Jacobian Matrices of Ray R̄_i with Respect to Incoming Ray R̄_i–1 and Boundary Variable Vector X̄_i

Abstract

In automated optical design systems, the Jacobian matrix of an optical quantity with respect to the system variables is generally estimated using the Finite Difference (FD) method.

Appendices

Appendix 1

To determine ${{\partial {\bar{\text{R}}}_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$ at a spherical boundary surface ${\bar{\text{r}}}_{\text{i}}$, the following terms are required in advance:

• (1) ${{\partial\upsigma_{\text{i}} } / {\partial \bar{\text{R}}_{{{\text{i}} - 1}} }}$, ${{\partial\uprho_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$ and ${{\partial\uptau_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$:

From Eq. (2.12), ${}^{\text{i}}{\bar{\text{P}}}_{\text{i}} = {}^{\text{i}}{\bar{\text{A}}}_{0} \,{\bar{\text{P}}}_{\text{i}} = ({}^{\text{0}}{\bar{\text{A}}}_{\text{i}} )^{ - 1} \,{\bar{\text{P}}}_{\text{i}} = \left[ {\begin{array}{*{20}c} {\upsigma_{\text{i}} } & {\uprho_{\text{i}} } & {\uptau_{\text{i}} } & 1 \\ \end{array} } \right]^{\text{T}}$. Thus,

$$ {\bar{\text{P}}}_{\text{i}} = {}^{\text{0}}{\bar{\text{A}}}_{\text{i}} \left[ {\begin{array}{*{20}c} {\upsigma_{\text{i}} } \\ {\uprho_{\text{i}} } \\ {\uptau_{\text{i}} } \\ 1 \\ \end{array} } \right]. $$

(7.56)

Differentiating Eq. (7.56) with respect to incoming ray ${\bar{\text{R}}}_{{{\text{i}} - 1}}$ yields

$$ \frac{{\partial {\bar{\text{P}}}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} = {}^{0}{\bar{\text{A}}}_{\text{i}} \left[ {\begin{array}{*{20}c} {{{\partial\upsigma_{\text{i}} } / {\partial {\bar{\text{R}}}_{{\text{i} - 1}} }}} \\ {{{\partial\uprho_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}} \\ {{{\partial\uptau_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}} \\ {\bar{0}} \\ \end{array} } \right], $$

(7.57)

where ${{\partial {\bar{\text{P}}}_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$ is given in Eq. (7.20). ${{\partial\upsigma_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$, ${{\partial\uprho_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$ and ${{\partial\uptau_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$ can then be determined from

$$ \left[ {\begin{array}{*{20}c} {{{\partial\upsigma_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} \, }}} \\ {{{\partial\uprho_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}} \\ {{{\partial\uptau_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}} \\ {\bar{0}} \\ \end{array} } \right] = \left( {{}^{0}{\bar{\text{A}}}_{\text{i}} } \right)^{ - 1} \,\frac{{\partial {\bar{\text{P}}}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}. $$

(7.58)

• (2) ${{\partial\upalpha_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$ and ${{\partial\upbeta_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$:

${{\partial\upalpha_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$ and ${{\partial\upbeta_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$ can be obtained by differentiating Eqs. (2.19) and (2.20), respectively, with respect to ${\bar{\text{R}}}_{{{\text{i}} - 1}}$, that is,

$$ \frac{{\partial {\upalpha }_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} = \frac{1}{{{\upsigma }_{\text{i}}^{ 2} + {\uprho }_{\text{i}}^{ 2} }}\left( {\upsigma}_{\text{i}} \frac{{\partial {\uprho }_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} - {\uprho }_{\text{i}} \frac{{\partial {\upsigma }_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} \right), $$

(7.59)

$$ \frac{{\partial\upbeta_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} = \frac{{\sqrt {\left( {\upsigma_{\text{i}}^{\text{2}} + \rho_{\text{i}}^{2} } \right)} }}{{(\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} +\uptau_{\text{i}}^{2} )}}\frac{{\partial\uptau_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} - \frac{{\tau_{\text{i}} }}{{(\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} +\uptau_{\text{i}}^{2} )\sqrt {\left( {\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} } \right)} }}\left( {\upsigma_{\text{i}} \frac{{\partial\upsigma_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} +\uprho_{\text{i}} \frac{{\partial\uprho_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}} \right). $$

(7.60)

• (3) ${{\partial {\bar{\text{n}}}_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$:

