Physical Chemistry Third Edition

1242 B Some Useful Mathematics z kAz kAz jAy 0 A 5 iAx 1 jAy 1 kAz iAx 1 jAy x iAx jy y Figure B.5 The Vector A Represented by t ...

B Some Useful Mathematics 1243 The scalar product of a vector with itself is the square of the magnitude of the vector: A·A|A|^ ...

1244 B Some Useful Mathematics or aCartesian tensor. Each of its components is multiplied by a product of two unit vectors. Thed ...

B Some Useful Mathematics 1245 B.4 Solution of a Differential Equation from the Two-Step Mechanism of Chapter 11 Equation (11.5- ...

1246 B Some Useful Mathematics Thedtterm gives no contribution on the second leg of the path sincetis constant and dtvanishes on ...

B Some Useful Mathematics 1247 where we use the common symbolrfor the magnitude ofz. If a complex number is written in the form ...

1248 B Some Useful Mathematics Therefore aj ∫ fj∗fjdqaj∗ ∫ fj∗fjdq (B-73) and aj∗aj (B-74) A quantity equals its complex conju ...

B Some Useful Mathematics 1249 B.7 Matrices and Determinants Amatrixis an array or list of quantities arranged in rows and colum ...

1250 B Some Useful Mathematics If two matrices are square, they can be multiplied together in either order. However, the multipl ...

B Some Useful Mathematics 1251 The trace of thenbynidentity matrix is equal ton. The trace is sometimes called the spur, from a ...

1252 B Some Useful Mathematics n−1byn−1 determinant. To determine the sign of a given term in the expansion, count the number of ...

B Some Useful Mathematics 1253 Property 7. The determinant of a matrix is equal to the determinant of the transpose of that matr ...

1254 B Some Useful Mathematics We have incorporated thea 0 term into the first sum, using the fact that cos(0)1. We have also u ...

B Some Useful Mathematics 1255 If convergence is fairly rapid, it is possible to approximate a Fourier series by one of its part ...

1256 B Some Useful Mathematics integrals converge. For the integral of Eq. (B-118) to converge, the following integral must conv ...

C. A Short Table of Integrals C.1 Indefinite Integrals ∫ sin^2 (x)dx x 2 − sin(2x) 4 ∫ cos^2 (x)dx x 2 + sin(2x) 4 ∫ cosn(x)si ...

1258 C A Short Table of Integrals ∫ xeaxdxeax ( x a − 1 a^2 ) ∫ x^2 eaxdxeax ( x^2 a − 2 x a^2 + 2 a^3 ) ∫ ln(x)dxxln(x)−x ∫ ...

C A Short Table of Integrals 1259 ∫∞ 0 x^2 e−ax 2 dx 1 4 √ π a^3 , a> 0 ∫∞ 0 x^3 e−ax 2 dx 1 2 a^2 , a> 0 ∫∞ 0 x^2 ne−ax ...

1260 C A Short Table of Integrals Table C.1 Table of Values of erf(z) z 0123456789 0.0 0.0 000 113 226 338 451 564 676 789 901 * ...

D. Some Derivations of Formulas and Methods This appendix expands on some of the things presented without derivation. D.1 Carath ...