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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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The small PCRk indicates that the process is likely to produce currents outside of the speci cation limits. From the normal distribution in Appendix Table II P1X P1X LSL2 USL2 P1Z P1Z 1110 190 1072 1.52 1072 1.52 P1Z P1Z 11.332 22 0

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For this example, the relatively large probability of exceeding the USL is a warning of potential problems with this criterion even if none of the measured observations in a preliminary sample exceed this limit. We emphasize that the fraction-nonconforming calculation assumes that the observations are normally distributed and the process is in control. Departures from normality can seriously affect the results. The calculation should be interpreted as an approximate guideline for process performance. To make matters worse, and need to be estimated from the data available and a small sample size can result in poor estimates that further degrade the calculation. Montgomery (2001) provides guidelines on appropriate values of the PCR and a table relating fallout for a normally distributed process in statistical control to the value of PCR. Many U.S. companies use PCR 1.33 as a minimum acceptable target and PCR 1.66 as a minimum target for strength, safety, or critical characteristics. Some companies require that internal processes and those at suppliers achieve a PCRk 2.0. Figure 16-15 illustrates a process with PCR PCRk 2.0. Assuming a normal distribution, the calculated fallout for this process is 0.0018 parts per million. A process with PCRk 2.0 is referred to as a sixsigma process because the distance from the process mean to the nearest speci cation is six standard deviations. The reason that such a large process capability is often required is that it

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(C.26)

1.5

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is dif cult to maintain a process mean at the center of the speci cations for long periods of time. A common model that is used to justify the importance of a six-sigma process is illustrated by referring to Fig. 16-15. If the process mean shifts off-center by 1.5 standard devia1.5. Assuming a normally distributed process, the tions, the PCRk decreases to 4.5 3 fallout of the shifted process is 3.4 parts per million. Consequently, the mean of a 6-sigma process can shift 1.5 standard deviations from the center of the speci cations and still maintain a fallout of 3.4 parts per million. In addition, some U.S. companies, particularly the automobile industry, have adopted the terminology Cp PCR and Cpk PCRk. Because Cp has another meaning in statistics (in multiple regression) we prefer the traditional notation PCR and PCRk. We repeat that process capability calculations are meaningful only for stable processes; that is, processes that are in control. A process capability ratio indicates whether or not the natural or chance variability in a process is acceptable relative to the specifications.

Now as in Appendix E, consider the flux of a quatemion function F through the surface of this hypercube:

EXERCISES FOR SECTION 16-7

+ 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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