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Definition: Direction fields are used in studying the solutions of differential equations in form of dy/dt = f(t,y). A direction field can be constructed by evaluating f at each point of a rectangular grid. At each point of the grid, a short line segment is drawn whose slope is the value of f at that point.
Source: Elementary Differential Equations and Boundary Value Problems - William Boyce & Richard DiPrima
Example: For the image : http://www.personal.psu.edu/ sul168/images/directionfield1.jpg