Due Friday October 27.

** Problem 1**

This problem explores the relationship between local extrema of functions and the geometry of flowlines for their gradients.

- a)
- Compute the gradient vector field
associated with the symmetric matrix

- b)
- Let and be orthogonal unit eigenvectors of A
with respective eigenvalues and . Verify that
is an integral curve for the gradient vector field

- c)
- When the eigenvalues and
are both nonzero with (say)
there are five cases:
- Improper Node Source:
- Proper Node Source:
- Saddle Equilibrium:
- (Here the relative magnitudes of and are not qualitatively very significant.)
- Improper Node Sink:
- Proper Node Sink:

- Give an example of a symmetric matrix whose eigenvalues satisfy the stated conditions.
- Decide what kind of critical point (local minimum, local maximum, or saddle) the origin is for the function .
- Using MacMath's phase plane program, plot some
flowlines for the gradient vector field

** Problem 2**

Read section 4.5 of Marsden and do (without a computer) Marsden page 297 number 1. This talks about a particle moving in a potential field on given by . ( The in problem 1 is an example ( with a change in sign ) of the kind of potential discussed in that section. )