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=lim :F(Q +
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n) - :F(Q).
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In vectorial calculus, such a directional derivative is given by the scalar product of the gradient with the unit vector that defines the given direction. Here, too, a quaternion gradient can be introduced in an analogous way. 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) as
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~: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)
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where S(AB) denotes the scalar part of the quatemion product AB and
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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 simply
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Now as in Appendix E, consider the flux of a quatemion function F through the surface of this hypercube:
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+ 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).
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Expanding 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 that
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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]:
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 when
dz I(z) =
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,
= 0.
Similarly, we call calF a right regular function in a domain D if for every closed hypersurface 8a in D,
= 0.
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 as
810 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 reads
8 I o =\7.f, 8xo 8f \710 = --8