Slope field
A slope field (also called a direction field[1]) is a graphical representation of the solutions to a first-order differential equation[2] of a scalar function.
Solutions to a slope field are functions drawn as solid curves.
A slope field shows the slope of a differential equation at certain vertical and horizontal intervals on the x-y plane, and can be used to determine the approximate tangent slope at a point on a curve, where the curve is some solution to the differential equation.
The slope field can be defined for the following type of differential equations which can be interpreted geometrically as giving the slope of the tangent to the graph of the differential equation's solution (integral curve) at each point (x, y) as a function of the point coordinates.
[3] It can be viewed as a creative way to plot a real-valued function of two real variables
is normalized to make the plot better looking for a human eye.
making a rectangular grid is typically used for the drawing.
Given a system of differential equations, the slope field is an array of slope marks 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, 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
The standard case, described above, represents
The general case of the slope field for systems of differential equations is not easy to visualize for
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.
If there is no explicit general solution, computers can use slope fields (even if they aren’t shown) to numerically find graphical solutions.
Different software packages can plot slope fields.
