The Chemistry Maths Book, Second Edition

250 Chapter 9Functions of several variables the slopes of the tangent lines to the curve DPE. The two tangent lines at point P i ...

9.3 Partial differentiation 251 EXAMPLES 9.2Partial differentiation (i) f(x,y,z) 1 = 1 x 2 1 + 12 y 2 1 + 13 z 2 1 + 14 xy 1 + 1 ...

252 Chapter 9Functions of several variables and each of these can be differentiated with respect to either variable to give four ...

9.4 Stationary points 253 EXAMPLES 9.3More partial derivatives (i) The nonzero partial derivatives ofu 1 = 1 x 1 + 1 y 2 1 + 12 ...

254 Chapter 9Functions of several variables EXAMPLE 9.4Find the stationary points of the function f(x, y) 1 = 1 x 3 1 + 16 xy 2 ...

9.4 Stationary points 255 The determination of the nature of these points is summarized in the following table. Table 9.1 xyf xx ...

256 Chapter 9Functions of several variables The optimization problem in this example has been simplified by using the constraint ...

9.4 Stationary points 257 where a 1 , 1 a 2 ,1=, 1 a m are constants. The procedure is to construct the auxiliary function (9.13 ...

258 Chapter 9Functions of several variables subject to the constraint where theC ij are constants (withC ij 1 = 1 C ji ). For ex ...

9.5 The total differential 259 For example, for the quadratic functionax 2 1 + 1 bxy 1 + 1 cy 2 , z p 1 = 1 f(x, y) 1 = 1 ax 2 1 ...

260 Chapter 9Functions of several variables If∆xand∆yare small enough, the terms quadratic in ∆are small compared with the linea ...

9.5 The total differential 261 (ii) u 1 = 1 (x 2 1 + 1 y 2 1 + 1 z 2 ) 122 (iii) x 1 = 1 r 1 sin 1 θ 1 cos 1 φ 0 Exercises 31–35 ...

262 Chapter 9Functions of several variables One of the principal uses of the total differential in the physical sciences is in t ...

9.6 Some differential properties 263 EXAMPLE 9.12Givenz 1 = 1 x 2 1 + 1 y 3 , wherex 1 = 1 e t andy 1 = 1 e −t , finddz 2 dt. (i ...

264 Chapter 9Functions of several variables A special case of the total derivative (9.21) is obtained if tis replaced by x. Then ...

9.6 Some differential properties 265 and, after division by ∆x, In the limit∆x 1 → 10 , the ratio∆y 2 ∆xapproaches the derivativ ...

266 Chapter 9Functions of several variables (9.28) the equation can also be written as (9.29) This form is sometimes called the ...

9.6 Some differential properties 267 The representative surface of the functionr 1 = 1 (x 2 1 + 1 y 2 ) 122 is a vertical cone w ...

268 Chapter 9Functions of several variables This expression is closely related to that in Example 9.13 (walking on a circle) for ...

9.6 Some differential properties 269 The inverse relationships are obtained in the same way from (9.32): (9.34a) (9.34b) EXAMPLE ...