${{\partial {\bar{\text{n}}}_{\text{i}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$ can be obtained by differentiating Eq. (2.10) with respect to ${\bar{\text{R}}}_{{{\text{i}} - 1}}$ to give

$$ \frac{{\partial {\bar{\text{n}}}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} = \left[ {\begin{array}{*{20}c} {{{\partial {\text{n}}_{\text{ix}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}} \\ {{{\partial {\text{n}}_{\text{iy}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}} \\ {{{\partial {\text{n}}_{\text{iz}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}} \\ {\bar{0}} \\ \end{array} } \right] = {}^{0}{\bar{\text{A}}}_{\text{i}} \,\frac{{\partial ({}^{\text{i}}{\bar{\text{n}}}_{\text{i}} )}}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}, $$

(7.61)

where ${{\partial ({}^{\text{i}}{\bar{\text{n}}}_{\text{i}} )} / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$ is obtained by differentiating Eq. (2.8) with respect to ${\bar{\text{R}}}_{{{\text{i}} - 1}}$, yielding

$$ \frac{{\partial ({}^{\text{i}}{\bar{\text{n}}}_{\text{i}} )}}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} = {\text{s}}_{\text{i}} \left[ {\begin{array}{*{20}c} { - {\text{S}}\upbeta_{\text{i}} {\text{C}}\upalpha_{\text{i}} } \\ { - {\text{S}}\upbeta_{\text{i}} {\text{S}}\upalpha_{\text{i}} } \\ {{\text{C}}\upbeta_{\text{i}} } \\ 0 \\ \end{array} } \right]\frac{{\partial\upbeta_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} + {\text{s}}_{\text{i}} \left[ {\begin{array}{*{20}c} { - {\text{C}}\upbeta_{\text{i}} {\text{S}}\upalpha_{\text{i}} } \\ {{\text{C}}\upbeta_{\text{i}} {\text{C}}\upalpha_{\text{i}} } \\ 0 \\ 0 \\ \end{array} } \right]\frac{{\partial\upalpha_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}. $$

(7.62)

• (4) ${{\partial ({\text{C}}\uptheta_{\text{i}} )} / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$:

${{\partial ({\text{C}}\uptheta_{\text{i}} )} / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$ can be computed directly from Eq. (2.21) as

$$ \begin{aligned} \frac{{\partial ({\text{C}}\uptheta_{\text{i}} )}}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} & = - \left( {\bar{\mathcal{\ell }}_{{{\text{i}}\text{ - 1}}} \cdot \frac{{\partial {\bar{\text{n}}}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} + \frac{{\partial \bar{\mathcal{\ell }}_{{{\text{i}}\text{ - 1}}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} \cdot {\bar{\text{n}}}_{\text{i}} } \right), \\ & = - \, \left( {\ell_{{{\text{i}}\text{ - 1}{\text{x}}}} \frac{{\partial {\text{n}}_{\text{ix}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} + \ell_{{\text{i - 1y}}} \frac{{\partial {\text{n}}_{\text{iy}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} + \ell_{{{\text{i}}\text{ - 1}{\text{z}}}} \frac{{\partial {\text{n}}_{\text{iz}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}} \right) - \left( {\frac{{\partial \ell_{{{\text{i}} - \text{1}{\text{x}}}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}{\text{n}}_{\text{ix}} + \frac{{\partial \ell_{{{\text{i}} - \text{1}{\text{y}}}} }}{{\partial \bar{\text{R}}_{{{\text{i}} - 1}} }}n_{{\text{iy}}} + \frac{{\partial \ell_{{{\text{i}} - \text{1}{\text{z}}}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}{\text{n}}_{\text{iz}} } \right) \\ \end{aligned} $$

(7.63)

where ${{\partial {\text{n}}_{\text{ix}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$, ${{\partial {\text{n}}_{\text{iy}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$ and ${{\partial {\text{n}}_{\text{iz}} } / {\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}$ are given in Eq. (7.61).

Appendix 2

To determine ${{\partial \bar{\text{R}}_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$ at spherical boundary surface ${\bar{\text{r}}}_{\text{ i}}$, the following terms are required in advance

• (1) ${{\partial ({}^{0}{\bar{\text{A}}}_{\text{i}} )} / {\partial {\bar{\text{X}}}_{\text{i}} }}$ (a matrix with dimensions $4 \times 4 \times 9$):

