MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 6 pps

... use the approximations sin 6$ N S+, sin 6Cp N 6Cp (10. 96) cos6$ N 1, cos6Cp _N 1 to find 10 R1R2 = [ 6+ 1 =R2R,. (10.97) 6$ -64 Note that in terms of the generators ... choosing three mutually orthogonal straight lines. A point is defined by giving its coordinates, (q,z2,q), or by using the position vector 163 188 COORDINATES AND TENSORS Hence aa a...

Ngày tải lên: 13/08/2014, 09:21

... Cancelling m on both sides and noting that (9. 166 ) (9. 167 ) and using Equation (9. 164 ) we finally write Similarly We now define the ladder operators L+ and L- for the m index of ... yiL (6) by 9 .6. 3 Spherical harmonics are defined as Ladder Operators for the Spherical Harmonics Using Equation (9. 162 ) we write Using (9. 162 ) (9. 163 ) (9. 164 ) (9...

Ngày tải lên: 13/08/2014, 09:21

... AND ITS LIE ALGEBRA 245 and and introduce the parameters (1 1. 166 ) 8= vf8:+8;+8,2 and (11. 167 ) (1 1. 168 ) so that we can summarize these results as L = X .68 + v-pp (11. 169 ) ... that remain invariant under similarity transformations. One such 254 CONTINUOUS GROUPS AND REPRESENTATIONS Y’ Fig. 11 .6 (z,y,z) and the (z‘,y’,z‘) coordinates we obtain...

Ngày tải lên: 13/08/2014, 09:21

... useful in finding solutions of Laplace equation in two dimensions. 2. The method of analytic continuation is very useful in finding solutions of differential equations and evaluating some ... complex techniques are very helpful in certain problems of physics and engineering, which are essentially problems defined in the real domain, complex numbers in quantum mech...

Ngày tải lên: 13/08/2014, 09:21

... INTEGRALS Many of the definite integrals encountered in physics and engineering can be evaluated by using the complex integral theorems and analytic continuation: I. Integrals of the form ... 0 and a-lrnl # 0, then is called a singular point of order m. Definition I11 Essential singular point: If m is infinity, then a is called an essential singular point....

Ngày tải lên: 13/08/2014, 09:21

... Schrodinger Equation in the Presence of Interactions 20.8 Feynman Phase Space Path Integral 61 0 61 1 61 3 61 5 61 5 61 6 61 8 62 1 62 4 62 6 63 3 63 3 63 5 64 9 65 0 65 0 65 5 65 5 ... 562 564 565 567 567 567 569 569 572 572 573 575 579 582 583 584 593 594 5 96 597 60 3 60 6 60 6 60 9 19.2.3 Compounding Propagators 60 9 67 2...

Ngày tải lên: 13/08/2014, 09:21

... write x = fl in the generating function Equation (2 .65 ) we find (2.85) Expanding the left-hand side by using the binomial formula and comparing equal powers oft, we obtain 9 (1) = 1 ... part of the learning process. In a vast area like mathematical methods in science and engineering, there is always room for new approaches, new applications, and new topics...

Ngày tải lên: 13/08/2014, 09:21

... that the value of the integral in Equation (3.38) can be obtained by expanding in powers of t and s and then by comparing the equal powers of tnsm with the left-hand side of Equation ... left-hand side of Equation (4 .61 ) explicitly we get 00 &e-t2+2tx -2+2sx -12 e ex (4 .62 ) = e2st (s + t) ,h, (4 .63 ) where we have defined u=x-(s+t). (4 .64 ) Ex...

Ngày tải lên: 13/08/2014, 09:21

... obtain the following recursion relations and First by adding and then by subtracting these equations we also obtain the relations m Jm-l(X) = -Jm(IL.) + J:,(X) (6. 52) X and (6. 53) ... equations in physics and engineering can be solved by the method of separation of variables. This method helps us to reduce a second-order partial differential equation into...

Ngày tải lên: 13/08/2014, 09:21

... (r2sinog) +ae d (TaB) r2sinOaW +a (WE)] After simplifying, the Laplacian is obtained as r2sin6 dr 34 r2sin28 a4 ' (10.239) Here we have obtained a well-known formula in ... be expressed in terms of the poten- tials -A' and 4 as In the Lorentz gauge 2 and 4 satisfy and (10.373) (10.374) (10.375) (10.3 76) (10.377) respectively. Defining...

Ngày tải lên: 13/08/2014, 09:21