Chapter 14.1, Problem 51E

Flow curves in the plane Let $F\left(x,y\right)=\left(f\left(x,y\right),g\left(x,y\right)\right)$ be defined on ${ℝ}^2$.

51.    Explain why the flow curves or streamlines of F satisfy $y^{\prime }=g\left(x,y\right)/f\left(x,y\right)$ and are everywhere tangent to the vector field.