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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

## 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.## Examples

## See also

## External links

## References

Blanchard, Paul; Devaney, Robert L.; and Hall, Glen R. (2002). Differential Equations (2nd ed.). Brooks/Cole: Thompson Learning. ISBN 0-534-38514-1

Given a system of differential equations,

- $frac\{du\}\{dt\}=f(t,u,...y,z)$

- $cdots$

- $frac\{dy\}\{dt\}=j(t,u,...y,z)$

- $frac\{dz\}\{dt\}=k(t,u,...y,z)$

- $begin\{pmatrix\}\; 1\; f(t,u,...y,z)\; cdots\; j(t,u,...y,z)\; k(t,u,...y,z)\; end\{pmatrix\}$.

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.

- Examples of differential equations
- Differential equations of mathematical physics
- Differential equations from outside physics
- Laplace transform applied to differential equations
- List of dynamical systems and differential equations topics

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Last updated on Sunday October 05, 2008 at 18:59:18 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Sunday October 05, 2008 at 18:59:18 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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