Jacobian Matrix of Boundary Variable Vector X̄_i with Respect to System Variable Vector X̄_sys

Abstract

The system variable vector ${\bar{\text{X}}}_{{\text{sys}}}$ of an optical system is, nearly always, different from the boundary variable vector ${\bar{\text{X}}}_{\text{i}}$ of a boundary surface. Furthermore, changes in the system variable vector may have a profound effect on the behavior of the rays as they propagate through the system. Therefore, the Jacobian matrix $\text{d}\bar{\text{X}}_{\text{i}} /\text{d}\bar{\text{X}}_{{\text{sys}}}$ of the boundary variable vector ${\bar{\text{X}}}_{\text{i}}$ with respect to the system variable vector ${\bar{\text{X}}}_{{\text{sys}}}$ is of crucial concern to optical systems designers.

Appendices

Appendix 1

When using Eqs. (8.24) and (8.44), it is first necessary to determine ${\text{d}}\left( {{}^{\text{0}}{\bar{\text{A}}}_{\text{ej}} } \right)/{\text{d}\bar{\text{X}}}_{\text{sys}}$ in advance, where ${}^{0}{\bar{\text{A}}}_{\text{ej}}$ is defined by Eqs. (3.1) and (3.25) as

$$ \begin{aligned} &{}^{0}{\bar{\text{A}}}_{\text{ej}} = {\text{tran}}({\text{t}}_{\text{ejx}} ,{\text{t}}_{\text{ejy}} ,{\text{t}}_{\text{ejz}} ){\text{rot}}({\bar{\text{z}}}{,\omega }_{\text{ejz}} ){\text{rot}}({\bar{\text{y}}},\upomega_{\text{ejy}} ){\text{rot}}({\bar{\text{x}}}{,\omega }_{\text{ejx}} ) \\ & = \left[ {\begin{array}{*{20}c} {{\text{C}}\upomega_{\text{ejy}} {\text{C}}\upomega_{\text{ejz}} } & {{\text{S}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejy}} {\text{C}}\upomega_{\text{ejz}} - {\text{C}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejz}} } & {{\text{C}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejy}} {\text{C}}\upomega_{\text{ejz}} - {\text{S}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejz}} } & {{\text{t}}_{\text{ejx}} } \\ {{\text{C}}\upomega_{\text{ejy}} {\text{S}}\upomega_{\text{ejz}} } & {{\text{C}}\upomega_{\text{ejx}} {\text{C}}\upomega_{\text{ejz}} + {\text{S}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejy}} {\text{S}}\upomega_{\text{ejz}} } & { - {\text{S}}\upomega_{\text{ejx}} {\text{C}}\upomega_{\text{ejz}} + {\text{C}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejy}} {\text{S}}\upomega_{\text{ejz}} } & {{\text{t}}_{\text{ejy}} } \\ { - {\text{S}}\upomega_{\text{ejy}} } & {{\text{S}}\upomega_{\text{ejx}} {\text{C}}\upomega_{\text{ejy}} } & {{\text{C}}\upomega_{\text{ejx}} {\text{C}}\upomega_{\text{ejy}} } & {{\text{t}}_{\text{ejz}} } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right] \, \\ & = \left[ {\begin{array}{*{20}c} {{\text{I}}_{\text{ejx}} } & {{\text{J}}_{\text{ejx}} } & {{\text{K}}_{\text{ejx}} } & {{\text{t}}_{\text{ejx}} } \\ {{\text{I}}_{\text{ejy}} } & {{\text{J}}_{\text{ejy}} } & {{\text{K}}_{\text{ejy}} } & {{\text{t}}_{\text{ejy}} } \\ {{\text{I}}_{\text{ejz}} } & {{\text{J}}_{\text{ejz}} } & {{\text{K}}_{\text{ejz}} } & {{\text{t}}_{\text{ejz}} } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right]. \\ \end{aligned} $$

A dimensional analysis reveals that the arguments of the rotation terms in matrix ${}^{0}{\bar{\text{A}}}_{\text{ej}}$ can be expressed as linear combinations of the components of ${\bar{\text{X}}}_{\text{sys}} = \left[ {{\text{x}}_{\text{v}} } \right]$, i.e.,

$$ \upomega_{\text{ejx}} = {\text{a}}_{0} + \sum\limits_{{{\text{v}} = 1}}^{{{\text{q}}_{\text{sys}} }} {{\text{a}}_{\text{v}} {\text{x}}_{\text{v}} } , $$

(8.45)

$$ \upomega_{\text{ejy}} = {\text{b}}_{0} + \sum\limits_{{{\text{v}} = 1}}^{{{\text{q}}_{\text{sys}} }} {{\text{b}}_{\text{v}} {\text{x}}_{\text{v}} } , $$

(8.46)

$$ \upomega_{\text{ejz}} = {\text{c}}_{0} + \sum\limits_{{{\text{v}} = 1}}^{{{\text{q}}_{\text{sys}} }} {{\text{c}}_{\text{v}} {\text{x}}_{\text{v}} } , $$

(8.47)

where ${\text{x}}_{\text{v}}$(${\text{v} \in }\left\{ {1,2, \ldots ,\text{q}_{{\text{sys}}} } \right\}$) is the vth component of ${\bar{\text{X}}}_{\text{sys}}$, and ${\text{a}}_{\text{v}}$, ${\text{b}}_{\text{v}}$ and ${\text{c}}_{\text{v}}$ (v = 0 to $\text{v} = \text{q}_{{\text{sys}}}$) are known constants. Therefore, $\text{d}\left( {{}^{\text{0}}{\bar{\text{A}}}_{\text{ej}} } \right)/\text{d}\bar{\text{X}}_{{\text{sys}}}$ can be determined directly by differentiating Eq. (3.1) (or Eq. (3.25)) with respect to ${\text{x}}_{\text{v}}$ to give

$$ \begin{aligned} \frac{{{\text{d}}\left( {{}^{0}{\bar{\text{A}}}_{\text{ej}} } \right)}}{{{\text{d}\bar{\text{X}}}_{\text{sys}} }} & = \left[ {\frac{{{\partial }\left( {{}^{0}{\bar{\text{A}}}_{\text{ej}} } \right)}}{{{\partial }{\text{x}}_{\text{v}} }}} \right]_{{4 \times 4 \times {\text{q}}_{\text{sys}} }} \\ &= \left[ {\begin{array}{*{20}l} {{\partial }{\text{I}}_{\text{ejx}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{J}}_{\text{ejx}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{K}}_{\text{ejx}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{t}}_{\text{ejx}} /{\partial }{\text{x}}_{\text{v}} } \hfill \\ {{\partial }{\text{I}}_{\text{ejy}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{J}}_{\text{ejy}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{K}}_{\text{ejy}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{t}}_{\text{ejy}} /{\partial }{\text{x}}_{\text{v}} } \hfill \\ {{\partial }{\text{I}}_{\text{ejz}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{J}}_{\text{ejz}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{K}}_{\text{ejz}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{t}}_{\text{ejz}} /{\partial }{\text{x}}_{\text{v}} } \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right],\end{aligned} $$

(8.48)

where

$$ \frac{{{\partial }{\text{I}}_{\text{ejx}} }}{{{\partial }{\text{x}}_{\text{v}} }} = - {\text{b}}_{\text{v}} {\text{S}}\upomega_{\text{ejy}} {\text{C}}\upomega_{\text{ejz}} - {\text{c}}_{\text{v}} {\text{C}}\upomega_{\text{ejy}} {\text{S}}\upomega_{\text{ejz}} , $$

(8.49)

$$ \frac{{{\partial }{\text{I}}_{\text{ejy}} }}{{{\partial }{\text{x}}_{\text{v}} }} = - {\text{b}}_{\text{v}} {\text{S}}\upomega_{\text{ejy}} {\text{S}}\upomega_{\text{ejz}} + {\text{c}}_{\text{v}} {\text{C}}\upomega_{\text{ejy}} {\text{C}}\upomega_{\text{ejz}} , $$

(8.50)

$$ \frac{{{\partial }{\text{I}}_{\text{ejz}} }}{{{\partial }{\text{x}}_{\text{v}} }} = - {\text{b}}_{\text{v}} {\text{C}}\upomega_{\text{ejy}} , $$

