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
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Examples
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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
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.