# Direction Field

(redirected from*Slope field*)

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## Direction Field

a set of points in the *xy*-plane such that at each point a certain direction is defined. The direction is usually represented by an arrow passing through the point.

If the equation *yʹ* = *f(x, y*) is given, then at each point (*x*_{0}, *y*_{0}) of some region in the *xy*-plane the value of the slope *k* = *f* (*x*_{0}, *y*_{0}) of the tangent to the integral curve passing through this point is known, so that the direction of the tangent can be represented by an arrow. Thus the differential equation defines a direction field; conversely, a direction field in some region of the *xy*-plane defines a differential equation of the form *yʹ* = *f(x, y).* By sketching sufficiently many isoclines (curves joining points at which the direction field *f* (*x, y*) = *C, C* a constant, has the same value), we can obtain approximations to the integral curves, that is, curves having the prescribed directions as tangents (the isocline method).

Thedirection field of the equation *yʹ* = *x*^{2} + *y*^{2} is shown in Figure 1. The light curves (circles) represent isoclines, and the thicker curves represent integral curves.

### REFERENCES

Stepanov, V. V.*Kurs differentsial’nykh uravnenii*, 8th ed. Moscow, 1959.

Petrovskii, I. G.

*Lektsii po teorii obyknovennykh differentsial’nykh uravnenii*, 6th ed. Moscow, 1970.