The Chemistry Maths Book, Second Edition

290 Chapter 9Functions of several variables 20.For the van der Waals equation Find (i) , (ii) , (iii) , (iv). Section 9.4 Find t ...

9.12 Exercises 291 Section 9.6 37.Givenz 1 = 1 x 2 1 + 12 xy 1 + 13 y 2 , wherex 1 = 1 (1 1 + 1 t) 122 andy 1 = 1 (1 1 − 1 t) 12 ...

292 Chapter 9Functions of several variables Show that the following functions of position in a plane satisfy Laplace’s equation: ...

9.12 Exercises 293 63.Evaluate on the circle with parametric equationsx 1 = 1 cos 1 θ, y 1 = 1 sin 1 θ, (i)from A(1, 0) to B( 0, ...

10 Functions in 3 dimensions 10.1 Concepts A function of three variables,f(x, y, z), in which the variables are the coordinates ...

10.2 Spherical polar coordinates 295 Other coordinate systems are normally defined in terms of the cartesian system. The system ...

296 Chapter 10Functions in 3 dimensions The conversion from cartesian coordinates to spherical polar coordinates makes use of th ...

10.3 Functions of position 297 The temperature field is therefore described equally well by the function g(r, 1 θ, 1 φ) 1 = 1 r ...

298 Chapter 10Functions in 3 dimensions and the value of this limit is the density at the point P. We note that although ρ is de ...

10.4 Volume integrals 299 The physical interpretation of an orbital is in terms of an electron probability density; for an elect ...

300 Chapter 10Functions in 3 dimensions The factorization of the triple integral is allowed because the integrand is the product ...

10.4 Volume integrals 301 EXAMPLE 10.6Find the volume∆vin Figure 10.6 and show that it reduces to the approximate expression (10 ...

302 Chapter 10Functions in 3 dimensions Integrals over all space When the region Vis the whole three-dimensional space, the volu ...

10.4 Volume integrals 303 Average values We shall see in Chapter 21 that if xis a continuous variable in the intervala 1 ≤ 1 x 1 ...

304 Chapter 10Functions in 3 dimensions and, after integration over the angles (see equation (10.13)), (ii) 0 Exercises 23, 24 1 ...

10.5 The Laplacian operator 305 of electromagnetic theory) and quantum mechanics (as in Schrödinger’s wave mechanics, see Chapte ...

306 Chapter 10Functions in 3 dimensions EXAMPLE 10.11Evaluate∇ 2 fforf(r, 1 θ, 1 φ) 1 = 1 re −r 22 sin 1 θ 1 cos 1 φ. Using the ...

10.6 Other coordinate systems 307 This result can be rearranged to give the equation and this is essentially the Schrödinger equ ...

308 Chapter 10Functions in 3 dimensions (parallel to the Oyz–plane), y 1 = 1 constant (parallel to the Ozx–plane), and z 1 = 1 c ...

10.6 Other coordinate systems 309 whose coordinate lines (and surfaces) are mutually perpendicular (orthogonal) at their point o ...