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Viewing F(Q) as F(qo, ql, q2, q3), where Q qO+ql i+q2j+q3k, we can expand:F( Q+ n) = :F(qo+ no, ql nl, q2+ n2, q3 + n3) in a Taylor series around:F(Q) = :F(qO, q}, q2, q3) and rewrite (C.23) asControl datamatrix data in .netto develop data matrix and datamatrix 2d barcode data, size, image with .net barcode sdk~:F(Q) = lim :F(Q+ n)-:F(Q) on -+0 o 0F o = lim {nooo:F +n 1 :F +n2 :F +n3 +O( 2)} 0 Xl 0 X2 0 X3 -+0 Xo =S(nO):F(Q), (C.24)Control data matrix barcodes data for vbdata matrix data for vb.netIOThat is, n is such that fin Code 39 Full ASCII barcode library on .netusing barcode generator for vs .net control to generate, create 3 of 9 barcode image in vs .net applications.= 1.Develop qr barcode in .netgenerate, create denso qr bar code none in .net projectsQUATERNIONS AND SPECIAL RELATIVITY Deploy ansi/aim code 128 for .netusing .net vs 2010 crystal toconnect code 128 code set b on asp.net web,windows applicationwhere S(AB) denotes the scalar part of the quatemion product AB and .net Framework barcode integration for .netusing barcode integrated for .net framework control to generate, create bar code image in .net framework applications.v == - f ) +Z-f) +J-f) + - f ) Xo Xl X2 X3 .net Framework planet encoder in .netgenerate, create postal alpha numeric encoding technique none with .net projects(C.25)Bar Code generation on visual c#.netuse visual studio .net barcode drawer tomake bar code with c#is the quatemion gradient. To obtain the quatemion analog of the Cauchy-Riemann equations, we first derive two quatemion integral theorems that will be very useful in 3. These are derived in a very similar way to our derivation of Gauss's theorem in Appendix E. Consider an infinitesimal hypercube (a four-dimensional cube) of sides dxo, dXl, dx 2, and dX3, whose lower comer is located at xo, Xl> X2, X3. To simplify the notation, from now on we drop Hamilton's historical symbols i, j, k and instead adopt the symbols il> i2, and i3 for the quatemion imaginary units. This not only prevents people confusing the quatemion i with the complex imaginary unit i, or the quatemion imaginary units i, j, k with the Cartesian unit vectors i, j, k, but also allows for much compacter expressions. The mUltiplication rules for the quatemion imaginary units, for example, become simplyControl datamatrix 2d barcode size for .net gs1 datamatrix barcode size on .net(C.26)Control code 128 data on visual basic.net code-128 data for visual basicNow as in Appendix E, consider the flux of a quatemion function F through the surface of this hypercube:Control gs1-128 data for excelto add ucc-128 and ucc ean 128 data, size, image with excel spreadsheets barcode sdk+ dx l dx 2dx 3 F(xo + dxo, Xl, X2, X3) -dxo dx 2dx 3 iIF(xo, Xl. X2, x3)+dxodx2dx3 iIF(xo, Xl + dxl. X2, X3) -dxodxldx3 i 2F(xo,xl. X2,X3)+dxodx l dx 3 i 2F(xo,Xl, X2 + dX2, X3) -dxodxldx2 i 3F(xo,XI,X2,X3)+dxodx l dx 2 i 3F(xo,Xl, X2, X3 + dX3).Control pdf-417 2d barcode data in .net barcode pdf417 data in .net-dXldx2dx3 F(xo, Xl. X2, X3)UPC Code barcode library for .netgenerate, create universal product code version a none in .net projects(C.27)Display universal product code version a in visual basicgenerate, create upc-a supplement 2 none in visual basic projectsExpanding terms such as F(xo + dxo, Xl, X2, X3) in a Taylor series about F(xo, Xl, X2, X3) and discarding terms higher than second order in the infinitesimal displacements, we find thatBarcode barcode library for javause ireport barcode implement toembed bar code with java(C.28)where d 4r == dxo dx l dX2 dX3. Notice that given a closed four-dimensional volume cr bounded by a hypersurface ocr, we can obtain the flux through Ocr adding infinitesimal hypercubes. The flux through adjacent hypercube walls will cancel out, leaving only the flux through the outer hypercube walls that form the hypersurface Ocr. Thus, we have obtained the following quatemion integral theorem [146]:(C.29)QUATERNION CALCULUS Repeating this derivation for the flux :F dQ through an infinitesimal hypercube, we obtain another integral theorem, [146] (C.30) In the complex plane, a function I(z) is analytic in a domain D whendz I(z) =(C.31)for every closed contour C in D. Analogously, we define for quaternion functions the ideas of left regular and right regular (now the order is important). We call calF a left regular function in a domain D if for every closed hypersurface 8a in D,loCT dQ:F = 0.(C.32)Similarly, we call calF a right regular function in a domain D if for every closed hypersurface 8a in D,loCT :FdQ = 0.(C.33)In complex analysis, (C.31) leads to the Cauchy-Riemann equations. Here, left regularity leads to O:F = 0, which can be written in terms of components as810 8xo \710= \7. f, = - - - \7 /\f,8f 8xo (C.34) (C.35)where :F == 10 + En inln and f is the three-dimensional vector formed with the imaginary components of the quaternion:F. Right regularity means that :FO = 0, which in terms of components reads8 I o =\7.f, 8xo 8f \710 = --8(C.36)