(8.51)

$$ \begin{aligned} \frac{{{\partial }{\text{J}}_{\text{ejx}} }}{{{\partial }{\text{x}}_{\text{v}} }} & = {\text{a}}_{\text{v}} {\text{C}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejy}} {\text{C}}\upomega_{\text{ejz}} + {\text{b}}_{\text{v}} {\text{S}}\upomega_{\text{ejx}} {\text{C}}\upomega_{\text{ejy}} {\text{C}}\upomega_{\text{ejz}} - {\text{c}}_{\text{v}} {\text{S}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejy}} {\text{S}}\upomega_{\text{ejz}} + {\text{a}}_{\text{v}} {\text{S}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejz}} \\ \, & \quad - \,{\text{c}}_{\text{v}} {\text{C}}\upomega_{\text{ejx}} {\text{C}}\upomega_{\text{ejz}} , \\ \end{aligned} $$

(8.52)

$$ \begin{aligned} \frac{{{\partial }{\text{J}}_{\text{ejy}} }}{{{\partial }{\text{x}}_{\text{v}} }} & = {\text{a}}_{\text{v}} {\text{C}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejy}} {\text{S}}\upomega_{\text{ejz}} + {\text{b}}_{\text{v}} {\text{S}}\upomega_{\text{ejx}} {\text{C}}\upomega_{\text{ejy}} {\text{S}}\upomega_{\text{ejz}} + {\text{c}}_{\text{v}} {\text{S}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejy}} {\text{C}}\upomega_{\text{ejz}} - {\text{a}}_{\text{v}} {\text{S}}\upomega_{\text{ejx}} {\text{C}}\upomega_{\text{ejz}} \\ \, & \quad - \,{\text{c}}_{\text{v}} {\text{C}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejz}} , \\ \end{aligned} $$

(8.53)

$$ \frac{{{\partial }{\text{J}}_{{\text{ejz}}} }}{{{\partial }{\text{x}}_{\text{v}} }} = {\text{a}}_{\text{v}} {\text{C}}\upomega_{\text{ejx}} {\text{C}}\upomega_{\text{ejy}} - {\text{b}}_{\text{v}} {\text{S}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejy}} , $$

(8.54)

$$ \begin{aligned} \frac{{{\partial }{\text{K}}_{\text{ejx}} }}{{{\partial }{\text{x}}_{\text{v}} }} & = - {\text{a}}_{\text{v}} {\text{S}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejy}} {\text{C}}\upomega_{\text{ejz}} + {\text{b}}_{\text{v}} {\text{C}}\upomega_{\text{ejx}} {\text{C}}\upomega_{\text{ejy}} C\upomega_{\text{ejz}} - {\text{c}}_{\text{v}} {\text{C}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejy}} {\text{S}}\upomega_{\text{ejz}} - {\text{a}}_{\text{v}} {\text{C}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejz}} \\ \, & \quad - \,{\text{c}}_{\text{v}} {\text{S}}\upomega_{\text{ejx}} {\text{C}}\upomega_{\text{ejz}} , \\ \end{aligned} $$

(8.55)

$$ \begin{aligned} \frac{{{\partial }{\text{K}}_{\text{ejy}} }}{{\partial x_{v} }} & = - {\text{a}}_{\text{v}} {\text{S}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejy}} {\text{S}}\upomega_{\text{ejz}} + {\text{b}}_{\text{v}} {\text{C}}\upomega_{\text{ejx}} {\text{C}}\upomega_{\text{ejy}} {\text{S}}\upomega_{\text{ejz}} + {\text{c}}_{\text{v}} {\text{C}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejy}} {\text{C}}\upomega_{\text{ejz}} - {\text{a}}_{\text{v}} {\text{C}}\upomega_{\text{ejx}} {\text{C}}\upomega_{\text{ejz}} \\ \, & \quad +\, {\text{c}}_{\text{v}} {\text{S}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejz}} , \\ \end{aligned} $$

(8.56)

$$ \frac{{{\partial }{\text{K}}_{\text{ejz}} }}{{{\partial }{\text{x}}_{\text{v}} }} = - {\text{a}}_{\text{v}} {\text{S}}\upomega_{\text{ejx}} {\text{C}}\upomega_{\text{ejy}} - {\text{b}}_{\text{v}} {\text{C}}\upomega_{\text{ejx}} {\text{S}}\upomega_{\text{ejy}} , $$

(8.57)

$$ \frac{{{\partial }{\text{t}}_{\text{ejx}} }}{{{\partial }{\text{x}}_{\text{v}} }} = \frac{{{\partial }{\text{t}}_{\text{ejx}} }}{{{\partial }{\text{x}}_{\text{v}} }}, $$

(8.58)

$$ \frac{{{\partial }{\text{t}}_{\text{ejy}} }}{{{\partial }{\text{x}}_{\text{v}} }} = \frac{{{\partial }{\text{t}}_{\text{ejy}} }}{{{\partial }{\text{x}}_{\text{v}} }}, $$

(8.59)

$$ \frac{{{\partial }{\text{t}}_{\text{ejz}} }}{{{\partial }{\text{x}}_{\text{v}} }} = \frac{{{\partial }{\text{t}}_{\text{ejz}} }}{{{\partial }{\text{x}}_{\text{v}} }}. $$

(8.60)

Equations (8.58), (8.59) and (8.60) indicate that ${\partial }{\text{t}}_{\text{ejx}} /{\partial }{\text{x}}_{\text{v}}$, ${\partial }{\text{t}}_{\text{ejy}} /{\partial }{\text{x}}_{\text{v}}$ and ${\partial }{\text{t}}_{\text{ejz}} /{\partial }{\text{x}}_{\text{v}}$ can be obtained simply by differentiating their corresponding expressions since they are always given in explicit form (e.g., ${\partial }{\text{t}}_{{{\text{e}}\text{2}{\text{x}}}} /{\partial }{\text{x}}_{25} = 1/2$, ${\partial }{\text{t}}_{{\text{e2y}}} /{\partial }{\text{x}}_{20} = 1$ and ${\partial }{\text{t}}_{{\text{e2z}}} /{\partial }{\text{x}}_{26} = - 1$ for ${}^{\text{0}}{\bar{\text{A}}}_{{\text{e2}}}$ in Example 3.11 and ${\bar{\text{X}}}_{\text{sys}}$ defined in Example 3.15).

Appendix 2

In using Eqs. (8.24) and (8.44), it is first necessary to determine ${\text{d}}\left( {{}^{{\text{ej}}}{\bar{\text{A}}}_{\text{i}} } \right)/{\text{d}\bar{\text{X}}}_{\text{sys}}$ in advance, where ${}^{\text{ej}}{\bar{\text{A}}}_{\text{i}}$ is defined by Eqs. (3.3) and (3.26) as

$$ \begin{aligned} {}^{\text{ej}}{\bar{\text{A}}}_{\text{i}} & = {\text{tran}}({}^{\text{ej}}{\text{t}}_{\text{ix}} ,{}^{\text{ej}}{\text{t}}_{\text{iy}} ,{}^{\text{ej}}{\text{t}}_{\text{iz}} ){\text{rot}}({\bar{\text{z}}}\text{,}{}^{\text{ej}}\upomega_{\text{iz}} ){\text{rot}}({\bar{\text{y}}},{}^{\text{ej}}\upomega_{\text{iy}} ){\text{rot}}({\bar{\text{x}}}\text{,}{}^{\text{ej}}\upomega_{\text{ix}} ) \\ \, & = \left[ {\begin{array}{*{20}c} {{}^{\text{ej}}{\text{I}}_{\text{ix}} } & {{}^{\text{ej}}{\text{J}}_{\text{ix}} } & {{}^{\text{ej}}{\text{K}}_{\text{ix}} } & {{}^{\text{ej}}{\text{t}}_{\text{ix}} } \\ {{}^{\text{ej}}{\text{I}}_{\text{iy}} } & {{}^{\text{ej}}{\text{J}}_{\text{iy}} } & {{}^{\text{ej}}{\text{K}}_{\text{iy}} } & {{}^{\text{ej}}{\text{t}}_{\text{iy}} } \\ {{}^{\text{ej}}{\text{I}}_{\text{iz}} } & {{}^{\text{ej}}{\text{J}}_{\text{iz}} } & {{}^{\text{ej}}{\text{K}}_{\text{iz}} } & {{}^{\text{ej}}{\text{t}}_{\text{iz}} } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right], \\ \end{aligned} $$

