\magnification 1800 \parskip 10pt \parindent 0pt \hoffset -0.3truein \hsize 7truein \def\N{{\bf N}} \def\S{\Sigma_{i=1}^n} \def\Sm{\Sigma_{i=1}^m} \def\R{{\bf R}} \def\C{{\bf C}} \def\x{{\bf x}} \def\a{{\bf a}} \def\y{{\bf y}} \def\e{{\bf e}} \def\i{{\bf i}} \def\j{{\bf j}} \def\k{{\bf k}} \def\ep{\epsilon} \def\f{\vskip 10pt} \def\u{\vskip -5pt} HW 2.1: 2, 8 (due Friday, next week) $f: \R^n \rightarrow \R$ is differentiable at $a$ if \vskip -5pt $$lim_{{\bf x} \rightarrow {\bf a}}{f({\bf x}) - f({\bf a}) - T({\bf x}- {\bf a}) \over ||{\bf x} - {\bf a}||} = 0$$ $$lim_{{\bf x} \rightarrow {\bf a}} {f({\bf x}) - f({\bf a}) - \Sigma b_i(x_i - a_i) \over ||{\bf x} - {\bf a}||} = 0$$ $y = f({\bf a}) + \Sigma b_i(x_i - a_i)$ approximates $y = f({\bf x})$ $f({\bf x}) = f({\bf a}) + T({\bf x}- {\bf a}) + ||{\bf x} - {\bf a}|| r({\bf x}, {\bf a})$ \rightline{where $lim_{{\bf x} \rightarrow {\bf a}} r({\bf x}, {\bf a}) = 0$} Thm 1.1: If $f$ is differentiable at $a$, then 1.) $f$ is continuous at $a$. 2.) All partial derivatives exist at $a$. 3.) $b_i = ( {\partial f \over \partial x_i})_a$ Proof: 1.) $lim_{{\bf x} \rightarrow {\bf a}}f({\bf x}) = lim_{{\bf x} \rightarrow {\bf a}}f({\bf a}) + T({\bf x}- {\bf a}) + ||{\bf x} - {\bf a}|| r({\bf x}, {\bf a}) =$ 2,3.) ${\partial f \over \partial x_j}({\bf a}) = lim_{h \rightarrow 0} { f({\bf a} + h {\bf e_j}) - f({\bf a}) \over h}$ $= lim_{h \rightarrow 0} { f({\bf a}) + T( {\bf a} + h {\bf e_j}- {\bf a}) + || {\bf a} + h {\bf e_j}- {\bf a}|| r( {\bf a} + h {\bf e_j}, {\bf a}) - f({\bf a}) \over h}$ $= lim_{h \rightarrow 0} { T( h {\bf e_j}) + |h| r( {\bf a} + h {\bf e_j}, {\bf a}) \over h}$ $= lim_{h \rightarrow 0} { h T( {\bf e_j}) + |h| r( {\bf a} + h {\bf e_j}, {\bf a}) \over h}$ Thm 1.3: If ${\partial f \over \partial x_j}$ exist for all $j$ in a nbhd of $a$ and if they are continuous at $a$, then $f$ is differentiable at $a$. \vskip 5pt \hrule \u Defn: Let $V$ be a nonempty open subset of $R^n$, $f: V \rightarrow R^m$, $p \in {\N}$. \u i.) $f$ is $C^p$ on $V$ is each partial derivative of order $k \leq p$ exists and is continuous on $V$. ii.) $f$ is $C^\infty$ on $V$ if $f$ is $C^p$ on $V$ for all $p \in {\N}$ ($f$ is {\it smooth}). \vskip 5pt \hrule \u Chain rule 1: Suppose $f: (a, b) \rightarrow \R^n$, $g: \R^n \rightarrow \R$, then ${d \over dt} (g \circ f)_{t_0} = D(G)_{f(t_0)} D(f)_{t_0} = (b_1, ..., b_n) \left(\matrix{ f_1'(t_0) \cr f_2'(t_0) \cr ...