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.