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\begin{center}
\textbf{Vector Fields}

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

\textbf{Question}

Describe the streamlines of the following velocity field.

$\un{v}(x,y) = x\un{i} +(x+y)\un{j}$, \it{Hint:} let $y=xv(x)$.


\textbf{Answer}

The field lines satisfy
\begin{eqnarray*}
\frac{dx}{x} & = & \frac{dy}{x+y}\\
\frac{dy}{dx} = \frac{ x+y}{x} \ \ \ \rm{Let \ }y=xv(x)\\
\Rightarrow \ \frac{dy}{dx} & = & v=+\frac{dv}{dx}\\
\Rightarrow V+ x\frac{dv}{dx} & = & \frac{x(1+v)}{x}\\
& = & 1 + v
\end{eqnarray*}
Thus $\frac{dv}{dx}=\frac{1}{x}$, and so $v(x) = \ln |x| + C$.

So the field lines have equations $y=x\ln |x| + Cx$.


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