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a/semester3/analysis-ii/cheat-sheet-rb/parts/05_diff.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/05_diff.tex index ba37544..151df30 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/parts/05_diff.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/parts/05_diff.tex @@ -100,10 +100,27 @@ $$ \lemma \textbf{Chain Rule} $\quad g \circ f \text{ is differentiable on } X$ \begin{align*} - & d(g \circ f)(x_0) &= dg\bigl( f(x_0) \bigr) \circ df(x_0) \\ - & \textbf{J}_{g \circ f}(x_0) &= \textbf{J}_g\bigl( f(x_0) \bigr) \cdot \textbf{J}_f(x_0) + & d(f \circ g)(x_0) &= df\bigl( g(x_0) \bigr) \circ dg(x_0) \\ + & \textbf{J}_{f \circ g}(x_0) &= \textbf{J}_f\bigl( g(x_0) \bigr) \cdot \textbf{J}_f(x_0) \end{align*} -\subtext{$X \subset \R^n \text{ open},\quad Y \subset \R^m \text{ open},\quad f: X \to Y, g: Y \to \R^p, f,g \text{ diff.-able}$} +\subtext{ + $X \subset \R^n \text{ open},\quad Y \subset \R^m \text{ open},\quad f: X \to Y, g: Y \to \R^p, f,g \text{ diff.-able}$\\ + $f \circ g$ meaning $f(g(x))$. +} + +\begin{footnotesize} + \textbf{Example:} Let $\nabla f(- \frac{\sqrt{3}}{2}, \frac{3}{2}, 7) = (6, 2, 0)$.\\ + We want $\partial_r f(\sqrt{3}, \frac{2}{3}\pi, 7)$ in cyl. coords, i.e. the 1st entry of $\textbf{J}_{f \circ g}$ with: + $$ + g(r, \theta, z) = (r\cos(\theta), r\sin(\theta), z) + $$ + Applying the Chain Rule: + $$ + \textbf{J}_{f \circ g} = \textbf{J}_f\bigr( g(x_0) \bigl) \cdot \textbf{J}_g(x_0) = \underbrace{\textbf{J}_{f}\bigl(g(\sqrt{3}, \frac{3}{2}\pi, 7)\bigr)}_{(6, 2, 0)} \cdot \textbf{J}_g + $$ + Where $\textbf{J}_g$ is the Jacobian for cylindrical coordinates. +\end{footnotesize} + \definition \textbf{Tangent Space} $$ @@ -111,6 +128,29 @@ $$ $$ \subtext{$X \subset \R^n \text{ open},\quad f: X \to \R^m \text{ diff.-able},\quad x_0 \in X,\quad u = df(x_0)$} +\begin{footnotesize} + \textbf{Example:} Find $T_f\underbrace{(1, 1, 1)}_{x_0}$ for $S = \{\ \underbrace{(x, y, e^{x-y})}_{f(x,y)} \sep (x,y) \in \R^2 \} \subset \R^3$. + $$ + u = df(x_0) = \textbf{J}_f(x_0) = + \begin{bmatrix} + 1 & 0 \\ + 0 & 1 \\ + e^{x-y} & -e^{x-y} + \end{bmatrix}(x_0) + = + \begin{bmatrix} + 1 & 0 \\ + 0 & 1 \\ + 1 & -1 + \end{bmatrix} + $$ + So we can define $T_f(x_0)$ as $g(x, y) = \underbrace{f(x_0)}_{(1, 1, 1)} + \textbf{J}_f(x_0) \cdot \begin{bmatrix} + x - x_{0, 1} \\ + y - y_{0, 2} + \end{bmatrix}$. +\end{footnotesize} + + \definition \textbf{Directional Derivative} $$ D_v f(x_0) = \underset{t \neq 0 \to 0}{\lim} \frac{f(x_0 + tv) - f(x_0)}{t} @@ -123,9 +163,11 @@ $$ $$ \subtext{$X \subset \R^n \text{ open},\quad f: X \to \R^m \text{ diff.-able},\quad v \neq 0 \in \R^n,\quad x_0 \in X$} -\remark $D_vf$ is linear w.r.t $v$, so: $D_{v_1 + v_2}f = D_{v_1}f + D_{v_2}f$ +\begin{footnotesize} + \remark $D_vf$ is linear w.r.t $v$, so: $D_{v_1 + v_2}f = D_{v_1}f + D_{v_2}f$ -\remark $D_vf(x_0) = \nabla f(x_0) \cdot v = \big\| \nabla f(x_0) \big\| \cos(\theta)$\\ + \remark $D_vf(x_0) = \nabla f(x_0) \cdot v = \big\| \nabla f(x_0) \big\| \cos(\theta)$\\ +\end{footnotesize} \subtext{In the case $f: X \to \R$, where $\theta$ is the angle between $v$ and $\nabla f(x_0)$} \newpage diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/06_int.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/06_int.tex index 051e8ad..552bd3c 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/parts/06_int.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/parts/06_int.tex @@ -304,4 +304,22 @@ $$ $$ \subtext{Same assumptions as above.} -\remark The \textit{Gauss-Ostrogradski} Formula exists for $\R^3$. \ No newline at end of file +\footnotesize +\remark The \textit{Gauss-Ostrogradski} Formula exists for $\R^3$. +\normalsize + +\begin{footnotesize} + \textbf{Example:} Finding an Area using Green: + $$ + \text{Area}(X) = \int_X 1\ dxdy + $$ + To use Green, we want $f$ s.t. $(\partial_x f_y - \partial_y f_x) = 1$. Common Choices: + \begin{enumerate} + \item $f(x,y) = (0, x)$ + \item $f(x,y) = (-y, 0)$ + \end{enumerate} + We can then use Green, assuming we have a $\gamma$ for the Boundary of $X$: + $$ + \text{Area}(X) = \int_X 1\ dxdy = \int_X \bigg(\partial_x f_y - \partial_y f_x \bigg) \ dxdy = \int_\gamma f(s)\ ds + $$ +\end{footnotesize} \ No newline at end of file