with

$$ {}^{\text{ej}}{\text{I}}_{\text{ix}} = {\text{C}}\left( {{}^{\text{ej}}\upomega_{\text{iy}} } \right){\text{C}}\left( {{}^{\text{ej}}\upomega_{\text{iz}} } \right), $$

$$ {}^{\text{ej}}{\text{I}}_{\text{iy}} = {\text{C}}\left( {{}^{\text{ej}}\upomega_{\text{iy}} } \right){\text{S}}\left( {{}^{\text{ej}}\upomega_{\text{iz}} } \right). $$

$$ {}^{\text{ej}}{\text{I}}_{\text{iz}} = - {\text{S}}\left( {{}^{\text{ej}}\upomega_{\text{iy}} } \right), $$

$$ {}^{\text{ej}}{\text{J}}_{\text{ix}} = {\text{S}}\left( {{}^{\text{ej}}\upomega_{\text{ix}} } \right){\text{S}}\left( {{}^{\text{ej}}\upomega_{\text{iy}} } \right){\text{C}}\left( {{}^{\text{ej}}\upomega_{\text{iz}} } \right) - {\text{C}}\left( {{}^{\text{ej}}\upomega_{\text{ix}} } \right){\text{S}}\left( {{}^{\text{ej}}\upomega_{\text{iz}} } \right), $$

$$ {}^{\text{ej}}{\text{J}}_{\text{iy}} = {\text{C}}\left( {{}^{\text{ej}}\upomega_{\text{ix}} } \right){\text{C}}\left( {{}^{\text{ej}}\upomega_{\text{iz}} } \right) + {\text{S}}\left( {{}^{\text{ej}}\upomega_{\text{ix}} } \right){\text{S}}\left( {{}^{\text{ej}}\upomega_{\text{iy}} } \right){\text{S}}\left( {{}^{\text{ej}}\upomega_{\text{iz}} } \right), $$

$$ {}^{\text{ej}}{\text{J}}_{\text{iz}} = {\text{S}}\left( {{}^{\text{ej}}\upomega_{\text{ix}} } \right){\text{C}}\left( {{}^{\text{ej}}\upomega_{\text{iy}} } \right), $$

$$ {}^{\text{ej}}{\text{K}}_{\text{ix}} = {\text{C}}\left( {{}^{\text{ej}}\upomega_{\text{ix}} } \right){\text{S}}\left( {{}^{\text{ej}}\upomega_{\text{iy}} } \right){\text{C}}\left( {{}^{\text{ej}}\upomega_{\text{iz}} } \right) - {\text{S}}\left( {{}^{\text{ej}}\upomega_{\text{ix}} } \right){\text{S}}\left( {{}^{\text{ej}}\upomega_{\text{iz}} } \right), $$

$$ {}^{\text{ej}}{\text{K}}_{\text{iy}} = - {\text{S}}\left( {{}^{\text{ej}}\upomega_{\text{ix}} } \right){\text{C}}\left( {{}^{\text{ej}}\upomega_{\text{iz}} } \right) + {\text{C}}\left( {{}^{\text{ej}}\upomega_{\text{ix}} } \right){\text{S}}\left( {{}^{\text{ej}}\upomega_{\text{iy}} } \right){\text{S}}\left( {{}^{\text{ej}}\upomega_{\text{iz}} } \right), $$

$$ {}^{\text{ej}}{\text{K}}_{\text{iz}} = {\text{C}}\left( {{}^{\text{ej}}\upomega_{\text{ix}} } \right){\text{C}}{}^{\text{ej}}\left( {\upomega_{\text{iy}} } \right). $$

A dimensional analysis again reveals that the arguments of the rotation terms in matrix ${}^{\text{ej}}{\bar{\text{A}}}_{\text{i}}$ can be expressed as linear combinations of the components of ${\bar{\text{X}}}_{\text{sys}} = \left[ {{\text{x}}_{\text{v}} } \right]$, i.e.,

$$ {}^{\text{ej}}\upomega_{\text{ix}} =\uprho_{0} + \sum\limits_{{{\text{v}} = 1}}^{{{\text{q}}_{\text{sys}} }} {\uprho_{\text{v}} {\text{x}}_{\text{v}} } , $$

(8.61)

$$ {}^{\text{ej}}\upomega_{\text{iy}} = {\text{g}}_{0} + \sum\limits_{{{\text{v}} = 1}}^{{{\text{q}}_{\text{sys}} }} {{\text{g}}_{\text{v}} {\text{x}}_{\text{v}} } , $$

(8.62)

$$ {}^{\text{ej}}\upomega_{\text{iz}} = {\text{h}}_{0} + \sum\limits_{{{\text{v}} = 1}}^{{{\text{q}}_{\text{sys}} }} {{\text{h}}_{\text{v}} {\text{x}}_{\text{v}} } , $$

(8.63)

where ${\text{x}}_{\text{v}}$ (${\text{v} \in }\left\{ {1,2, \ldots ,\text{q}_{{\text{sys}}} } \right\}$) is the vth component of ${\bar{\text{X}}}_{\text{sys}}$, and $\uprho_{\text{v}}$, ${\text{g}}_{\text{v}}$ and ${\text{h}}_{\text{v}}$ (v = 0 to $\text{v} = \text{q}_{{\text{sys}}}$) are known constants. Therefore, ${\partial }\left( {{}^{\text{ej}}{\bar{\text{A}}}_{\text{i}} } \right)/{\partial }{\text{x}}_{\text{v}}$ can be determined directly by differentiating Eq. (3.3) (or Eq. (3.26)) with respect to ${\text{x}}_{\text{v}}$ to give

$$ \begin{aligned}\frac{{{\text{d}}\left( {{}^{\text{ej}}{\bar{\text{A}}}_{\text{i}} } \right)}}{{{\text{d}\bar{\text{X}}}_{\text{sys}} }} & = \left[ {\frac{{{\partial }\left( {{}^{\text{ej}}{\bar{\text{A}}}_{\text{i}} } \right)}}{{{\partial }{\text{x}}_{\text{v}} }}} \right]_{{4 \times 4 \times {\text{q}}_{\text{sys}} }} \\ &= \left[ {\begin{array}{*{20}l} {{\partial }({}^{\text{ej}}{\text{I}}_{\text{ix}} )/{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }({}^{\text{ej}}{\text{J}}_{\text{ix}} )/{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }({}^{\text{ej}}{\text{K}}_{\text{ix}} )/{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }({}^{\text{ej}}{\text{t}}_{\text{ix}} )/{\partial }{\text{x}}_{\text{v}} } \hfill \\ {{\partial }({}^{\text{ej}}{\text{I}}_{\text{iy}} )/{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }({}^{\text{ej}}{\text{J}}_{\text{iy}} )/{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }({}^{\text{ej}}{\text{K}}_{\text{iy}} )/{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }({}^{\text{ej}}{\text{t}}_{{\text{iy}}} )/{\partial }{\text{x}}_{\text{v}} } \hfill \\ {{\partial }({}^{\text{ej}}{\text{I}}_{\text{iz}} )/{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }({}^{\text{ej}}{\text{J}}_{\text{iz}} )/{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }({}^{\text{ej}}{\text{K}}_{\text{iz}} )/{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }({}^{\text{ej}}{\text{t}}_{\text{iz}} )/{\partial }{\text{x}}_{\text{v}} } \hfill \\ \qquad\quad0 \hfill &\qquad\quad 0 \hfill &\qquad\quad 0 \hfill &\qquad\quad 0 \hfill \\ \end{array} } \right], \end{aligned} $$

(8.64)

where

$$ \frac{{{\partial }({}^{\text{ej}}{\text{I}}_{\text{ix}} )}}{{{\partial }{\text{x}}_{\text{v}} }} = - {\text{g}}_{\text{v}} {\text{S}}({}^{\text{ej}}\upomega_{\text{iy}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iz}} ) - {\text{h}}_{\text{v}} {\text{C}}({}^{\text{ej}}\upomega_{\text{iy}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iz}} ), $$

(8.65)

$$ \frac{{{\partial }({}^{\text{ej}}{\text{I}}_{\text{iy}} )}}{{{\partial }{\text{x}}_{\text{v}} }} = - {\text{g}}_{\text{v}} {\text{S}}({}^{\text{ej}}\upomega_{\text{iy}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iz}} ) + {\text{h}}_{\text{v}} {\text{C}}({}^{\text{ej}}\upomega_{\text{iy}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iz}} ), $$

