### 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
(

*x*_{0},

*y*_{0}) 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 (
*x*_{0},*y*_{0})
can be chosen in either of two ways:
- Enter the
values directly into the text input fields marked
"
*x*_{0}=" and
"*y*_{0}="
- Click and drag the red point on the graph

(Example:
(*x*_{0},*y*_{0})=(1,0))
- The value for
*t*_{max} (the maximum value for
the parameter *t*) can be changed by entering a new value directly
into the text input field marked "*t*_{max}="
(Example: *t*_{max}=6.28)
- To erase the graph and the text input fields for
*f* (*x*,*y*) and *g*(*x*,*y*), and
set *x*_{0}, *y*_{0}, and
*t*_{max}
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 *x*_{0} and
*y*_{0} 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
"Δ_{x}=" and
"Δ_{y}=" 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
del_{x}=1.0, del_{y}=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
(*x*_{0},*y*_{0}) 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)=*x*_{0} and
*y*(0)=*y*_{0}.

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.