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\begin_body

\begin_layout Title
Math Review #4 Supplement (Math Camp 2007)
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\begin_layout Author
John Morrow
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\begin_layout Date
8/27/07
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\begin_layout Standard
What follows is a rough sketch of material for Day 4.
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\begin_layout Section
Day 4
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\begin_layout Enumerate
Multivariate Considerations
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\begin_deeper
\begin_layout Enumerate
Homogeneous functions.
\end_layout

\begin_layout Enumerate
Partial derivative notations, 
\begin_inset Formula $dx$
\end_inset

 and 
\begin_inset Formula $Df(x)$
\end_inset

.
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\begin_layout Enumerate
Derivative as a linear operator
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\begin_inset Formula \begin{alignat*}{3}
f(x) & =\sum_{i=1}^{N}i\cdot x^{i} & \; & \; & f(x) & =\sum_{i=1}^{N}e^{i\beta x}\end{alignat*}

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\begin_layout Enumerate
Partial derivatives 
\begin_inset Formula \begin{alignat*}{3}
f(x,y) & =x^{\alpha}y^{\beta} & \; & \; & f(x_{1},\ldots,x_{N}) & =(\sum\alpha_{i}x_{i}^{\rho})^{\frac{1}{\rho}}\end{alignat*}

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\begin_layout Enumerate
Definition of derivative for 
\begin_inset Formula $\mathbb{R}^{N}\rightarrow\mathbb{R}$
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, 
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\begin_deeper
\begin_layout Enumerate
How it looks.
 
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\begin_layout Enumerate
\begin_inset Quotes eld
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Hessian
\begin_inset Quotes erd
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.
 
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\begin_layout Enumerate
\begin_inset Quotes eld
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Young's Theorem
\begin_inset Quotes erd
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.
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\end_deeper
\begin_layout Enumerate
Hessian and Strict Concavity/Convexity.
\end_layout

\begin_layout Enumerate
Sufficient conditions for multivariate maxima.
\end_layout

\end_deeper
\begin_layout Enumerate
More Linear algebra 
\end_layout

\begin_deeper
\begin_layout Enumerate
Dot product, vector 
\begin_inset Quotes eld
\end_inset

multiplication
\begin_inset Quotes erd
\end_inset

 and distance.
\end_layout

\begin_layout Enumerate
Solving 
\begin_inset Formula $Ax=b$
\end_inset

, inverses.
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\begin_layout Enumerate
Eigenvectors/Eigenvalues and Definiteness.
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