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zNmo}Z40;}B{6x`8LQ+U`LHu7_yG7p{H*kifLk*ec85G%6gM|MTMBA_j=yXE30n;(W)IiD vDcO<#5}LbihRaEg&tvtMyTKD36^fauj%AliW{%57BqJk1#KWVe4<`D5dyred diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/01_partial_derivatives.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/01_partial_derivatives.tex index 92b1702..19eb1a1 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/01_partial_derivatives.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/01_partial_derivatives.tex @@ -2,9 +2,12 @@ \subsection{Partial derivatives} \shortdef $X \subseteq \R^n$ \bi{open} if for any $x = (x_1, \ldots, x_n) \in X$ $\exists \delta > 0$ s.t. $\{ y = (y_1, \ldots, y_n) \in \R^n : |x_i - y_i| < \delta \smallhspace \forall i \}$ is contained in $X$. -(= changing a coordinate of $x$ by $< \delta \rightarrow x' \in X$) -\shortproposition $X \subseteq \R^n$ open $\Leftrightarrow$ \bi{complement} $Y = \{ x \in \R^n : x \notin X \}$ is closed -\shortcorollary If $f: \R^n \rightarrow \R^m$ cont. and $Y \subseteq \R^m$ open, then $f^{-1}(Y)$ is open in $\R^n$ +(= changing a coordinate of $x$ by $< \delta \rightarrow x' \in X$)\\ +% +\shortproposition $X \subseteq \R^n$ open $\Leftrightarrow$ \bi{complement} $Y = \{ x \in \R^n : x \notin X \}$ is closed\\ +% +\shortcorollary If $f: \R^n \rightarrow \R^m$ cont. and $Y \subseteq \R^m$ open, then $f^{-1}(Y)$ is open in $\R^n$\\ +% \shortex \bi{(1)} $\emptyset$ and $\R^n$ are both open and closed. \bi{(2)} Open ball $D = \{ x \in \R^n : ||x - x_0|| < r \}$ is open in $\R^n$ ($x_0$ the center and $r$ radius) \bi{(3)} $I_1 \times \dots \times I_n$ is open in $\R^n$ for $I_i$ open diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/02_differential.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/02_differential.tex index 56ef61e..7f4b9f0 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/02_differential.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/02_differential.tex @@ -47,12 +47,12 @@ The Jacobi matrix is $J_{g \circ f}(x_0) = J_g(f(x_0)) J_f(x_0)$ (RHS is a matri % ──────────────────────────────────────────────────────────────────── \stepLabelNumber{all} \compactdef{Directional derivative} $f$ has a directional derivative $w \in \R^m$ in the direction of $v \in \R^n$, -if the function $g$ defined on the set $I = \{ t \in \R : x_0 + tv \in X \}$ by $g(t) = f(x_0 + tv)$ has a derivative at $t = 0$ and is equal to $w$ +if the function $g$ defined on the set $I = \{ t \in \R : x_0 + tv \in X \}$ by $g(t) = f(x_0 + tv)$ has a derivative at $t = 0$ and is equal to $w$\\ % ──────────────────────────────────────────────────────────────────── -\shortremark Because $X$ is open, the set $I$ contains an open interval $]-\delta, \delta[$ for some $\delta > 0$. +\shortremark Because $X$ is open, the set $I$ contains an open interval $]-\delta, \delta[$ for some $\delta > 0$.\\ % ──────────────────────────────────────────────────────────────────── \shortproposition Let $f$ as previously be differentiable. Then for any $x \in X$ and non-zero $v \in \R^n$, -$f$ has a directional derivative at $x_0$ in the direction of $v$, given by$\dx f(x_0)(v)$ +$f$ has a directional derivative at $x_0$ in the direction of $v$, given by$\dx f(x_0)(v)$\\ % ──────────────────────────────────────────────────────────────────── \shortremark The values of the above directional derivative are linear with respect to the vector $v$. Suppose we know the dir. der. $w_1$ and $w_2$ in directions $v_1$ and $v_2$, then the directional derivative in direction $v_1 + v_2$ is $w_1 + w_2$ diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex index 511c395..22a81ab 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex @@ -1,7 +1,7 @@ \newsectionNoPB \subsection{Critical points} \stepLabelNumber{all} -\compactdef{Critical Point} For $f: X \rightarrow \R^n$ differentiable, $x_0 \in X$ is called a \bi{critical point} of $f$ if $\nabla f(x_0) = 0$ +\compactdef{Critical Point} For $f: X \rightarrow \R^n$ differentiable, $x_0 \in X$ is called a \bi{critical point} of $f$ if $\nabla f(x_0) = 0$\\ \shortremark As in 1 dimensional case, check edges of the interval for the critical point.\\ % To determine the kind of critical point, we need to determine if $H_f(x_0)$ is definite: diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/00_line_integrals.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/00_line_integrals.tex index abfc078..67c8629 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/00_line_integrals.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/00_line_integrals.tex @@ -16,7 +16,8 @@ We usually call $f : X \rightarrow \R^n$ a \bi{vector field}, which maps each point $x \in X$ to a vector in $\R^n$, displayed as originating from $x$\\ Often, we use $V$ instead of $f$ to denote the vector field. Ideally, to compute a line integral, we compute the derivative of $\gamma$ and $V(\gamma(t))$ separately, then simply do the integral after. -Be careful with hat functions like $|x|$, we need two separate integrals for each side of the center! +\hl{Be careful with hat functions} like $|x|$, we need two separate integrals for each side of the center! +Alternatively to using a line integral, see section \ref{sec:green-formula} for a faster way \setLabelNumber{all}{4} \compactdef{Oriented reparametrization} of $\gamma$ is parametrized curve $\sigma : [c, d] \rightarrow \R^n$ s.t $\sigma = \gamma \circ \varphi$, with $\varphi : [c, d] \rightarrow I$ cont. map, @@ -79,5 +80,3 @@ Below a chart to figure out some properties: \end{tikzpicture} \end{center} \dnrmvspace -% TODO: Some tips and tricks -% With the line integral, we can compute the length of the curve, as defined by the function. diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex index 01d839f..41a9e5e 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex @@ -1,5 +1,6 @@ \newsection \subsection{The Green Formula} +\label{sec:green-formula} \compactdef{Simple parametrized curve} $\gamma : [a, b] \rightarrow \R^2$ is a closed parametrized curve s.t. $\gamma(t) \neq \gamma(s)$ (if $s \neq t$ and $\{ s, t \} = \{ a, b \}$), s.t. $\gamma'(t) \neq 0$ for $a < t < b$. If $\gamma$ only piecewise in $C^1$ in $]a, b[$, then only apply when $\gamma'(t)$ exists.