(8.66)

$$ \frac{{{\partial }({}^{\text{ej}}{\text{I}}_{\text{iz}} )}}{{{\partial }{\text{x}}_{\text{v}} }} = - {\text{g}}_{\text{v}} {\text{C}}({}^{\text{ej}}\upomega_{\text{iy}} ), $$

(8.67)

$$ \begin{aligned} \frac{{{\partial }({}^{\text{ej}}{\text{J}}_{\text{ix}} )}}{{{\partial }{\text{x}}_{\text{v}} }} & =\uprho_{\text{v}} {\text{C}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iy}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iz}} ) + {\text{g}}_{v} {\text{S}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iy}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iz}} ) \\ & \quad -\, {\text{h}}_{\text{v}} {\text{S}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iy}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iz}} ) +\uprho_{\text{v}} {\text{S}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iz}} ) - {\text{h}}_{\text{v}} {\text{C}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iz}} ), \\ \end{aligned} $$

(8.68)

$$ \begin{aligned} \frac{{{\partial }({}^{\text{ej}}{\text{J}}_{\text{iy}} )}}{{{\partial }{\text{x}}_{\text{v}} }} & =\uprho_{\text{v}} {\text{C}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iy}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iz}} ) + {\text{g}}_{\text{v}} {\text{S}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iy}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iz}} ) \\ & \quad +\, {\text{h}}_{\text{v}} {\text{S}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iy}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iz}} ) -\uprho_{\text{v}} {\text{S}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iz}} ) - {\text{h}}_{\text{v}} {\text{C}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iz}} ), \\ \end{aligned} $$

(8.69)

$$ \frac{{{\partial }({}^{\text{ej}}{\text{J}}_{\text{iz}} )}}{{{\partial }{\text{x}}_{\text{v}} }} =\uprho_{\text{v}} {\text{C}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iy}} ) - {\text{g}}_{\text{v}} {\text{S}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iy}} ), $$

(8.70)

$$ \begin{aligned} \frac{{{\partial }({}^{\text{ej}}{\text{K}}_{\text{ix}} )}}{{{\partial }{\text{x}}_{\text{v}} }} & = -\uprho_{\text{v}} {\text{S}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iy}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iz}} ) + {\text{g}}_{\text{v}} {\text{C}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iy}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iz}} ) \\ & \quad - \,{\text{h}}_{\text{v}} {\text{C}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iy}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iz}} ) -\uprho_{\text{v}} {\text{C}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iz}} ) - {\text{h}}_{\text{v}} {\text{S}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iz}} ), \\ \end{aligned} $$

(8.71)

$$ \begin{aligned} \frac{{{\partial }({}^{\text{ej}}{\text{K}}_{\text{iy}} )}}{{{\partial }{\text{x}}_{\text{v}} }} & = -\uprho_{\text{v}} {\text{S}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iy}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iz}} ) + {\text{g}}_{\text{v}} {\text{C}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iy}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iz}} ) \\ & \quad +\, {\text{h}}_{\text{v}} {\text{C}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iy}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iz}} ) -\uprho_{\text{v}} {\text{C}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iz}} ) + {\text{h}}_{\text{v}} {\text{S}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iz}} ), \\ \end{aligned} $$

(8.72)

$$ \frac{{{\partial }({}^{\text{ej}}{\text{K}}_{\text{iz}} )}}{{{\partial }{\text{x}}_{\text{v}} }} = -\uprho_{\text{v}} {\text{S}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{C}}({}^{\text{ej}}\upomega_{\text{iy}} ) - {\text{g}}_{\text{v}} {\text{C}}({}^{\text{ej}}\upomega_{\text{ix}} ){\text{S}}({}^{\text{ej}}\upomega_{\text{iy}} ). $$

(8.73)

Note that ${}^{\text{ej}}{\text{t}}_{\text{ix}}$, ${}^{\text{ej}}{\text{t}}_{\text{iy}}$ and ${}^{\text{ej}}{\text{t}}_{\text{iz}}$ are always given in explicit form. As a result, ${{{\partial }({}^{\text{ej}}{\text{t}}_{\text{ix}} )} / {{\partial }{\text{x}}_{\text{v}} }}$, ${{{\partial }({}^{\text{ej}}{\text{t}}_{\text{iy}} )} / {{\partial }{\text{x}}_{\text{v}} }}$ and ${{{\partial }({}^{\text{ej}}{\text{t}}_{\text{iz}} )} / {{\partial }{\text{x}}_{\text{v}} }}$ can be obtained simply by differentiating their corresponding expressions, i.e.,

$$ \frac{{{\partial }({}^{\text{ej}}{\text{t}}_{\text{ix}} )}}{{{\partial }{\text{x}}_{\text{v}} }} = \frac{{{\partial }({}^{\text{ej}}{\text{t}}_{\text{ix}} )}}{{{\partial }{\text{x}}_{\text{v}} }}, $$

(8.74)

$$ \frac{{{\partial }({}^{\text{ej}}{\text{t}}_{\text{iy}} )}}{{{\partial }{\text{x}}_{\text{v}} }} = \frac{{{\partial }({}^{\text{ej}}{\text{t}}_{\text{iy}} )}}{{{\partial }{\text{x}}_{\text{v}} }}, $$

(8.75)

$$ \frac{{{\partial }({}^{\text{ej}}{\text{t}}_{\text{iz}} )}}{{{\partial }{\text{x}}_{\text{v}} }} = \frac{{{\partial }({}^{\text{ej}}{\text{t}}_{\text{iz}} )}}{{{\partial }{\text{x}}_{\text{v}} }}. $$

(8.76)

For example, ${\partial }({}^{{{\text{e}}1}}{\text{t}}_{{\text{2}{\text{y}}}} )/{\partial }{\text{x}}_{11} = - 1$, ${\partial }({}^{{{\text{e}}1}}{\text{t}}_{{\text{2}{\text{y}}}} )/{\partial }{\text{x}}_{19} = 1$ and ${\partial }({}^{{{\text{e}}1}}{\text{t}}_{{\text{2}{\text{y}}}} )/{\partial }{\text{x}}_{12} = 1$ for ${}^{{{\text{e}}\text{1}}}{\bar{\text{A}}}_{\text{2}}$ in Example 3.12 and ${\bar{\text{X}}}_{\text{sys}}$ defined in Example 3.15.

Having obtained numerical values of ${}^{0}{\bar{\text{A}}}_{\text{ej}}$, ${}^{\text{ej}}{\bar{\text{A}}}_{\text{i}}$, ${\partial }({}^{0}{\bar{\text{A}}}_{\text{ej}} )/{\partial }{\text{x}}_{\text{v}}$ and ${\partial }({}^{\text{ej}}{\bar{\text{A}}}_{\text{i}} )/{\partial }{\text{x}}_{\text{v}}$, the Jacobian matrices ${\partial }({\bar{\text{r}}}_{\text{ i}} )/{\partial }{\bar{\text{X}}}_{\text{sys}}$ and ${\text{d}}({}^{0}{\bar{\text{A}}}_{\text{i}} )/{\text{d}\bar{\text{X}}}_{\text{sys}}$ can be computed from Eqs. (8.24) and (8.44), respectively, using the methods described in Appendices 3 and 4 of this chapter.

Appendix 3

It is noted from Eq. (8.24) that to compute ${\text{d}}({\bar{\text{r}}}_{\text{ i}} )/{\text{d}\bar{\text{X}}}_{\text{sys}}$, it is first necessary to have the numerical values of ${}^{\text{0}}{\bar{\text{A}}}^{\prime }_{\text{ej}}$, ${}^{\text{ej}}{\bar{\text{r}}}_{\text{ i}}$, ${\text{d}}({}^{0}{\bar{\text{A}}}^{\prime }_{\text{ej}} )/{\text{d}\bar{\text{X}}}_{\text{sys}}$ and ${\text{d}}({}^{\text{ej}}{\bar{\text{r}}}_{\text{ i}} )/{\text{d}\bar{\text{X}}}_{\text{sys}}$. The related expressions are presented in the following.

