Reading Assignment:
  Read 7.1 and 7.2 which will be covered on Wednesday.


Problems:
  Section 13.7 : 1,3,5,6,8,13,17

Notes for Reading (care of John & Alex):

Notes for Section 7.1:
  1) a gentle reading of the first few examples is fine, but  
      look carefully at p. 337-339 
  2) how do we transform a single equation into a system of 
      equations? 
  3) in the context of systems of equations what do the following 
      words mean: solution, linear, nonlinear, homogeneous,
      nonhomogeneous 
  4) what do Theorems 7.1.1 & 7.1.2 say? 

Notes for Section 7.2:
You must read this section. There is a lot of vocabulary. 
  1) What do the following words mean: matrix, transpose, 
      conjugate, adjoint, length, magnitude, orthogonal, identity, 
      inverse, nonsingular, singular, row reduction 
  2) Do you understand the following properties/operations of matrices:
      equality, zero, addition, scalar multiplication, subtraction, 
      multiplication, vector multiplication, identity, inverses
  3) What are matrix functions? What about derivatives and integrals of
      matrix functions? 


Think about the following rephrasing of Theorem 7.1.1 (John Finn):

We consider the Initial value problem given by the system of
differential equations 

                      x_1' = F_1(t, x_1, ..., x_n),          (12)
                      x_2' = F_2(t, x_1, ..., x_n),
                                   ...
                      x_n' = F_n(t, x_1, ..., x_n),

and the initial conditions 
                        x_1(t_0) = x_1^0,                       (14)
                        x_2(t_0) = x_2^0,
                           ...
                        x_n(t_0) = x_n^0.
Suppose that the F_n's are continuous, and have continuous first
partials with respect to the x's, on the box
           B =  [a,b] X [a_1,b_1] X ... X [a_n, b_n]     
in t, x_1, ..., x_n space, and that 
           X_0  = (t_0, x_(1,0), x_(2,0), ..., x_(n,0)) 
is a point in B.   

Then there exists a unique solution 
                        x_1 = phi_1(t),
                        x_2 = phi_2(t),
                          ...
                        x_n = phi_n 
to the initial value problem given by (12) and (14) on some interval 
[t_0 - h, t_0 + h] about t_0.