its derivative only at certain points. In such cases the preceding operational rules for derivatives, including the chain rule, can be apphed numerically. 

Partial derivatives of and may be related to each other by differentiating (5.187) using the chain rule... 

The chain rule of differentiation applied to (A.8) provides, in rectangular Cartesian coordinates,... 

For the input-to-hidden connections, we use the chain rule to find that... 

An alternative, simpler expression can be obtained as follows. Using the chain rule, the temperature derivative of the volume for constant value of the relaxation time is expressed as... 

Again using Euler s chain rule, the value in brackets equals the negative of the isochoric derivative of temperature with respect to pressure. Thus, Equation 24.16 simplifies to... 

Similar expressions can be derived for second spatial derivatives. The final form of the equations that result after a generalized coordinate transformation depends on the degree of differentiation by using the chain rule, i.e. on the treatment of the metrics x, x, and y. For more details we refer to the... 

The expressions for the various vector operators in spherical coordinates can be derived with the use of the chain rule. Thus, for example,... 

Cauchy function 276 Cauchy s ratio test 35-36 central forces 107,132-135 spherical harmonics 134-135 spherical polar coordinates 132-133 chain rule 37, 57, 160 character 153,195,197 orthogonality 197, 204 tables 198-200... 

This expression is a simple generalization of the argument deve Section 2.7. It, and its extension to functions of any number of is referred to as the chain rule . In many applications it is customs) one or more subscripts to the partial derivatives to specify the one variables that were held constant As an example, Eq. (45) becomes... 

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),... 

Before the results obtained in the previous section can be applied, it is necessary to describe briefly the method of underdetertmned multipliers. Given a function /(xi, 2,..., 0 of variables Xj, for which it is desired to find stationary values, the chain rule leads to the expression... 

Expressing Eqs. (32) and (33) with the use of the chain rule leads to three relations which must be simultaneously satisfied, viz. 

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