Vector Fields and Phase Plots in Three Dimensions
Graphs flow curves for the vector field in three dimensions
given by the vectorvalued
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
(
x_{0},
y_{0},
z_{0})
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
(x_{0},y_{0},z_{0})
can be chosen by entering the
values directly into the text fields marked
"x_{0}=",
"y_{0}=", and
"z_{0}=".
(Example:
(x_{0},y_{0},z_{0})=(3,0,0))
 The value of t_{max}
can be chosen by entering the value directly
into the text input field marked
"t_{max}="
(Example: t_{max}=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 x_{0}, y_{0}, z_{0}, and
t_{max}
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 x_{0}, y_{0},
and z_{0} can accept real numbers
in decimal notation.
Examples
Helix:
x'=y
y'= x
z'=0.5
(x_{0},y_{0},z_{0})=(3,0,0)
t_{max}=20

Twisted Circle:
x'=y
y'= x
z'=x^{.}y
(x_{0},y_{0},z_{0})=(3,0,0)
t_{max}=7

Lorenz Attractor:
x'=10(yx)
y'=28xy5xz
z'=5xy(8/3)z
(x_{0},y_{0},z_{0})=(1,0,0)
t_{max}=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 vectorvalued
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
(x_{0},y_{0},z_{0})
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 t_{max}.
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)=x_{0},
y(0)=y_{0}, and z(0)=z_{0}.
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)
RungeKutta Method.