### Vector Fields and Phase Plots

Graphs the vector field in the plane given by the vector-valued function F (x,y)=<f (x,y),g(x,y)> and flow curves given parametrically as (x(t),y(t)) from starting point (x0,y0) associated with the value t=0.
How to use   ||   Examples   ||   Other Notes

How to use
• Enter the function f (x,y) for x' in the text input field marked "x'=f (x,y)=" (Example: x'=y)
• Enter the function g(x,y) for y' in the text input field marked "y'=g(x,y)=" (Example: y'= -x)
• Click the "Graph" button (this button also refreshes the graph)
• The initial value point (x0,y0) can be chosen in either of two ways:
• Enter the values directly into the text input fields marked "x0=" and "y0="
• Click and drag the red point on the graph
(Example: (x0,y0)=(1,0))
• The value for tmax (the maximum value for the parameter t) can be changed by entering a new value directly into the text input field marked "tmax=" (Example: tmax=6.28)
• To erase the graph and the text input fields for f (x,y) and g(x,y), and set x0, y0, and tmax to default values, click the "Clear" button (this also sets the fields in the "Field" window to default values -- see below)
The text input fields for f (x,y) and g(x,y) can accept a wide variety of expressions to represent functions, and the buttons under the graph allow various manipulations of the graph coordinates. The text input fields for x0 and y0 can accept real numbers in decimal notation.

The "Field" button displays a window which allows manipulations of the vectors shown in green on the graph: the values in the "Δx=" and "Δy=" fields in the window determine the x and y spacing (respectively) of the grid points at which the vectors are plotted, and the value in the "scale=" field determines a scaling factor by which the vectors can be reduced or extended. Standard values (set by the "Std" button in the window) are delx=1.0, dely=1.0, and scale=0.1 (these are also the default values set by the "Clear" button). These text input fields can accept any real number in decimal notation.

Examples
 Circles: x'=y y'= -x (x0,y0)=(1,0)     tmax=6.28 Predator-Prey Model: x'=x(1-y) y'=y(x-1) (x0,y0)=(3,1) tmax=5

Other Notes
The graph shows the vector field in the plane given by the vector-valued function F (x,y)=<f (x,y),g(x,y)> and flow curves given parametrically as (x(t),y(t)) from initial point (x0,y0) associated with the value t=0. The flow curve is graphed in blue, starting from the given initial point (associated with t=0) and continuing to the t value given in the "tmax=" text input field. The vectors <f (x,y),g(x,y)> associated with various points on the plane are shown in green.

This is equivalent to graphing a phase portrait and solution curve for the system of differential equations x'=f (x,y) and y'=g(x,y), derivatives with respect to t, having solutions of the form x(t) and y(t) with initial values given by x(0)=x0 and y(0)=y0.

The particular solution curve (x(t),y(t)) is graphed by determining a numerical approximation to the curve using the classical (order four) Runge-Kutta Method.