The components of ${{\partial ({}^{0}{\bar{\text{A}}}_{\text{i}} )} / {\partial {\bar{\text{X}}}_{\text{i}} }}$ can be determined by differentiating ${}^{0}{\bar{\text{A}}}_{\text{i}}$ given in Eq. (2.9) with respect to boundary variable vector (Eq. (2.30))

$$ \bar{\text{X}}_{\text{i}} = \left[ {\begin{array}{*{20}c} {{\text{t}}_{\text{ix}} } & {{\text{t}}_{\text{iy}} } & {{\text{t}}_{\text{iz}} } & {\upomega_{\text{ix}} } & {\upomega_{\text{iy}} } & {\upomega_{\text{iz}} } & {\upxi_{{{\text{i}} - 1}} } & {{\upxi}_{\text{i}} } & {{\text{R}}_{\text{i}} } \\ \end{array} } \right]^{\text{T}} , $$

to give

$$ \frac{{\partial ({}^{0}{\bar{\text{A}}}_{\text{i}} )}}{{\partial {\text{t}}_{\text{ix}} }} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(7.64)

$$ \frac{{\partial ({}^{0}{\bar{\text{A}}}_{\text{i}} )}}{{\partial {\text{t}}_{\text{iy}} }} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(7.65)

$$ \frac{{\partial ({}^{0}{\bar{\text{A}}}_{\text{i}} )}}{{\partial {\text{t}}_{\text{iz}} }} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(7.66)

$$ \frac{{\partial ({}^{0}{\bar{\text{A}}}_{\text{i}} )}}{{\partial\upomega_{\text{ix}} }} = \left[ {\begin{array}{*{20}c} 0 & {{\text{C}}\upomega_{\text{iz}} {\text{S}}\upomega_{\text{iy}} {\text{C}}\upomega_{\text{ix}} + {\text{S}}\upomega_{\text{iz}} {\text{S}}\upomega_{\text{ix}} } & { - {\text{C}}\upomega_{\text{iz}} {\text{S}}\upomega_{\text{iy}} {\text{S}}\upomega_{\text{ix}} + {\text{S}}\upomega_{\text{iz}} {\text{C}}\upomega_{\text{ix}} } & 0 \\ 0 & {\text{S}\omega_{{\text{iz}}} \text{S}\omega_{{\text{iy}}} \text{C}\omega_{{\text{ix}}} - {\text{C}}\upomega_{\text{iz}} {\text{S}}\upomega_{\text{ix}} } & { - {\text{S}}\upomega_{\text{iz}} {\text{S}}\upomega_{\text{iy}} {\text{S}}\upomega_{\text{ix}} - {\text{C}}\upomega_{\text{iz}} {\text{C}}\upomega_{\text{ix}} } & 0 \\ 0 & {{\text{C}}\upomega_{\text{iy}} {\text{C}}\upomega_{\text{ix}} } & { - {\text{C}}\upomega_{\text{iy}} {\text{S}}\upomega_{\text{ix}} } & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(7.67)

$$ \frac{{\partial ({}^{0}{\bar{\text{A}}}_{\text{i}} )}}{{\partial\upomega_{\text{iy}} }} = \left[ {\begin{array}{*{20}c} { - {\text{C}}\upomega_{\text{iz}} {\text{S}}\upomega_{\text{iy}} } & {{C\omega }_{\text{iz}} {\text{C}}\upomega_{\text{iy}} {\text{S}}\upomega_{\text{ix}} } & {{\text{C}}\upomega_{\text{iz}} {\text{C}}\upomega_{\text{iy}} {\text{C}}\upomega_{\text{ix}} } & 0 \\ { - {\text{S}}\upomega_{\text{iz}} {S\omega }_{\text{iy}} } & {{\text{S}}\upomega_{\text{iz}} {\text{C}}\upomega_{\text{iy}} {\text{S}}\upomega_{\text{ix}} } & {{\text{S}}\upomega_{\text{iz}} {\text{C}}\upomega_{\text{iy}} {\text{C}}\upomega_{\text{ix}} } & 0 \\ { - {\text{C}}\upomega_{\text{iy}} } & { - {\text{S}}\upomega_{\text{iy}} {\text{S}}\upomega_{\text{ix}} } & { - {\text{S}}\upomega_{\text{iy}} {\text{C}}\upomega_{\text{ix}} } & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(7.68)

