% This texfile was generated by executing Latify on file op3d_act.tex
\documentclass[12pt]{sep2e}
\begin{document}

\input{macros_act}
\chapter{Tau operator in 3-D}
\label{chap:op3d}

\beqa
\tau(z,x,y) &=& \int_0^z 2 S_m(z',x,y) dz' \\
x'  &=& x \\
y'  &=& y 
\eeqa

\beqa
\frac{\partial t}{\partial z} &=&
\frac{\partial t}{\partial \tau} 
\frac{\partial \tau}{\partial z}  +
\frac{\partial t}{\partial x'} 
\frac{\partial y'}{\partial z}  +
\frac{\partial t}{\partial y'} 
\frac{\partial y'}{\partial z}  = 
\frac{\partial t}{\partial \tau} 2\smf \\  
\frac{\partial t}{\partial x} &=&
\frac{\partial t}{\partial \tau} 
\frac{\partial \tau}{\partial x}  +
\frac{\partial t}{\partial x'} 
\frac{\partial x'}{\partial x}  +
\frac{\partial t}{\partial y'} 
\frac{\partial y'}{\partial x}  = 
\frac{\partial t}{\partial x'} + 
\frac{\partial t}{\partial y'} + 
\frac{\partial t}{\partial \tau} \int_0^z \frac{\partial}{\partial x} 2s_m(z',x,y) dz'
\\
\frac{\partial t}{\partial y} &=&
\frac{\partial t}{\partial \tau} 
\frac{\partial \tau}{\partial y}  +
\frac{\partial x'}{\partial y}  +
\frac{\partial t}{\partial y'} 
\frac{\partial y'}{\partial y}  = 
\frac{\partial t}{\partial y'} + 
\frac{\partial t}{\partial \tau} \int_0^z \frac{\partial}{\partial y} 2s_m(z',x,y) dz'
\eeqa

\beqa
\sigma_x(z,x,y) &=& \int_0^z \frac{\partial}{\partial x} 2s_m(z',x,y) dz' \\
\sigma_y(z,x,y) &=& \int_0^z \frac{\partial}{\partial y} 2s_m(z',x,y) dz' 
\eeqa

\beqa
dz &=& \frac{d \tau}{2 s_m } -\frac{ \sigma_x dx'}{2 s_m} - \frac{\sigma_y dy
'}{2 s_m}\\
dx &=& dx'\\
dy &=& dy' 
\eeqa

\begin{equation}
dt = \sqrt{ \left(\widetilde{S_f} \widetilde{dx'} \right)^2 + \left( \widetilde{S_f} \widetilde{dy'}\right) +
\left(\frac{\widetilde{d \tau}- widetilde{\sigma_x} \widetilde{dx'}- 
\widetilde{\sigma_y} \widetilde{dy} }{2}\right)}
\end{equation}
\clearpage

\end{document}