\cr f_n'(t_0)}\right) $ \vskip 15pt \rightline{$= \S ({\partial g \over \partial x_i})_{f({t_0})} ({df_i \over dt})_{t_0}$} Ex: $f(t) = (t^2, sin(t))$, $D(f) = \left(\matrix{ 2t \cr cos(t) }\right)$ $g(x, y) = x + y^3$, $D(g) = (1, 3y^2)$ $(g \circ f)({t}) = g(t^2, sin(t)) = t^2 + sin^3(t)$ $(g \circ f)'({t}) = 2t_0 + 3sin^2(t_0)cos(t_0)$ $ D(g)_{f(t_0)} D(f)_{t_0} = (1, 3sin^2(t_0))\left(\matrix{ 2t_0 \cr cos(t_0) }\right) $, Defn: $U$ is starlike with respect to {\bf a} if ${\bf x} \in U$ implies $\overline{\bf ax} \subset U$ Thm 1.5 (Mean Value Theorem) Let $g$ by a differentiable function on an open set $U \subset \R^n$. Let ${\bf a} \in U$ and suppose $U$ is starlike with respect to {\bf a}. Then given ${\bf x} \in U$, there exists $c \in \R$, $0 < t_0 < 1$ such that $g({\bf x}) - g({\bf a}) = \S ({\partial g \over \partial x_i})_{\bf p} (x_i - a_i)$ where ${\bf p} = {\bf a} + t_0({\bf x} - {\bf a})$ \vfill \f Cor 1.6: If $|{\partial g \over \partial x_i}| < K$ on $U$ for all $i$, then for all $\x \in U$, $$|g(\x) - g(\a)| < K \sqrt{n} ||\x - \a||$$ \vfill \f Cor 1.7 If $f \in C^r$ on $U$, then ${\partial^k g \over \partial x_{i_1} \partial x_{i_2} ... \partial x_{i_k} }$ = ${\partial^k g \over \partial x_{j_1} \partial x_{j_2} ... \partial x_{j_k} }$ where $(j_1, j_2, ... , j_k)$ is a permutation of $(i_1, i_2, ... , i_k)$ \vfill \eject 2.2: $f: \R^n \rightarrow \R^m$ Let $ \pi_i: \R^m \rightarrow \R, \pi_i(\x) = x_i$ $f = (f_1, ..., f_m)$ where $f_i = \pi_i \circ f$ $f$ continuous iff $f_i$ continuous for all $i$ $f \in C^r$ iff $f_i \in C^r$ for all $i$ $f \in C^\infty$ iff $f_i \in C^\infty$ for all $i$ Defn: The {\bf Jacobian matrix of $f$ at a} is $$\left[\matrix{{\partial f_i \over \partial x_j}({\bf a})}\right]_{m \times n} = \left[\matrix{ {\partial f_1 \over \partial x_1}({\bf a}) & ... & {\partial f_1 \over \partial x_n}({\bf a}) \cr . & .~~~~ & . \cr . & ~~.~~ & . \cr . & ~~~~. & . \cr {\partial f_m \over \partial x_1}({\bf a}) & ... & {\partial f_m \over \partial x_n}({\bf a}) \cr }\right]$$ 2.1 Let $V$ be an open subset of $R^n$, ${\bf a} \in V$, $f: V \rightarrow R^m$. Then $f$ is differentiable at {\bf a} if and only if there is a matrix $T$ and a function $\epsilon: R^n \rightarrow R^m$ such that $lim_{{\bf h} \rightarrow 0} \epsilon({\bf h}) = {\bf 0}$ and $$f({\bf a} + {\bf h}) - f({\bf a}) = T({\bf h}) + ||{\bf h}|| \epsilon({\bf h})$$ Or equivalently, there exists an $m$-tuple, \hfil \break $R(\x, \a) = (r_1(\x, \a), r_2(\x, \a), ..., r_m(\x, \a)$ \hfil \break such that $lim_{{\bf x} \rightarrow a} ||R(\x, \a)|| = 0$ and $$f({\bf x}) = f({\bf a}) + T({\bf x} - \a) + ||\x - \a|| R(\x, \a)$$ Thm 2.2: Let $f$ by a differentiable function on an open set $U \subset \R^n$. Let ${\bf a} \in U$ and suppose $U$ is starlike with respect to {\bf a}. If $|{\partial f_i \over \partial x_i}| < K$ on $U$ for all $i, j$, then