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# 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 achieved without solving the differential equation analytically, and thence it is useful. The representation may be used to qualitatively visualise solutions, or to numerically approximate them.

## Definition

Given a system of differential equations,

$frac\left\{du\right\}\left\{dt\right\}=f\left(t,u,...y,z\right)$
$cdots$
$frac\left\{dy\right\}\left\{dt\right\}=j\left(t,u,...y,z\right)$
$frac\left\{dz\right\}\left\{dt\right\}=k\left(t,u,...y,z\right)$
the slope field is an array of slope marks in the phase space (the preceding equations imply seven dimensions, but can be any number depending on the number of relevant variables; for example, two in the case of a first-order linear ODE, as seen to the right). Each slope mark is centered at a point (t,u,...y,z) and is parallel to the vector

$begin\left\{pmatrix\right\} 1 f\left(t,u,...y,z\right) cdots j\left(t,u,...y,z\right) k\left(t,u,...y,z\right) end\left\{pmatrix\right\}$.
The number, position, and length of the slope marks can be arbitrary. The positions are usually chosen as (t,u,...y,z)=(aΔt, bΔu, ... eΔy, fΔz) for arbitrary (but usually equal) Δt, Δu, ... Δy, and Δz, and for all integers a, b, ... e, and f that produce points within the chosen t, u, ... y, and z intervals. The length of the slope marks is usually uniform throughout, and unitary or no greater than the least of Δt, Δu, ... Δy, and Δz.

## General application

With computers, complicated slope fields can be quickly made without tedium, and so an only recently practical application is to use them merely to get the feel for what a solution should be before an explicit general solution is sought. Of course, computers can also just solve for one, if it exists.

If there is no explicit general solution, computers can use slope fields (even if they aren’t shown) to numerically find graphical solutions. Examples of such routines are Euler's method, or better, the Runge-Kutta methods.