By A.L. Septier
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In Appendix B, we have examined the roots of complex number equations like z2 + 1 = 0 in 2-D space and found the roots to be at ; and -i. Using the analogous equation in 4-D space, we would consider Q2 + 1 = 0 and find an infinite number of solutions. We could draw the locus of these solutions in 3-D space when there was no real part (a = 0) for the quaternion with no real part, Q - ib + jc + kd and b2 + c2 + d2 = 1. These solutions form a unitary sphere centered on zero in the 3-D pure imaginary subspace of quaternions.
4f уЛ „(dfz\,. \dRRj . d/ . d/ R dR R dR dR \dRRJ CONCLUSION The gradient of a function of Я is a vector («„ /Ш) in the positive or negative radial direction. Now, consider a space point P + ΔΡ, which is displaced by dl = âxdx + â,dy + й-dz from the point P. 63) which can be written as the scalar product of two vectors: dV = (à, dV/dx + âv dV/dy+â. 64) 42 Chapter 2 Vector Analysis Geometrical Representation of the Gradient Operator in 2-D Suppose V(x, >·) describes a surface above the x,y plane.
Torrance (Eugene, OR: Wipf and Stock Publishers, 1996). A commemorative reprint. iii. org. Gravity Probe B, P. S. Wesson and M. Anderson, Nov. 3, 2008. iv. html. v. Murray R. Spiegel, Shaum's Outline series, Complex Variables: With an Introduction to Conformai Mapping and Its Applications (McGraw-Hill, 1999). Chapter Σ Vector Analysis LEARNING OBJECTIVES • Use vector algebra to carry out addition, subtraction, and the dot product and cross product, of vectors and to understand the results pictorially • Use fundamental orthogonal coordinate systems—Cartesian, cylindrical, and spherical coordinates—in the description of geometric configurations commonly encountered in the study offieldsand convert from one system to another • Use and interpret the "del" or V operator in computing spatial derivatives involving vectors, that is, the gradient, divergence, curl, and Laplacian • Derive and understand the divergence theorem and Stokes's theorem INTRODUCTION In electromagnetic engineering, in addition to scalar quantities, there are many quantities defined not only by their amplitudes but also by their directions, for instance, the electric field intensity, E(x, y, z, t).