Lecture20

# Lecture20 - \documentclass[12pt,letterpaper]cfw_article

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\documentclass[12pt,letterpaper]{article} \ \usepackage{amsmath} \usepackage{amssymb} \usepackage{latexsym} \usepackage{array} \usepackage{enumerate} \usepackage{amsthm} \usepackage{amscd} \ \newtheorem{lemma}{Lemma} \newtheorem{theorem}{Theorem} \newtheorem{definition}{Definition} \ \newcommand{\bC}{{\mathbb C}} \newcommand{\bH}{{\mathbb H}} \newcommand{\bR}{{\mathbb R}} \newcommand{\bN}{{\mathbb N}} \newcommand{\ra}{\rightarrow} \newcommand{\Ra}{\Rightarrow} \ \date{} \ \begin{document} \ \title{\bf Lecture 20} \author{} \maketitle \ Let $U \subset X_1 \times X_2$ and let $F: U \ra Y$ be twice-differentiable at $a \ in U$ (and differentiable on $U$). \\ i \noindent {\bf Recall:} $x \in U, (b_1,b_2) \in X_1 \times X_2$ $F'_x(b_1,b_2) = (\partial_1F)_x(b_1) + (\partial_2F)_x(b_2)$ \ If we write $\Lambda_1: X_1 \ra X_1 \times X_2: b_1 \mapsto (b_1, 0)$ $\Lambda_2: X_2 \ra X_1 \times X_2: b_2 \mapsto (0, b_2)$ then $(\partial_jF)_x(b_j) = F'_x(\Lambda_jb_j)$ or $(\partial_jF)_x = F'_x \circ \Lambda_j;$ that is, $\partial_jF = R_{\Lambda_j} \circ F'$ where $R_{\Lambda_j}: L(X,Y) \ra L(X_j,Y): T \mapsto T \circ \Lambda_j$ is bounded linear. Since $F'$ is differentiable at $a$ and $R_{\Lambda_j}$ is differentiable everywhere, it follows that $\partial_jF$ is differentiable at $a$ and the chain rule gives \begin{eqnarray*} \end{eqnarray*} As $(\partial_jF)$ is differentiable at $a$, its partial derivatives

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$\partial_j(\partial_jF)$ exist at $a$, and $(\partial_jF)'_a(b) = \partial_1(\partial_jF)_a(b_1) + \partial_2(\partial_jF)_a(b_2)$ or \begin{eqnarray*} \end{eqnarray*}
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## This note was uploaded on 11/09/2011 for the course MAT 6932 taught by Professor Staff during the Spring '10 term at University of Florida.

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Lecture20 - \documentclass[12pt,letterpaper]cfw_article

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