Slope field

In mathematics, a slope field (or direction field) is a graphical representation of the solutions of a first-order differential equation. It is useful because it can be created without solving the differential equation analytically. The representation may be used to qualitatively visualize solutions, or to numerically approximate them.

Definition

Standard case

The slope field is traditionally defined for the following type of differential equations

It can be viewed as a creative way to plot a real-valued function of two real variables as a planar picture. Specifically, for a given pair , a vector with the components is drawn at the point on the -plane. Sometimes, the vector is normalized to make the plot better looking for a human eye. A set of pairs making a rectangular grid is typically used for the drawing.

An Isocline (a series of lines with the same slope) is often used to supplement the slope field. In an equation of the form , the isocline is a line in the -plane plane obtained by setting equal to a constant.