This worksheet does not need to be handed in, but you are responsible
for the material. The ideas laid out here will be used in next week's
lab (which will be graded). Some of the instructions for this lab are
taken from J. C. Polking, *MATLAB Manual: Ordinary Differential
Equations*, Prentice-Hall, 1994.

Gradients appear everywhere in nature. Newton's law of graviation involve the gradient of a ``gravitational potential function.'' The equations of electromagnetism (Maxwell's equations) and fluid dynamics (Navier-Stokes equations) also involve gradient terms. Furthermore, gradient differential equations are used in science and engineering to optimize functions of several variables. Understanding the geometry of gradients is one of the most important goals of this course.

The purpose of the lab is

- to learn how to numerically produce solutions to a planar differential equation using Matlab and PPLANE.
- to explore the geometry of gradient differential equations.
- to better understand the distinction between graphs and images of functions.

In this lab we will study gradient flows while simultaneously
learning to use PPLANE.
To get started, invoke matlab by typing
`matlab` in a Unix shell window.
When the matlab prompt appears, type `pplane`
to launch the ``phase plane'' module.
The **PPLANE Setup** window will appear.

In this lab we will examine the gradient flow for the following function:

This function contains several critical points of varying types.

The gradient flow of a function *f* is the differential equation defined
by , . For functions of two
variables, this can be rewritten as

Since the gradient vector always points in the direction in which *f*
increases the fastest (or, more correctly, the direction in which the
linear approximation to *f* increases the fastest), trajectories in a
gradient differential equation ``flow uphill.'' You can think of the
trajectories as the paths in (*x*,*y*)-space traced out by a robot that
is programmed to always walk in the steepest direction.

For our example function, and
. Therefore the gradient differential equation
determined by the function *f* is

Note that for each point (*x*,*y*), the right-hand side of the
differential equation, , defines a vector. We therefore
say that the right-hand side of the differential equation defines a
*vector field*.

PPLANE has a default differential equation. Delete it, and type the right-hand side of the differential equation above into the first two fields of the PPLANE Setup window. In the lower-left corner of the setup window, change the minimum and maximum values of the display window to and . Now hit the

Inside this window is a representation of the vector field for this
differential equation. The vector field at (*x*,*y*) must be tangent to
the solution curve passing through (*x*,*y*). To compute and plot a
solution curve from an initial point, move the mouse to that point,
and click the left mouse button. The solution will be computed and
plotted: first in the direction of *increasing* time, then in the
direction of *decreasing* time.

- away from a critical point
- near a maxima
- near a minima
- near a saddle point

A trajectory is a parametrized curve, , such that the tangent vector to the curve at is exactly equal to the vector field evaluated at . Thus differential equations give us another opportunity to study parametrized curves.

Note that once a trajectory is numerically determined, we can view the trajectory in several ways:

- as a parametrized curve in the (
*x*,*y*)-plane. (This is what we have been doing so far today). - as the graph
*t*versus*x*(*t*) and the graph*t*versus*y*(*t*). - as the graph of the function in three-dimensional space

The first point of view is the view adopted in dynamical systems,
and leads to a rich geometric understanding of differential
equations. This is the point of view we will typically use in this class. The
second point of view is used when it is possible to explicitly solve
a differential equation for the functions *x*(*t*) and *y*(*t*). We will
see some of this in the next quarter of this course. The third point of view unites
the previous two views, and leads to a better understanding of the
difference between ``graph'' and ``image.''

- as
*t*increases, what value does*x*(*t*) approach? - as
*t*increases, what value does*y*(*t*) approach? - What do these two values mean geometrically?

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Last modified: Oct 23 1996

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