• (1) Determination of ${}^{\text{0}}{\bar{\text{A}}}^{\prime }_{\text{ej}}$ from ${}^{\text{0}}{\bar{\text{A}}}_{\text{ej}}$:

  If ${}^{\text{0}}{\bar{\text{A}}}_{\text{ej}}$ is given as Eq. (3.1) (or Eq. (3.25)), then ${}^{\text{0}}{\bar{\text{A}}}^{\prime }_{\text{ej}}$ can be determined by Eq. (1.35) with h = 0 and g = ej to give Eqs. (8.20), (8.21), (8.22) and (8.23):

  $$ {}^{0}{\bar{\text{A}}}^{\prime }_{\text{ej}} = \left[ {\begin{array}{*{20}c} {{\text{I}}_{\text{ejx}} } & {{\text{J}}_{\text{ejx}} } & {{\text{K}}_{\text{ejx}} } & 0 \\ {{\text{I}}_{\text{ejy}} } & {{\text{J}}_{\text{ejy}} } & {{\text{K}}_{\text{ejy}} } & 0 \\ {{\text{I}}_{\text{ejz}} } & {{\text{J}}_{\text{ejz}} } & {{\text{K}}_{\text{ejz}} } & 0 \\ {{\text{f}}_{\text{ejx}} } & {{\text{f}}_{\text{ejy}} } & {{\text{f}}_{\text{ejz}} } & 1 \\ \end{array} } \right], $$

  with

  $$ {\text{f}}_{\text{ejx}} = - ({\text{I}}_{\text{ejx}} {\text{t}}_{\text{ejx}} + {\text{I}}_{\text{ejy}} {\text{t}}_{\text{ejy}} + {\text{I}}_{\text{ejz}} {\text{t}}_{\text{ejz}} ), $$
  $$ {\text{f}}_{\text{ejy}} = - ({\text{J}}_{\text{ejx}} {\text{t}}_{\text{ejx}} + {\text{J}}_{\text{ejy}} {\text{t}}_{\text{ejy}} + {\text{J}}_{\text{ejz}} {\text{t}}_{\text{ejz}} ), $$
  $$ {\text{f}}_{\text{ejz}} = - ({\text{K}}_{\text{ejx}} {\text{t}}_{\text{ejx}} + {\text{K}}_{\text{ejy}} {\text{t}}_{\text{ejy}} + {\text{K}}_{\text{ejz}} {\text{t}}_{\text{ejz}} ). $$

• (2) Determination of ${\text{d}}({}^{0}{\bar{\text{A}}}^{\prime }_{\text{ej}} )/{\text{d}\bar{\text{X}}}_{\text{sys}}$ from ${\text{d}}({}^{0}{\bar{\text{A}}}_{\text{ej}} )/{\text{d}\bar{\text{X}}}_{\text{sys}}$:

  Equations (8.20)–(8.23) indicate that if ${\text{d}}({}^{0}{\bar{\text{A}}}_{\text{ej}} )/{\text{d}\bar{\text{X}}}_{\text{sys}}$ (see Appendix 1 of this chapter) is known, then ${\text{d}}({}^{0}{\bar{\text{A}}}^{\prime }_{\text{ej}} )/{\text{d}\bar{\text{X}}}_{\text{sys}}$ can be determined as

  $$ \frac{{{\text{d}}({}^{0}{\bar{\text{A}}}^{\prime }_{\text{ej}} )}}{{{\text{d}\bar{\text{X}}}_{\text{sys}} }} = \left[ {\frac{{{\partial }({}^{0}{\bar{\text{A}}}^{\prime }_{\text{ej}} )}}{{{\partial }{\text{x}}_{\text{v}} }}} \right]_{{4 \times 4 \times {\text{q}}_{\text{sys}} }} = \left[ {\begin{array}{*{20}l} {{\partial }{\text{I}}_{\text{ejx}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{J}}_{\text{ejx}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{K}}_{\text{ejx}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {\bar{0}} \hfill \\ {{\partial }{\text{I}}_{\text{ejy}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{J}}_{\text{ejy}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{K}}_{\text{ejy}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {\bar{0}} \hfill \\ {{\partial }{\text{I}}_{\text{ejz}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{J}}_{\text{ejz}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{K}}_{\text{ejz}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {\bar{0}} \hfill \\ {{\partial }{\text{f}}_{\text{ejx}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{f}}_{\text{ejy}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{f}}_{\text{ejz}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {\bar{0}} \hfill \\ \end{array} } \right], $$

  (8.77)

  with (see Eqs. (8.21), (8.22) and (8.23))

  $$ \frac{{{\partial }{\text{f}}_{\text{ejx}} }}{{{\partial }{\text{x}}_{\text{v}} }} = - \left( {\frac{{{\partial }{\text{I}}_{\text{ejx}} }}{{{\partial }{\text{x}}_{\text{v}} }}{\text{t}}_{\text{ejx}} + \frac{{{\partial }{\text{I}}_{\text{ejy}} }}{{{\partial }{\text{x}}_{\text{v}} }}{\text{t}}_{\text{ejy}} + \frac{{{\partial }{\text{I}}_{\text{ejz}} }}{{{\partial }{\text{x}}_{\text{v}} }}{\text{t}}_{\text{ejz}} + {\text{I}}_{\text{ejx}} \frac{{{\partial }{\text{t}}_{\text{ejx}} }}{{{\partial }{\text{x}}_{\text{v}} }} + {\text{I}}_{\text{ejy}} \frac{{{\partial }{\text{t}}_{\text{ejy}} }}{{{\partial }{\text{x}}_{\text{v}} }} + {\text{I}}_{\text{ejz}} \frac{{{\partial }{\text{t}}_{\text{ejz}} }}{{{\partial }{\text{x}}_{\text{v}} }}} \right), $$

  (8.78)
  $$ \frac{{{\partial }{\text{f}}_{\text{ejy}} }}{{{\partial }{\text{x}}_{\text{v}} }} = - \left( {\frac{{{\partial }{\text{J}}_{\text{ejx}} }}{{{\partial }{\text{x}}_{\text{v}} }}{\text{t}}_{\text{ejx}} + \frac{{{\partial }{\text{J}}_{\text{ejy}} }}{{{\partial }{\text{x}}_{\text{v}} }}{\text{t}}_{\text{ejy}} + \frac{{{\partial }{\text{J}}_{\text{ejz}} }}{{{\partial }{\text{x}}_{\text{v}} }}{\text{t}}_{\text{ejz}} + {\text{J}}_{\text{ejx}} \frac{{{\partial }{\text{t}}_{\text{ejx}} }}{{{\partial }{\text{x}}_{\text{v}} }} + {\text{J}}_{\text{ejy}} \frac{{{\partial }{\text{t}}_{{\text{ejy}}} }}{{{\partial }{\text{x}}_{\text{v}} }} + {\text{J}}_{\text{ejz}} \frac{{{\partial }{\text{t}}_{\text{ejz}} }}{{{\partial }{\text{x}}_{\text{v}} }}} \right), $$

  (8.79)
  $$ \frac{{{\partial }{\text{f}}_{\text{ejz}} }}{{{\partial }{\text{x}}_{\text{v}} }} = - \left( {\frac{{{\partial }{\text{K}}_{\text{ejx}} }}{{{\partial }{\text{x}}_{\text{v}} }}{\text{t}}_{\text{ejx}} + \frac{{{\partial }{\text{K}}_{\text{ejy}} }}{{{\partial }{\text{x}}_{\text{v}} }}{\text{t}}_{\text{ejy}} + \frac{{{\partial }{\text{K}}_{\text{ejz}} }}{{{\partial }{\text{x}}_{\text{v}} }}{\text{t}}_{\text{ejz}} + {\text{K}}_{\text{ejx}} \frac{{{\partial }{\text{t}}_{\text{ejx}} }}{{{\partial }{\text{x}}_{\text{v}} }} + {\text{K}}_{\text{ejy}} \frac{{{\partial }{\text{t}}_{\text{ejy}} }}{{{\partial }{\text{x}}_{\text{v}} }} + {\text{K}}_{\text{ejz}} \frac{{{\partial }{\text{t}}_{\text{ejz}} }}{{{\partial }{\text{x}}_{\text{v}} }}} \right). $$

  (8.80)

• (3) Determination of ${}^{\text{ej}}{\bar{\text{r}}}_{{\,{\text{i}}}}$ from ${}^{\text{ej}}{\bar{\text{A}}}_{\text{i}}$:

  From Sect. 2.3, it is known that for a flat boundary surface, the boundary surface is expressed as ${}^{\text{i}}{\bar{\text{r}}}_{{\,{\text{i}}}} = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 \\ \end{array} } \right]^{\text{T}}$ when referred to boundary coordinate frame $({\text{xyz}})_{\text{i}}$. From Eq. (1.34), the expression ${}^{\text{ej}}{\bar{\text{r}}}_{{\,{\text{i}}}}$ for this flat boundary surface is given as

  $$ {}^{\text{ej}}{\bar{\text{r}}}_{{\,{\text{i}}}} = {}^{\text{ej}}{\bar{\text{A}}}^{\prime }_{\text{i}} \left[ {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ 0 \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{}^{\text{ej}}{\text{J}}_{\text{ix}} } \\ {{}^{\text{ej}}{\text{J}}_{\text{iy}} } \\ {{}^{\text{ej}}{\text{J}}_{\text{iz}} } \\ {{}^{\text{ej}}{\text{e}}_{\text{i}} } \\ \end{array} } \right], $$

  (8.81)

  where ${}^{\text{ej}}{\bar{\text{A}}}_{\text{i}}$ is given in Eq. (3.3) (or Eq. (3.26)), and

  $$ {}^{\text{ej}}{\text{e}}_{\text{i}} = - \left[ {({}^{\text{ej}}{\text{J}}_{\text{ix}} )({}^{\text{ej}}{\text{t}}_{\text{ix}} ) + ({}^{\text{ej}}{\text{J}}_{\text{iy}} )({}^{\text{ej}}{\text{t}}_{\text{iy}} ) + ({}^{\text{ej}}{\text{J}}_{\text{iz}} )({}^{\text{ej}}{\text{t}}_{\text{iz}} )} \right]. $$

  (8.82)

• (4) Determination of ${\text{d}}({}^{\text{ej}}{\bar{\text{r}}}_{{\,{\text{i}}}} )/{\text{d}\bar{\text{X}}}_{\text{sys}}$ from ${\text{d}}({}^{\text{ej}}{\bar{\text{A}}}_{\text{i}} )/{\text{d}\bar{\text{X}}}_{\text{sys}}$:

Once the numerical values of the components of ${\text{d}}({}^{\text{ej}}{\bar{\text{A}}}_{\text{i}} )/{\text{d}\bar{\text{X}}}_{\text{sys}}$ have been determined (see Appendix 2 of this chapter), ${\text{d}}({}^{\text{ej}}{\bar{\text{r}}}_{{\,{\text{i}}}} )/{\text{d}\bar{\text{X}}}_{\text{sys}}$ can be obtained by differentiating Eq. (8.81) to give

$$ \frac{{{\text{d}}\left( {{}^{\text{ej}}{\bar{\text{r}}}_{{\,{\text{i}}}} } \right)}}{{{\text{d}\bar{\text{X}}}_{\text{sys}} }} = \left[ {\begin{array}{*{20}l} {{\text{d}}({}^{\text{ej}}{\text{J}}_{\text{ix}} )/{\text{d}\bar{\text{X}}}_{\text{sys}} } \hfill \\ {{\text{d}}({}^{\text{ej}}{\text{J}}_{\text{iy}} )/{\text{d}\bar{\text{X}}}_{\text{sys}} } \hfill \\ {{\text{d}}({}^{\text{ej}}{\text{J}}_{\text{iz}} )/{\text{d}\bar{\text{X}}}_{\text{sys}} } \hfill \\ {{\text{d}}({}^{\text{ej}}{\text{e}}_{\text{i}} )/{\text{d}\bar{\text{X}}}_{\text{sys}} } \hfill \\ \end{array} } \right], $$

(8.83)

with

$$ \begin{aligned} \frac{{{\text{d}}({}^{\text{ej}}{\text{e}}_{\text{i}} )}}{{{\text{d}\bar{\text{X}}}_{\text{sys}} }} & = - \left[ {\frac{{{\text{d}}({}^{\text{ej}}{\text{J}}_{\text{ix}} )}}{{{\text{d}\bar{\text{X}}}_{\text{sys}} }}({}^{\text{ej}}{\text{t}}_{\text{ix}} ) + ({}^{\text{ej}}{\text{J}}_{\text{ix}} )\frac{{{\text{d}}({}^{\text{ej}}{\text{t}}_{\text{ix}} )}}{{{\text{d}\bar{\text{X}}}_{\text{sys}} }} + \frac{{{\text{d}}({}^{\text{ej}}{\text{J}}_{\text{iy}} )}}{{{\text{d}\bar{\text{X}}}_{\text{sys}} }}({}^{\text{ej}}{\text{t}}_{\text{iy}} ) + ({}^{\text{ej}}{\text{J}}_{\text{iy}} )\frac{{{\text{d}}({}^{\text{ej}}{\text{t}}_{\text{iy}} )}}{{{\text{d}\bar{\text{X}}}_{\text{sys}} }}} \right. \\ & \quad \left. + \,{ \frac{{{\text{d}}({}^{\text{ej}}{\text{J}}_{\text{iz}} )}}{{{\text{d}\bar{\text{X}}}_{\text{sys}} }}({}^{\text{ej}}{\text{t}}_{\text{iz}} ) + ({}^{\text{ej}}{\text{J}}_{\text{iz}} )\frac{{{\text{d}}({}^{\text{ej}}{\text{t}}_{\text{iz}} )}}{{{\text{d}\bar{\text{X}}}_{\text{sys}} }}} \right]. \\ \end{aligned} $$

(8.84)

Having obtained numerical values of ${}^{0}{\bar{\text{A}}}_{\text{ej}}$, ${}^{\text{ej}}{\bar{\text{r}}}_{{\,{\text{i}}}}$, ${\text{d}}({}^{0}{\bar{\text{A}}}_{\text{ej}} )/{\text{d}\bar{\text{X}}}_{\text{sys}}$ and ${\text{d}}({}^{\text{ej}}{\bar{\text{r}}}_{{\,{\text{i}}}} )/{\text{d}\bar{\text{X}}}_{\text{sys}}$, ${\text{d}}({\bar{\text{r}}}_{{\,{\text{i}}}} )/{\text{d}\bar{\text{X}}}_{\text{sys}}$ for a flat boundary surface can be computed directly from Eq. (8.24).

Appendix 4

Let the elements of an optical system be labeled from j = 1 to j = k sequentially, and the boundary surfaces be marked from i = 1 to i = n. In addition, let pose matrices ${}^{\text{0}}{\bar{\text{A}}}_{\text{ej}}$ (j = 1 to j = k) (Eqs. (3.1) and (3.25)) and ${}^{{\text{ej}}}{\bar{\text{A}}}_{\text{i}}$ (i = 1 to i = n) (Eqs. (3.3) and (3.26)) define the poses of the elements and boundary surfaces, respectively. The matrices ${}^{\text{0}}{\bar{\text{A}}}_{\text{i}}$ (i = 1 to i = n) required to perform raytracing can be obtained as (Eqs. (8.40), (8.41), (8.42) and (8.43))

$$ \begin{aligned} {}^{\text{0}}{\bar{\text{A}}}_{\text{i}} & = \text{tran}({\text{t}}_{\text{ix}} ,{\text{t}}_{{\text{iy}}} ,{\text{t}}_{{\text{i}z}} )\text{rot}(\bar{\text{z}},\upomega_{\text{iz}} ){\text{rot}}({\bar{\text{y}}},\upomega_{\text{iy}} ){\text{rot}}(\bar{\text{x}},\upomega_{\text{ix}} ) = {}^{\text{0}}{\bar{\text{A}}}_{{\text{ej}}} \, {}^{{\text{ej}}}{\bar{\text{A}}}_{\text{i}} \\ & = \left[ {\begin{array}{*{20}c} {{\text{I}}_{\text{ejx}} } & {{\text{J}}_{\text{ejx}} } & {{\text{K}}_{\text{ejx}} } & {{\text{t}}_{\text{ejx}} } \\ {{\text{I}}_{\text{ejy}} } & {{\text{J}}_{\text{ejy}} } & {{\text{K}}_{\text{ejy}} } & {{\text{t}}_{\text{ejy}} } \\ {{\text{I}}_{\text{ejz}} } & {{\text{J}}_{\text{ejz}} } & {{\text{K}}_{\text{ejz}} } & {{\text{t}}_{\text{ejz}} } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{}^{\text{ej}}{\text{I}}_{\text{ix}} } & {{}^{\text{ej}}{\text{J}}_{\text{ix}} } & {{}^{\text{ej}}{\text{K}}_{\text{ix}} } & {{}^{\text{ej}}{\text{t}}_{\text{ix}} } \\ {{}^{\text{ej}}{\text{I}}_{\text{iy}} } & {{}^{\text{ej}}{\text{J}}_{\text{iy}} } & {{}^{\text{ej}}{\text{K}}_{\text{iy}} } & {{}^{\text{ej}}{\text{t}}_{\text{iy}} } \\ {{}^{\text{ej}}{\text{I}}_{\text{iz}} } & {{}^{\text{ej}}{\text{J}}_{\text{iz}} } & {{}^{\text{ej}}{\text{K}}_{\text{iz}} } & {{}^{\text{ej}}{\text{t}}_{\text{iz}} } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right] \\ &= \left[ {\begin{array}{*{20}c} {{\text{I}}_{\text{ix}} } & {{\text{J}}_{\text{ix}} } & {{\text{K}}_{\text{ix}} } & {{\text{t}}_{\text{ix}} } \\ {{\text{I}}_{\text{iy}} } & {{\text{J}}_{\text{iy}} } & {{\text{K}}_{\text{iy}} } & {{\text{t}}_{\text{iy}} } \\ {{\text{I}}_{\text{iz}} } & {{\text{J}}_{\text{iz}} } & {{\text{K}}_{\text{iz}} } & {{\text{t}}_{\text{iz}} } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right], \\ \end{aligned} $$

