By G. Stephenson
This textbook presents a superior starting place to a few very important themes in arithmetic of curiosity to technology and engineering scholars. incorporated are tensor algebra, usual differential equations, contour integration, Laplace and Fourier transforms, partial differential equations and the calculus of adaptations. The authors' procedure is easy and direct with an emphasis at the analytical realizing of the cloth. The textual content is almost selfcontained, assuming in simple terms that the scholar has a superb figuring out of ancillary arithmetic. every one bankruptcy incorporates a huge variety of labored examples, and concludes with difficulties for resolution, with solutions behind the book
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Extra info for Advanced mathematical methods for engineering and science students
Tensor algebra and its associated calculus are important tools in the study of continuum mechanics and in the general theory of relativity. Problems 1 1. Write out aikXiXk in expanded form, assuming aik — akiy and i, k = 1, 2, 3. 2. Over which indices (if any) in the following expressions is summation implied? (i) dijbj, (ii) dijbjj, (iii) atibny (iv) au = bu. 3. Find the values of 6iy6iy, <5,y(5yVAm<5/m, ejklAkAh and dikeikmy all indices ranging from 1 to 3. 4. Evaluate €iklejki and €ijkeijky all indices ranging from 1 to 3.
186) We also have the following integral: f1 j-i 2 Pn(x)Pm(x)dx=——dnm. 7. Further, the associated Legendre functions are defined by dm+n - n v-/ where 2 n n! 188) i AZ. In Chapter 8 we shall meet the equation The solutions are bounded only ifA = - « ( « 4 - l ) and have the form y = PjT'Ccos 6). 190) 48 Special functions 4. Laguerre polynomials The Laguerre equation x2 —^ + (1 - x) - / + ny = 0, dx dx where n = 0, 1, 2, . . 191) called the n! dx In particular L0(JC) = 1, L^*) = 1 - x9 2 L2(x) = \(x - 4x + 2).
For example, A1 ± Bl = —. 156) so that if A1 and Bl are contravariant vectors their sum and difference are also contravariant vectors. Similarly, it can be seen from the appropriate transformation law that the sum and difference of two tensors of the same type are tensors of that type. An important property of all tensor equations is that if the components of a tensor are zero in one coordinate system, then they remain zero in any other coordinate system (as can be seen from the transformation laws).
Advanced mathematical methods for engineering and science students by G. Stephenson