= Direction field
{title2=$y'=f(t,y)$}
= Slope field
{synonym}
A <direction field> for $y'=f(t,y)$ places a short segment with slope $f(t,y)$ at each sampled point where $f$ is defined. A <function graph> of a <solution of a differential equation> follows these segments: its <tangent line> has exactly the prescribed slope. Segment length is a drawing choice, rather than a speed; arrows, if used, point toward increasing $t$. An <isocline> collects points with one common slope.
For <Newton's law of cooling>, $y'=-0.1(y-20)$ with time in minutes and <temperature> in degrees Celsius, the slopes at $y=80,40,20,10$ are respectively $-6,-2,0,1$ degrees Celsius per minute. Each horizontal row is an <isocline>, because this <autonomous differential equation> has no explicit time dependence. Above $20$ the <solutions of a differential equation> descend, below $20$ they rise, and $y=20$ is an <equilibrium point>.
\Image[/ode-cooling-direction-field.png]
{title=Cooling trajectories follow the direction field toward the ambient temperature}
The highlighted <initial condition> $y(0)=80$ selects one curve from the family. The picture reveals approach to the <equilibrium point> before any algebra is done. Under the hypotheses of the <Picard-Lindelöf theorem>, distinct <function graphs> of solutions cannot cross or even meet in the time-state plane. A <direction field> alone does not guarantee uniqueness: several curves can share a <tangent line> at the same point when those hypotheses fail. See the delayed-start example under <initial value problem>.
This geometric interpretation is developed in <18.03SC Differential Equations, Fall 2011/Direction Fields, Isoclines, Integral Curves>.
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