for all $\x \in U$, $$||f(\x) - f(\a)|| < K \sqrt{nm} ||\x - \a||$$ Proof: $||f(\x) - f(\a)|| = \sqrt{ \Sm (f_i(\x) - f_i(\a))^2}$ $ < \sqrt{ \Sm ( K\sqrt{n} ||\x - \a|| )^2}$ $ = \sqrt{ m (K \sqrt{n} ||\x - \a|| )^2}$ \rightline{$ = K \sqrt{nm} ||\x - \a||$} \vfill Thm 2.3 (Chain rule): Suppose $U \subset R^m$ is open and $f: U \rightarrow V \subset \R^m$, $g: V \rightarrow \R^p$. Let $h = g \circ f$. Suppose $f$ is differentiable at $a \in U$ and $g$ is differentiable at $f(a) \in V$. Then $h$ is differentiable at $a \in U$ and $D(h)_a = D(G)_{f(a)} D(f)_a$. \vfill Cor 2.4: If $f, g \in C^r$ on $U, V$ respectively, then $h = g \circ f \in \C^r$. \end Lemma 11.22: $D(f \cdot g)({\bf a}) = g({\bf a})Df({\bf a}) + f({\bf a}) Dg({\bf a})$, etc. Thm 11.12: Let $V$ be a nonempty open subset of $R^n$, $f: V \rightarrow R^m$. If $f$ is differentiable at some point ${\bf a} \in V$, then all first order partial derivatives of $f$ exist at {\bf a} and the matrix which represents the total derivative of $f$ at ${\bf a}$ is the Jacobian matrix of $f$ at ${\bf a}$. If $m = 1$, $Df({\bf a}) = \left[ {\partial f_1 \over \partial x_1} ({\bf a}) ... {\partial f_1 \over \partial x_n}({\bf a})\right]$ In this case, the gradient of $f$ is denoted by $$\nabla f({\bf a}) = \left({\partial f_1 \over \partial x_1}({\bf a}), ..., {\partial f_1 \over \partial x_n}({\bf a})\right)$$ Thm H: Let $V$ be an open subset of $R^n$, ${\bf a} \in V$, $f: V \rightarrow R^m$. $f = (f_1, ..., f_m)$ is differentiable at {\bf a} if and only if for all $i$, $$lim_{{\bf h} \rightarrow 0} {f_i({\bf a} + {\bf h}) - f_i({\bf a}) - \nabla f_i({\bf a}) \cdot {\bf h} \over ||{\bf h}||} = 0$$ Ex: The Jacobian of $f(x, y) = \cases{ x + y & if $xy = 0$ \cr 1 & otherwise}$ exists, but $f$ is not differentiable at (0, 0). \vskip 8pt Thm: Let $V$ be an open subset of $R^n$, ${\bf a} \in V$, $f: V \rightarrow R^m$. If all first order partial derivatives of $f$ exist in $V$ and are continuous at {\bf a}, then $f$ is differentiable at {\bf a}. \vfill \vskip 5pt \hrule \vfill Lemma 11.21: Let $V$ be an open subset of $R^n$, ${\bf a} \in V$, $f: V \rightarrow R^m$. Then $f$ is differentiable at {\bf a} if and only if there is a linear function $T \in {\cal L}(R^n; R^m)$ and a function $\epsilon: R^n \rightarrow R^m$ such that $lim_{{\bf h} \rightarrow 0} \epsilon({\bf h}) = {\bf 0}$ and $$f({\bf a} + {\bf h}) - f({\bf a}) = T({\bf h}) + ||{\bf h}|| \epsilon({\bf h})$$ for sufficiently small $h$ (i.e., there exists an $r > 0$ such that the above holds for $||{\bf h}|| < r$). Lemma 11.22: $D(f \cdot g)({\bf a}) = g({\bf a})Df({\bf a}) + f({\bf a}) Dg({\bf a})$, etc. \end \end