**Week 10**

This week we covered material in Spivak,
Chapter 5. The main theorem is the following (Spivak, Theorem 5.2).

*Theorem 1*. Suppose *f*: *U* --> R^*n*
is a function defined on an
open set *U* in R^*n*. Suppose *x*_0 is a point of *U*
and that the closed
ball *B* = *B*_{2*a*}(*x*_0) is contained in *U*,
for some *a* > 0. Suppose that
*f* satisfies a Lipschitz condition on *B*,
i.e. that there exists a number *K* such that

|*f*(*x*) - *f*(*y*)| < *K* |*x* - *y*|

for every *x*,*y* in *B*.
Such a function is automatically continuous
on *B*. Let *L* be an upper bound for |*f*| on *B*, and
choose *b* > 0 such that *b* < = *a*/*L*
and that *b* < 1/*K*.

Then for each *x* in the closed ball *B*_*a*(*x*_0)
there exists a
unique solution *alpha*_*x* : (-*b*,*b*)
--> *U* to the initial value
problem

*alpha*'(*t*) = *f*(*alpha*(*t*)),

*alpha*(0) = *x*.

This theorem is proved using the contraction lemma (similarly to its use in the proof of the inverse function theorem).

We also proved the following (Spivak, Theorem 5.4).

*Theorem 2*. With the same notation, the map

*alpha* : (-*b*,*b*) x *B*_*a*(*x*_0)
--> *U*

defined by *alpha*(*t*,*x*) =
*alpha*_*x*(*t*) is continuous in both
variables.

And we agreed to accept without proof the theorem (see
Lang1 or Lang2) that if
*f* is of class C^*k*, then so is *alpha*.

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