\documentclass[a4paper,12pt]{article}
\usepackage{epsfig}
\newcommand{\un}{\underline}
\begin{document}
\parindent=0pt


\begin{center}
\textbf{Vector Fields}

\textit{\textbf{Scalar and Vector Fields}}
\end{center}

\textbf{Question}

Describe the streamlines of the following velocity field.

$\un{v}(x,y,z) = y\un{i} -x\un{j} +\un{k}$


\textbf{Answer}

The streamlines satisfy $\frac{dx}{y}=-\frac{dy}{x}=dz$.
Thus $xdx+ydy=0$, so $x^2+y^2 = C_1^2$.

Therefore $$\frac{dz}{dx}=\frac{1}{y}=\frac{1}{\sqrt{C_1^2-x^2}}.$$
This implies that $z=\sin^{-1} \frac{x}{C_1} + C_2$.

The streamlines are the spirals in which the surfaces
$x=C_1 \sin (z-C_2)$ intersect the cylinders $x^2+y^2=C_1^2$.

\end{document}