The Chemistry Maths Book, Second Edition

270 Chapter 9Functions of several variables wheref 1 = 1 f(x, y)is a function of the cartesian coordinates of a point in a plane ...

9.6 Some differential properties 271 Then Similarly, Therefore 0 Exercise 49 EXAMPLE 9.19Show that the functionf 1 = 1 x 2 1 − 1 ...

272 Chapter 9Functions of several variables Therefore, 0 Exercises 50–52 EXAMPLE 9.20Show that the functionf 1 = 1 ln 1 r, where ...

9.7 Exact differentials 273 is exact when there exists a functionz 1 = 1 z(x, y)such that (9.44) The differential can then be eq ...

274 Chapter 9Functions of several variables EXAMPLE 9.22Maxwell relations It follows from equations (9.42) that so that This is ...

9.8 Line integrals 275 9.8 Line integrals Consider the functionF(x)and the integral (9.47) The integral was interpreted in Secti ...

276 Chapter 9Functions of several variables Line integrals are important in several branches of the physical sciences. For examp ...

9.8 Line integrals 277 (see Example 6.10 for ). The result is different from that obtained in Example 9.23. 0 Exercises 59, 60 T ...

278 Chapter 9Functions of several variables Therefore (ii) PathC 3 1 + 1 C 4 The work done along path C 3 is at constant pressur ...

9.8 Line integrals 279 the graphical representation of z(x, y)discussed in Section 9.2, the line integral is the change in ‘heig ...

280 Chapter 9Functions of several variables EXAMPLE 9.27Change in entropy in thermodynamics The change in the entropy when a the ...

9.9 Multiple integrals 281 Also, the Maxwell relation derived from the differential Gibbs energy is Therefore, Experimentally, t ...

282 Chapter 9Functions of several variables This is called a double integraland is evaluated by integrating first with respect t ...

9.10 The double integral 283 9.10 The double integral The double integral can be defined as the limit of a (double) sum in the s ...

284 Chapter 9Functions of several variables that is, the volume between the representative surface of the functionf(x,y)and the ...

9.11 Change of variables 285 and the integral (the sum of horizontal strips) is (9.58b) EXAMPLE 9.30Evaluate the integral off(x, ...

286 Chapter 9Functions of several variables such that Changes of variable are even more important for double (and higher) integr ...

9.11 Change of variables 287 The integral (9.59) is then EXAMPLE 9.31Evaluate the integral off(r, 1 θ) 1 = 1 e −r 2 sin 2 1 θove ...

288 Chapter 9Functions of several variables The general case The transformation from cartesian to polar coordinates is a special ...

9.12 Exercises 289 because the value of a definite integral does not depend on the symbol used for the variable of integration. ...