By G. Stephenson

This textbook offers an excellent starting place to a couple of very important issues in arithmetic of curiosity to technological know-how and engineering scholars. integrated are tensor algebra, usual differential equations, contour integration, Laplace and Fourier transforms, partial differential equations and the calculus of diversifications. The authors' strategy is straightforward and direct with an emphasis at the analytical knowing of the cloth. The textual content is almost selfcontained, assuming simply that the scholar has an effective realizing of ancillary arithmetic. every one bankruptcy features a huge variety of labored examples, and concludes with difficulties for resolution, with solutions behind the ebook.

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**Example text**

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.

157) can be solved in terms of the modified Bessel functions. 158) /0(JC) and K0(x). y(x) = Axko(x) + Bx±K0(x). 160) is y(x) =Ax%(x/2) + Bx*Kv(x/2). 163) Summary of the main properties of special functions 1. Bessel functions Bessel's equation of order v is x2 —^ + x j - + (x2 - v2)y = 0. 103)) v-2 x4 2 2. 4 , 2v „ x6 " 2 . 4 2 . 24)). The generating function is e4x(/-i/«)= 2 *"/„(*). 171) (note dJ0(x)/dx = -J{(x)). 174) 2. 175) are the modified Bessel functions Iv(x) and Kv{x), where Iv(x) = rvJv(ix), Ky(x) = - iv+1[iYv(ix) + Jv(ix)].

V)B, where A and B are vectors. 9. If fk = xixjeijk + XiXiXk> show that dfk/dxs = 2xsxk + XiXtdte. Find also d2fk/dxr dxs, and deduce that d2fkldx2s = 2xk + 4xs8ksy no summation over s being implied. Verify the last result directly for the cases k = 1, s = 1, and k = 1, s = 2. 10. If Ak =xnxnxmxmxk, show that dAk/dxp = ocxpxkxmxm + xnxnxmxmdkp, and determine the constant a. Show also that = fixnxnxmxm and determine /3 for the cases (i) when all indices range from 1 to 3, and (ii) when all indices range from 1 to 7.