Direction Field

(redirected from Slope field)
Also found in: Wikipedia.

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 = f(x, y) is given, then at each point (x0, y0) of some region in the xy-plane the value of the slope k = f (x0, y0) 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 = 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).

Figure 1

Thedirection field of the equation = x2 + y2 is shown in Figure 1. The light curves (circles) represent isoclines, and the thicker curves represent integral curves.


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.
References in periodicals archive ?
Earlier this summer, BP and Prudhoe Bay co-owner ExxonMobil asked for Alaska Oil and Gas Conservation Commission approval of a plan to substantially increase the amount of gas allowed to be produced and sold from the North Slope field.
The sensor payloads produce imagery and data for processing into 3D computerized models of roads, pads and pipelines, and other actionable information, including precision volumetric measurement and topographic analysis of gravel pits at the North Slope field.
As for piping the condensate away, they plan to install a 22-mile elevated pipeline to the Badami field, at present the easternmost North Slope field.
Clay slope field investigations by direct cut may help to judge about the stability of the slope and a possible slip of it in the nearest future.
Plotting both solutions in Figure 5 superimposed against the slope-field for differential equation (2) shows that solution (12) given by DSolve goes against the slope field.
The customers will be North Slope field operating companies and contractors.
But make no mistake, establishing any flow at all from the remote eastern North Slope field will stand as a major victory for the state.
Calculus teachers used Wikki Stix and yarn to plot slope fields.
It will allow us to keep our North Slope fields in production for another 50 years.
In undergraduate Differential Equations courses, once slope fields are understood, a graphing calculator (or program such as ODE Architect, etc.
It can create several types of graphs: Cartesian graphs, graphs of tables, polar, parametric and implicit functions, inequalities and slope fields.
Many solutions can be displayed simultaneously and viewed as a flow, which is easier to understand than the more traditional slope fields.