### Vector Fields and Phase Plots in Three Dimensions

Graphs flow curves for the vector field in three dimensions given by the vector-valued function F (x,y,z)=<f (x,y,z),g(x,y,z),h(x,y,z)>, given parametrically as (x(t),y(t),z(t)) from starting point (x0,y0,z0) associated with the value t=0.
How to use   ||   Examples   ||   Other Notes

How to use
• Enter the function f (x,y,z) in the text input field marked "x'=f (x,y,z)=" (Example: x'=y)
• Enter the function g(x,y,z) in the text input field marked "y'=g(x,y,z)=" (Example: y'=-x)
• Enter the function h(x,y,z) in the text input field marked "z'=h(x,y,z)=" (Example: z'=0.5)
• Click the "Graph" button (this button also refreshes the graph). Be patient -- the graph takes a few seconds to be fully generated.
• The initial value point (x0,y0,z0) can be chosen by entering the values directly into the text fields marked "x0=", "y0=", and "z0=". (Example: (x0,y0,z0)=(3,0,0))
• The value of tmax can be chosen by entering the value directly into the text input field marked "tmax=" (Example: tmax=20)
• To erase the graph and the text input fields for f (x,y,z), g(x,y,z), and h(x,y,z), and set x0, y0, z0, and tmax to default values, click the "Clear" button
The text input fields for f (x,y,z), g(x,y,z), and h(x,y,z) 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, y0, and z0 can accept real numbers in decimal notation.
Examples
 Helix: x'=y y'= -x z'=0.5 (x0,y0,z0)=(3,0,0)     tmax=20 Twisted Circle: x'=y y'= -x z'=x.y (x0,y0,z0)=(3,0,0)     tmax=7 Lorenz Attractor: x'=10(y-x) y'=28x-y-5xz z'=5xy-(8/3)z (x0,y0,z0)=(1,0,0) tmax=30
(Again, be patient -- these examples take a few seconds to load and generate the graph.)
Other Notes
The graph shows flow curves for the vector field in three dimensions given by the vector-valued function F (x,y,z)=<f (x,y,z),g(x,y,z),h(x,y,z)>, given parametrically as (x(t),y(t),z(t)) from starting point (x0,y0,z0) associated with the value t=0. The flow curve will be graphed in blue, starting from the given initial point (associated with t=0) and continuing to the t value given by tmax.

This is equivalent to graphing a phase portrait and solution curve for the system of differential equations x'=f (x,y,z), y'=g(x,y,z), z'=h(x,y,z), derivatives with respect to t, having solutions of the form x(t), y(t), z(t)< with initial values given by x(0)=x0, y(0)=y0, and z(0)=z0.

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