$$ \frac{{\partial ({}^{0}{\bar{\text{A}}}_{\text{i}} )}}{{\partial\upomega_{\text{iz}} }} = \left[ {\begin{array}{*{20}c} { - {\text{S}}\upomega_{\text{iz}} {\text{C}}\upomega_{\text{iy}} } & { - {\text{S}}\upomega_{\text{iz}} {\text{S}}\upomega_{\text{iy}} {\text{S}}\upomega_{\text{ix}} - \text{C}\omega_{{\text{iz}}} \text{C}\omega_{{\text{ix}}} } & { - {\text{S}}\upomega_{\text{iz}} {\text{S}}\upomega_{\text{iy}} {\text{C}}\upomega_{\text{ix}} + {\text{C}}\upomega_{\text{iz}} {\text{S}}\upomega_{\text{ix}} } & 0 \\ {{\text{C}}\upomega_{\text{iz}} {\text{C}}\upomega_{\text{iy}} } & {{\text{C}}\upomega_{\text{iz}} {\text{S}}\upomega_{\text{iy}} {\text{S}}\upomega_{\text{ix}} - {\text{S}}\upomega_{\text{iz}} {\text{C}}\upomega_{\text{ix}} } & {{\text{C}}\upomega_{\text{iz}} {\text{S}}\upomega_{\text{iy}} {\text{C}}\upomega_{\text{ix}} + {\text{S}}\upomega_{\text{iz}} {\text{S}}\upomega_{\text{ix}} } & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(7.69)

$$ \frac{{\partial ({}^{0}{\bar{\text{A}}}_{\text{i}} )}}{{\partial\upxi_{{{\text{i}}\text{ - 1}}} }} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(7.70)

$$ \frac{{\partial ({}^{0}{\bar{\text{A}}}_{\text{i}} )}}{{\partial\upxi_{\text{i}} }} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(7.71)

$$ \frac{{\partial ({}^{0}{\bar{\text{A}}}_{\text{i}} )}}{{\partial {\text{R}}_{\text{i}} }} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right]. $$

(7.72)

• (2) ${{\partial\upsigma_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$, ${{\partial\uprho_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$ and ${{\partial\uptau_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$:

From ${}^{\text{i}}{\bar{\text{P}}}_{\text{i}} = {}^{\text{i}}{\bar{\text{A}}}_{0} {\bar{\text{P}}}_{\text{i}} = ({}^{\text{0}}{\bar{\text{A}}}_{\text{i}} )^{ - 1} \,{\bar{\text{P}}}_{\text{i}}$ given in Eq. (2.12), it can be shown that

$$ {\bar{\text{P}}}_{\text{i}} = {}^{\text{0}}{\bar{\text{A}}}_{\text{i}} \, \left[ {\begin{array}{*{20}c} {\upsigma_{\text{i}} } \\ {\uprho_{\text{i}} } \\ {\uptau_{\text{i}} } \\ 1 \\ \end{array} } \right]. $$

(7.73)

Differentiating Eq. (7.73) with respect to ${\bar{\text{X}}}_{\text{i}}$ gives

$$ \frac{{\partial {\bar{\text{P}}}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} = \frac{{\partial ({}^{0}{\bar{\text{A}}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} }}\left[ {\begin{array}{*{20}c} {\upsigma_{\text{i}} } \\ {\uprho_{\text{i}} } \\ {\uptau_{\text{i}} } \\ 1 \\ \end{array} } \right] + {}^{0}{\bar{\text{A}}}_{\text{i}} \left[ {\begin{array}{*{20}c} {{{\partial\upsigma_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}} \\ {{{\partial\uprho_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}} \\ {{{\partial\uptau_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}} \\ {\bar{0}} \\ \end{array} } \right], $$

(7.74)

where ${{\partial {\bar{\text{P}}}_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$ is given in Eq. (7.43) and the components of ${{\partial ({}^{0}{\bar{\text{A}}}_{\text{i}} )} / {\partial {\bar{\text{X}}}_{\text{i}} }}$ are listed in Eqs. (7.64)–(7.72) of this appendix. ${{\partial\upsigma_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$, ${{\partial\uprho_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$ and ${{\partial {\uptau }_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$ can then be obtained from

$$ \left[ {\begin{array}{*{20}c} {{{\partial\upsigma_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} \, }}} \\ {{{\partial\uprho_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}} \\ {{{\partial\uptau_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}} \\ {\bar{0}} \\ \end{array} } \right] = \left( {{}^{0}{\bar{\text{A}}}_{\text{i}} } \right)^{ - 1} \left( {\frac{{\partial {\bar{\text{P}}}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} - \frac{{\partial ({}^{0}{\bar{\text{A}}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} }}\left[ {\begin{array}{*{20}c} {\upsigma_{\text{i}} } \\ {\uprho_{\text{i}} } \\ {\uptau_{\text{i}} } \\ 1 \\ \end{array} } \right]} \right). $$

(7.75)

• (3) ${{\partial\upalpha_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$ and ${{\partial\upbeta_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$:

