Slope field - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a slope field (or direction field) is a graphical representation of the solutions of a first-order differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

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

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

Isocline

thumb|right|300px|Fig. 1: Isoclines , slope field , and some solution curves of y'=xyAn Isocline is a curve through points at which the parent function's slope will always be the same, regardless of initial conditions...

(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.

General case of a system of differential equations

Given a system of differential equations,

the slope field is an array of slope marks in the phase space

Phase space

In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...

(in any number of dimensions depending on the number of relevant variables; for example, two in the case of a first-order linear ODE

Ordinary differential equation

In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....

, as seen to the right). Each slope mark is centered at a point

and is parallel to the vector

.
The number, position, and length of the slope marks can be arbitrary. The positions are usually chosen such that the points

make a uniform grid. The standard case, described above, represents

. The general case of the slope field for systems of differential equations is not easy to visualize for

General application

With computers, complicated slope fields can be quickly made without tedium, and so 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.

Example code in GNU Octave
GNU Octave
GNU Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command-line interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB...

/MATLAB
MATLAB
MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...

Ffun = @(X,Y)X.*Y; % function f(x,y)=xy
[X,Y]=meshgrid(-2:.3:2,-2:.3:2); % choose the plot sizes
DY=Ffun(X,Y); DX=ones(size(DY)); % generate the plot values
quiver(X,Y,DX,DY); % plot the direction field
hold on;
contour(X,Y,DY,[-6 -2 -1 0 1 2 6]); %add the isoclines
title('Slope field and isoclines for f(x,y)=xy')

Standard case

General case of a system of differential equations

General application

See also