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 (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 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).

Figure 1

Thedirection field of the equation yʹ = x² + y² 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.

单词	direction field
释义	Direction Field 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 (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 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). Figure 1 Thedirection field of the equation yʹ = x² + y² 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.
随便看	bonac bon-accord bonaci bona dea bona dea festival bonae fidei contractum bon a enlever bona fide bonafide bona fide benefit plan bona fide bid bona fide determination bona fide error bona fide errors bona fide foreign resident bona-fide foreign resident bona fide foreign residents bona-fide foreign residents bona fidei bona fide kennel club bona fide occupational qualification bona fide occupational qualifications bona fide occupational requirement bona fide percepta et consumpta bona fide prospective purchaser