${{\partial\upalpha_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$ and ${{\partial\upbeta_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$ can be obtained by differentiating Eqs. (2.19) and (2.20), respectively, with respect to ${\bar{\text{X}}}_{\text{i}}$ to give

$$ \frac{{\partial\upalpha_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} = \frac{1}{{\upsigma_{\text{i}}^{ 2} +\uprho_{\text{i}}^{ 2} }}\left( {\upsigma_{\text{i}} \frac{{\partial\uprho_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} -\uprho_{\text{i}} \frac{{\partial\upsigma_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right), $$

(7.76)

$$ \frac{{\partial\upbeta_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} = \frac{{\sqrt {\left( {\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} } \right)} }}{{(\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} +\uptau_{\text{i}}^{2} )}}\frac{{\partial\uptau_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} - \frac{{\uptau_{\text{i}} }}{{(\sigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} +\uptau_{\text{i}}^{2} )\sqrt {\left( {\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} } \right)} }}\left( {\upsigma_{\text{i}} \frac{{\partial\upsigma_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} +\uprho_{\text{i}} \frac{{\partial\uprho_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right). $$

(7.77)

• (4) ${{\partial {\bar{\text{n}}}_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$:

${{\partial {\bar{\text{n}}}_{\text{i}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$ can be obtained by differentiating Eq. (2.10) with respect to ${\bar{\text{X}}}_{\text{i}}$ to give

$$ \frac{{\partial {\bar{\text{n}}}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} = \left[ {\begin{array}{*{20}c} {{{\partial {\text{n}}_{\text{ix}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}} \\ {{{\partial {\text{n}}_{\text{iy}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}} \\ {{{\partial {\text{n}}_{\text{iz}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}} \\ {\bar{0}} \\ \end{array} } \right] = \frac{{\partial ({}^{0}{\bar{\text{A}}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} }}{}^{\text{i}}{\bar{\text{n}}}_{\text{i}} + {}^{0}{\bar{\text{A}}}_{\text{i}} \, \frac{{\partial ({}^{\text{i}}{\bar{\text{n}}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} }}, $$

(7.78)

where ${{\partial ({}^{\text{i}}{\bar{\text{n}}}_{\text{i}} )} / {\partial {\bar{\text{X}}}_{\text{i}} }}$ is obtained by differentiating Eq. (2.8) to give

$$ \frac{{\partial ({}^{\text{i}}{\bar{\text{n}}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} }} = {\text{s}}_{\text{i}} \left[ {\begin{array}{*{20}c} { - {\text{S}}\upbeta_{\text{i}} {\text{C}}\upalpha_{\text{i}} } \\ { - {\text{S}}\upbeta_{\text{i}} {\text{S}}\upalpha_{\text{i}} } \\ {{\text{C}}\upbeta_{\text{i}} } \\ 0 \\ \end{array} } \right]\frac{{\partial\upbeta_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + {\text{s}}_{\text{i}} \left[ {\begin{array}{*{20}c} { - {\text{C}}\upbeta_{\text{i}} {\text{S}}\upalpha_{\text{i}} } \\ {{\text{C}}\upbeta_{\text{i}} {\text{C}}\upalpha_{\text{i}} } \\ 0 \\ 0 \\ \end{array} } \right]\frac{{\partial\upalpha_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}. $$

(7.79)

• (5) ${{\partial ({\text{C}}\uptheta_{\text{i}} )} / {\partial {\bar{\text{X}}}_{\text{i}} }}$

${{\partial ({\text{C}}\uptheta_{\text{i}} )} / {\partial {\bar{\text{X}}}_{\text{i}} }}$ can be computed directly from Eq. (2.21) as

$$ \frac{{\partial ({\text{C}}\uptheta_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} }} = - \left({\ell_{{{\text{i}}\text{- 1}{\text{x}}}} \frac{{\partial {\text{n}}_{\text{ix}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + \ell_{{{\text{i}}\text{- 1}{\text{y}}}} \frac{{\partial {\text{n}}_{\text{iy}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + \ell_{{{\text{i}}\text{- 1}{\text{z}}}} \frac{{\partial {\text{n}}_{\text{iz}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right), $$

(7.80)

where ${{\partial {\text{n}}_{\text{ix}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$, ${{\partial {\text{n}}_{\text{iy}} } / {\partial {\bar{\text{X}}}_{\text{i}} }}$ and ${{{\partial}{\text{n}}_{\text{iz}} } / {{\partial}{\bar{\text{X}}}_{\text{i}} }}$ are given in Eq. (7.78).