isocline

Drawing a slope field by hand, point by point, is tedious — but there is a clever shortcut. Instead of hopping from point to point, hunt for whole CURVES along which the prescribed slope is the same constant everywhere. Along such a curve every little segment of the field is parallel, all tilted at the identical angle, so you can rake in a whole swath of the field at once. These curves of constant slope are called isoclines, from Greek for 'equal incline.'

Precisely, an isocline of the equation y' = f(x, y) is the set of points where f(x, y) = m for a chosen constant m: the curve along which every solution crosses with slope exactly m. To use it, pick a value of m, draw the curve f(x, y) = m, and then hatch short segments of slope m all along that curve. Repeat for several values of m and the slope field assembles itself efficiently. For y' = x + y, the isocline of slope m is the line x + y = m — a family of parallel lines, each carrying field segments of its own constant tilt.

Isoclines are mainly a practical drawing aid, but they also sharpen your intuition: one special isocline, the one where the slope is zero (f(x, y) = 0), is the locus of all horizontal tangents — where solution curves reach a maximum, minimum, or level off. That zero-slope isocline is closely tied to equilibrium behaviour and to where solutions turn around. So isoclines do double duty: they make the slope field drawable by hand, and they pinpoint the special places where solutions flatten out.