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Slope Field

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SlopeField

Given an ordinary differential equation y^'=f(x,y), the slope field for that differential equation is the vector field that takes a point (x,y) to a unit vector with slope f(x,y). The vectors in a slope field are usually drawn without arrowheads, indicating that they can be followed in either direction. Using a visualization of a slope field, it is easy to graphically trace out solution curves to initial value problems. For example, the illustration above shows the slope field for the equation f(x,y)=x+y together with solution curves for various initial values of (x,y).

See also

Isocline, Pólya Plot, Slope, Unit Vector, Vector Field Explore this topic in the MathWorld classroom

Portions of this entry contributed by John Renze

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More things to try:

  • {sin(x), cos(y)}
  • {x^2 - y^2, x^2 + y^2}
  • {x sin(y), y cos(x)}

References

Thomas, G. B. Jr. and Finney, R. L. "Slope Fields and Picard's Theorem." §15.8 in Calculus and Analytic Geometry, 8th ed. Reading, MA: Addison-Wesley, pp. 1088-1089 and 1101, 1992.

Referenced on Wolfram|Alpha

Slope Field

Cite this as:

Weisstein, Eric W., with contributions by John Renze. "Slope Field." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SlopeField.html

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