Session 2 Event 3
Session Two Event Three: Individual Cost
All of your behaviors have consequences. The violations you commit are not victimless.
Your choices have consequences. In this event you will be given a chance to quantify
your costs for your beha
Denition 1. Let M be a metric space. A contraction on M is a function
: M M such that there exists a positive constant k < 1 with
a, b M = d(a), (b) kd(a, b).
Theorem 1. (Contraction Principle): Each contraction on a complete
metric space M ha
Theorem 1. Let X be a Banach space. The map
exp : L(X )L(X ) : A eA
is dierentiable at 0 and
exp0 = idL(X ) .
eH e0 idL(X ) H
Proof. We must estimate
Because e =
H , we have that
e I H =
H = H2
(n + 2)!
Dierentiability (of inversion)
First suppose F is dierentiable at A G(X ). From
F (A)A = I
we dierentiate by the Leibniz rule and obtain
FA (H )A + F (A)H = 0
FA (H ) = F (A)HA1 = A1 HA1 .
This shows what the inverse is, assuming its existence.
Denition 1. Let U be open in X . When F : U Y is dierentiable at all
points a U , its derivative is the map
F : U L(X, Y ) : a Fa
and we call F continuously dierentiable when F is continuous.
Theorem 1. Suppose U contained in X1 X2 ; then F : U
Theorem 1. DetA (H ) = T r(AH )
Proof. It is enough to see both sides agree when H is a standard basis matrix
Left Hand Side:
DetA (Eij ) =
+ tEij )|t=0
+ (1)i+j tAij )|t=0 (where Aij is the minor of A)
DetA (H ) = (Det A) Tr(A1 H ) if A Mn (R) is invertible and H Mn (R).
with columns a1 , . . . , an Rn . In particular,
for the standard basis e1 , . . . , en of Rn . Since Det A is multilinear in a
Exam Chapter 8 TakeHome Test
Test the claim. (traditional method you may use the pvalue if you wish.
State the claim in symbols.
Identify the null and the alternative hypotheses.
Sketch the curve and label the critical value(s) an
L = [a, b] U Y
F dierentiable at each point of L
p L such that (F (b) F (a) = (Fp (b a)
Theorem 1 (MVT). If F : U Y is dierentiable at each point of [a, b] U
then there exists p (a, b) such that
F (b) F (a) Fp (b a)
Example: Recall that if F : U Y and G : U Z are dierentiable at
a U , then so is
H = (F, G) : U Y Z
Ha (h) = (Fa (h), Ga (h).
Conversely, if H is dierentiable at a then so are F and G.
[Let P : Y Z Y be rst-factor projection: P (y,
Spatial Laplacian in Spherical Coordinates
Denition 1. The Laplacian is dened as
+ 2+ 2
The Laplacian is seen in equations such as
Laplaces Equation: f = 0
Heat Equation1 : f =
Wave Equation1 : f =
(such f are called harmonic)
Invariant Bilinear Forms on VN :
= x1 y 2 x2 y 1 .
(x, y ) = Tr(xy )
x, y sl2
To see that such is invariant, compute (z x, y ) and (x, z y ) and add:
(z x, y ) =
(x, z y ) =
Tr([z, x]y )
Tr(zxy ) Tr(xzy )