where

$$ {\text{t}}_{\text{ix}} = {\text{I}}_{\text{ejx}} ({}^{\text{ej}}{\text{t}}_{\text{ix}} ) + {\text{J}}_{\text{ejx}} ({}^{\text{ej}}{\text{t}}_{\text{iy}} ) + {\text{K}}_{\text{ejx}} ({}^{\text{ej}}{\text{t}}_{\text{iz}} ) + {\text{t}}_{\text{ejx}} , $$

$$ {\text{t}}_{\text{iy}} = {\text{I}}_{\text{ejy}} ({}^{\text{ej}}{\text{t}}_{\text{ix}} ) + {\text{J}}_{\text{ejy}} ({}^{\text{ej}}{\text{t}}_{\text{iy}} ) + {\text{K}}_{\text{ejy}} ({}^{\text{ej}}{\text{t}}_{\text{iz}} ) + {\text{t}}_{\text{ejy}} , $$

$$ {\text{t}}_{\text{iz}} = {\text{I}}_{\text{ejz}} ({}^{\text{ej}}{\text{t}}_{\text{ix}} ) + {\text{J}}_{\text{ejz}} ({}^{\text{ej}}{\text{t}}_{\text{iy}} ) + {\text{K}}_{\text{ejz}} ({}^{\text{ej}}{\text{t}}_{\text{iz}} ) + {\text{t}}_{\text{ejz}} . $$

The numerical values of the six boundary pose variables (${\text{t}}_{\text{ix}} ,\,{\text{t}}_{\text{iy}} ,\,{\text{t}}_{\text{iz}} ,\,\upomega_{\text{ix}} ,\,\upomega_{\text{iy}}$ and $\upomega_{\text{iz}}$) for each boundary surface can then be determined from Eqs. (1.38) to (1.43). Furthermore, ${\text{d}}({}^{\text{0}}{\bar{\text{A}}}_{\text{i}} )/{\text{d}\bar{\text{X}}}_{\text{sys}}$ can be computed from Eq. (8.44) as

$$ \frac{{{\text{d}}({}^{\text{0}}{\bar{\text{A}}}_{\text{i}} )}}{{{\text{d}\bar{\text{X}}}_{\text{sys}} }} = \left[ {\frac{{{\partial }({}^{\text{0}}{\bar{\text{A}}}_{\text{i}} )}}{{{\partial }{\text{x}}_{\text{v}} }}} \right]_{{4 \times 4 \times {\text{q}}_{\text{sys}} }} = \left[ {\begin{array}{*{20}l} {{\partial }{\text{I}}_{\text{ix}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{J}}_{\text{ix}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{K}}_{\text{ix}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{t}}_{\text{ix}} /{\partial }{\text{x}}_{\text{v}} } \hfill \\ {{\partial }{\text{I}}_{\text{iy}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{J}}_{\text{iy}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{K}}_{\text{iy}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{t}}_{\text{iy}} /{\partial }{\text{x}}_{\text{v}} } \hfill \\ {{\partial }{\text{I}}_{\text{iz}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{J}}_{\text{iz}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{K}}_{\text{iz}} /{\partial }{\text{x}}_{\text{v}} } \hfill & {{\partial }{\text{t}}_{\text{iz}} /{\partial }{\text{x}}_{\text{v}} } \hfill \\ \qquad0 \hfill &\qquad 0 \hfill &\qquad 0 \hfill &\qquad 0 \hfill \\ \end{array} } \right]. $$

Now the Jacobian matrix ${\text{d}}\upomega_{\text{iz}} /{\text{d}\bar{\text{X}}}_{\text{sys}} = \left[ {{\partial}{\upomega}_{\text{iz}} /{\partial }{\text{x}}_{\text{v}} } \right]$ can be computed by differentiating Eq. (1.38) to obtain

$$ \frac{{{\partial} {\upomega}_{\text{iz}} }}{{{\partial }{\text{x}}_{\text{v}} }} = \frac{\text{E}}{\text{F}}, $$

(8.85)

where

$$ {\text{F}} = {\text{I}}_{\text{ix}}^{2} + {\text{I}}_{\text{iy}}^{2} , $$

(8.86)

$$ {\text{E}} = {\text{I}}_{\text{ix}} \frac{{{\partial }{\text{I}}_{\text{iy}} }}{{{\partial }{\text{x}}_{\text{v}} }} - {\text{I}}_{\text{iy}} \frac{{{\partial }{\text{I}}_{\text{ix}} }}{{{\partial }{\text{x}}_{\text{v}} }}. $$

(8.87)

Similarly, ${\text{d}}\upomega_{\text{iy}} /{\text{d}\bar{\text{X}}}_{\text{sys}} = \left[ {{\partial} {\upomega}_{\text{iy}} /{\partial }{\text{x}}_{\text{v}} } \right]$ can be obtained by differentiating Eq. (1.39) to give

$$ \frac{{{\partial} {\upomega}_{\text{iy}} }}{{{\partial }{\text{x}}_{\text{v}} }} = \frac{{{\text{G}} + {\text{H}}}}{\text{L}}, $$

(8.88)

in which

$$ {\text{L}} = {\text{I}}_{\text{iz}}^{2} + ({\text{I}}_{\text{ix}} {\text{C}}\upomega_{\text{iz}} + {\text{I}}_{\text{iy}} {\text{S}}\upomega_{\text{iz}} )^{2} , $$

(8.89)

$$ {\text{G}} = {\text{I}}_{\text{iz}} \left[ {( - {\text{I}}_{\text{ix}} {\text{S}}\upomega_{\text{iz}} + {\text{I}}_{\text{iy}} {\text{C}}\upomega_{\text{iz}} )\frac{{{\partial} {\upomega}_{\text{iz}} }}{{{\partial }{\text{x}}_{\text{v}} }} + \left( {\frac{{{\partial }{\text{I}}_{\text{ix}} }}{{{\partial }{\text{x}}_{\text{v}} }}C\upomega_{\text{iz}} + \frac{{{\partial }{\text{I}}_{\text{iy}} }}{{{\partial }{\text{x}}_{\text{v}} }}{\text{S}}\upomega_{\text{iz}} } \right)} \right], $$

(8.90)

$$ {\text{H}} = - \left( {{\text{I}}_{\text{ix}} {\text{C}}\upomega_{\text{iz}} + {\text{I}}_{\text{iy}} {\text{S}}\upomega_{\text{iz}} } \right)\frac{{{\partial }{\text{I}}_{\text{iz}} }}{{{\partial }{\text{x}}_{\text{v}} }}. $$

(8.91)

Finally, ${\text{d}}\upomega_{\text{ix}} /{\text{d}\bar{\text{X}}}_{\text{sys}} = \left[ {{\partial}\upomega_{\text{ix}} /{\partial }{\text{x}}_{\text{v}} } \right]$ can be obtained by differentiating Eq. (1.40) to give

$$ \frac{{{\partial} {\upomega}_{\text{ix}} }}{{{\partial }{\text{x}}_{\text{v}} }} = \frac{{{\text{PT}} - {\text{UQ}}}}{\text{M}}, $$

(8.92)

where

$$ {\text{M}} = ({\text{K}}_{\text{ix}} {\text{S}}\upomega_{\text{iz}} - {\text{K}}_{\text{iy}} {\text{C}}\upomega_{\text{iz}} )^{2} + ( - {\text{J}}_{\text{ix}} {\text{S}}\upomega_{\text{iz}} + {\text{J}}_{\text{iy}} {\text{C}}\upomega_{\text{iz}} )^{2} , $$

(8.93)

$$ {\text{P}} = - {\text{J}}_{\text{ix}} {\text{S}}\upomega_{\text{iz}} + {\text{J}}_{\text{iy}} {\text{C}}\upomega_{\text{iz}} , $$

(8.94)

$$ {\text{T}} = ({\text{K}}_{\text{ix}} {\text{C}}\upomega_{\text{iz}} + {\text{K}}_{\text{iy}} {\text{S}}\upomega_{\text{iz}} )\frac{{{\partial} {\upomega}_{\text{iz}} }}{{{\partial }{\text{x}}_{\text{v}} }} + \left( {\frac{{{\partial }{\text{K}}_{\text{ix}} }}{{{\partial }{\text{x}}_{\text{v}} }}{\text{S}}\upomega_{\text{iz}} - \frac{{{\partial }{\text{K}}_{\text{iy}} }}{{{\partial }{\text{x}}_{\text{v}} }}{\text{C}}\upomega_{\text{iz}} } \right), $$

(8.95)

$$ {\text{U}} = {\text{K}}_{\text{ix}} {\text{S}}\upomega_{\text{iz}} - {\text{K}}_{\text{iy}} {\text{C}}\upomega_{\text{iz}} , $$

(8.96)

$$ {\text{Q}} = ( - {\text{J}}_{\text{ix}} {\text{C}}\upomega_{\text{iz}} - {\text{J}}_{\text{iy}} {\text{S}}\upomega_{\text{iz}} )\frac{{{\partial} {\upomega}_{\text{iz}} }}{{{\partial }{\text{x}}_{\text{v}} }} + \left( { - \frac{{{\partial }{\text{J}}_{\text{ix}} }}{{{\partial }{\text{x}}_{\text{v}} }}{\text{S}}\upomega_{\text{iz}} + \frac{{{\partial }{\text{J}}_{\text{iy}} }}{{{\partial }{\text{x}}_{\text{v}} }}{\text{C}}\upomega_{\text{iz}} } \right). $$

(8.97)

It is noted from Eqs. (8.58) to (8.60) of Appendix 1 and Eqs. (8.74)–(8.76) of Appendix 2 that the (1,4)th, (2,4)th and (3,4)th components of matrices ${}^{0}{\bar{\text{A}}}_{\text{ej}}$ and ${}^{\text{ej}}{\bar{\text{A}}}_{\text{i}}$ are always given in explicit form. As a result, their Jacobian matrices with respect to ${\bar{\text{X}}}_{\text{sys}}$ can be determined simply by direct differentiation. Consequently, ${\text{dt}}_{\text{ix}} /{\text{d}\bar{\text{X}}}_{\text{sys}}$, ${\text{dt}}_{\text{iy}} /{\text{d}\bar{\text{X}}}_{\text{sys}}$, and ${\text{dt}}_{\text{iz}} /{\text{d}\bar{\text{X}}}_{\text{sys}}$ can be obtained from Eqs. (8.41) to (8.43) as

$$ \begin{aligned}\frac{{{\partial }{\text{t}}_{\text{ix}} }}{{{\partial }{\text{x}}_{\text{v}} }} &= \frac{{{\partial }{\text{I}}_{\text{ejx}} }}{{{\partial }{\text{x}}_{\text{v}} }}\left({{}^{\text{ej}}{\text{t}}_{\text{ix}} } \right) + \frac{{{\partial }{\text{J}}_{\text{ejx}} }}{{{\partial }{\text{x}}_{\text{v}} }}\left({{}^{\text{ej}}{\text{t}}_{\text{iy}} } \right) + \frac{{{\partial }{\text{K}}_{\text{ejx}} }}{{{\partial }{\text{x}}_{\text{v}} }}\left({{}^{\text{ej}}{\text{t}}_{\text{iz}} } \right)\\ & \quad + \, \frac{{{\partial }\left({{\text{t}}_{\text{ejx}} } \right)}}{{{\partial }{\text{x}}_{\text{v}} }} + {\text{I}}_{\text{ejx}} \frac{{{\partial }\left({{}^{\text{ej}}{\text{t}}_{\text{ix}} } \right)}}{{{\partial }{\text{x}}_{\text{v}} }} + {\text{J}}_{\text{ejx}} \frac{{{\partial }\left({{}^{\text{ej}}{\text{t}}_{\text{iy}} } \right)}}{{{\partial }{\text{x}}_{\text{v}} }} + {\text{K}}_{\text{ejx}} \frac{{{\partial }\left({{}^{\text{ej}}{\text{t}}_{\text{iz}} } \right)}}{{{\partial }{\text{x}}_{\text{v}} }}, \end{aligned} $$

(8.98)

$$ \begin{aligned}\frac{{\partial \text{t}_{\text{iy}} }}{{\partial \text{x}_{\text v} }} &= \frac{{{\partial }{\text{I}}_{\text{ejy}} }}{{{\partial }{\text{x}}_{\text{v}} }}\left({{}^{\text{ej}}{\text{t}}_{\text{ix}} } \right) + \frac{{{\partial }{\text{J}}_{\text{ejy}} }}{{{\partial }{\text{x}}_{\text{v}} }}\left({{}^{\text{ej}}{\text{t}}_{\text{iy}} } \right) + \frac{{{\partial }{\text{K}}_{\text{ejy}} }}{{{\partial }{\text{x}}_{\text{v}} }}\left({{}^{\text{ej}}{\text{t}}_{\text{iz}} } \right)\\ & \quad + \, \frac{{{\partial }\left({{\text{t}}_{\text{ejy}} } \right)}}{{{\partial }{\text{x}}_{\text{v}} }} + {\text{I}}_{\text{ejy}} \frac{{{\partial }\left({{}^{\text{ej}}{\text{t}}_{\text{ix}} } \right)}}{{{\partial }{\text{x}}_{\text{v}} }} + {\text{J}}_{\text{ejy}} \frac{{{\partial }\left({{}^{\text{ej}}{\text{t}}_{\text{iy}} } \right)}}{{{\partial }{\text{x}}_{\text{v}} }} + {\text{K}}_{\text{ejy}} \frac{{{\partial }\left({{}^{\text{ej}}{\text{t}}_{\text{iz}} } \right)}}{{{\partial }{\text{x}}_{\text{v}} }},\end{aligned} $$

(8.99)

$$ \begin{aligned} \frac{{{\partial }{\text{t}}_{\text{iz}} }}{{{\partial }{\text{x}}_{\text{v}} }} & = \frac{{{\partial }{\text{I}}_{\text{ejz}} }}{{{\partial }{\text{x}}_{\text{v}} }}\left({{}^{\text{ej}}{\text{t}}_{\text{ix}} } \right) + \frac{{{\partial }{\text{J}}_{\text{ejz}} }}{{{\partial }{\text{x}}_{\text{v}} }}\left({{}^{\text{ej}}{\text{t}}_{\text{iy}} } \right) + \frac{{{\partial }{\text{K}}_{\text{ejz}} }}{{{\partial }{\text{x}}_{\text{v}} }}\left({{}^{\text{ej}}{\text{t}}_{\text{iz}} } \right)\\ & \quad + \, \frac{{{\partial }\left({{\text{t}}_{\text{ejz}} } \right)}}{{{\partial }{\text{x}}_{\text{v}} }} + {\text{I}}_{\text{ejz}} \frac{{{\partial }\left({{}^{\text{ej}}{\text{t}}_{\text{ix}} } \right)}}{{{\partial }{\text{x}}_{\text{v}} }} + {\text{J}}_{\text{ejz}} \frac{{{\partial }\left({{}^{\text{ej}}{\text{t}}_{\text{iy}} } \right)}}{{{\partial }{\text{x}}_{\text{v}} }} + {\text{K}}_{\text{ejz}} \frac{{{\partial }\left({{}^{\text{ej}}{\text{t}}_{\text{iz}} } \right)}}{{{\partial }{\text{x}}_{\text{v}} }}.\end{aligned} $$

(8.100)