diff --git a/semester3/analysis-ii/cheat-sheet-rb/main.pdf b/semester3/analysis-ii/cheat-sheet-rb/main.pdf index 9306199..60ab0f9 100644 Binary files a/semester3/analysis-ii/cheat-sheet-rb/main.pdf and b/semester3/analysis-ii/cheat-sheet-rb/main.pdf differ diff --git a/semester3/analysis-ii/cheat-sheet-rb/main.tex b/semester3/analysis-ii/cheat-sheet-rb/main.tex index 49f4e7b..81ff7eb 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/main.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/main.tex @@ -16,12 +16,16 @@ \section{Differential Equations} \input{parts/02_diffeq.tex} +\newpage +\section{Solutions to Differential Equations} +\input{parts/03_diffeq_sol.tex} + \newpage \section{Continuous functions in $\R^n$} -\input{parts/03_cont.tex} +\input{parts/04_cont.tex} \newpage \section{Differential Calculus in $\R^n$} -\input{parts/04_diff.tex} +\input{parts/05_diff.tex} \end{document} diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/02_diffeq.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/02_diffeq.tex index 2f4abbc..f5a9628 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/parts/02_diffeq.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/parts/02_diffeq.tex @@ -83,139 +83,4 @@ $\S_b(F) := \{ f + f_0 \sep f \in \S(F),\ f_0 \text{ is a particular sol.} \}$\\ \remark \textbf{Applications of Linearity}\\ If $f_1$ solves $F$ for $b_1$, and $f_2$ for $b_2$: $f_1 + f_2$ solves $b_1 + b_2$. \\ -Follows from: $D(f_1) + D(f_2) = b_1 + b_2$. - -\newpage -\subsection{Linear Solutions: First Order} -\subtext{ $I \subset \R, \quad a,b: I \to \R$ } - -\textbf{Form:} -$$ y' + ay = b $$ -\textbf{Approach:} -\begin{enumerate} - \item Hom. Solution $f_1$ for: $y' + ay = 0$\\ - \subtext{Note that $\S$ has $\dim(\S) = 1$, so $f_1 \neq 0$ is a Basis for $\S$} - \item Part. Solution $f_0$ for $y' + ay = b$ -\end{enumerate} -\textbf{Solutions:} $ f_0 + zf_1 \quad \text{ for } z \in \C $ - -\begin{subbox}{Explicit Homogeneous Solution} - \smalltext{$A(x)$ is a primitive of $a$, $f(x_0) = y_0$} - \begin{align*} - f_1(x) &= z \cdot \exp(-A(x)) \\ - f_1(x) &= y_0 \cdot \exp(A(x_0) - a(x)) - \end{align*} -\end{subbox} - -Variation of Constants: Treating $z$ as $z(x)$ yields: - -\begin{subbox}{Explicit Inhomogeneous Solution} - \smalltext{$A(x)$ is a primitive of $a$} - $$ - f_0(x) = \underbrace{\left(\int b(x)\cdot\exp(A(x)) \right)}_{z(x)} \cdot \exp\left(-A(x)\right) - $$ -\end{subbox} - -\method \textbf{Educated Guess}\\ -Usually, $y$ has a similar form to $b$: - -\begin{tabular}{ll} - \hline - $b(x)$ & \text{Guess} \\ - \hline - $a \cdot e^{\alpha x}$ & $b \cdot e^{\alpha x}$ \\ - $a \cdot \sin(\beta x)$ & $c\sin(\beta x) + d\cos(\beta x)$\\ - $b \cdot \cos(\beta x)$ & $c\sin(\beta x) + d\cos(\beta x)$\\ - $ae^{\alpha x} \cdot \sin(\beta x)$ & $e^{\alpha x}\left(c\sin(\beta x) + d\cos(\beta x)\right)$\\ - $be^{\alpha x} \cdot \cos(\beta x)$ & $e^{\alpha x}\left(c\sin(\beta x) + d\cos(\beta x)\right)$\\ - $P_n(x) \cdot e^{\alpha x}$ & $R_n(x) \cdot e^{\alpha x}$\\ - $P_n(x) \cdot e^{\alpha x}\sin(\beta x)$ & $e^{\alpha x}\left( R_n(x) \sin(\beta x) + S_n(x) \cos(\beta x) \right)$\\ - $P_n(x) \cdot e^{\alpha x}\cos(\beta x)$ & $e^{\alpha x}\left( R_n(x) \sin(\beta x) + S_n(x) \cos(\beta x) \right)$\\ - \hline -\end{tabular} - -\remark If $\alpha, \beta$ are roots of $P(X)$ with multiplicity $j$, multiply guess with a $P_j(x)$. - -\subsection{Linear Solutions: Constant Coefficients} -\textbf{Form:} -$$ - y^{(k)} + a_{k-1}y^{(k-1)} + \ldots + a_1y' + a_0y = b -$$ -\subtext{Where $a_0, \ldots, a_{k-1} \in \C$ are constants, $b(x)$ is continuous.} - -\subsubsection{Homogeneous Equations} - -The idea is to find a Basis of $\S$: - -\definition \textbf{Characteristic Polynomial} $P(X) = \prod_{i=1}^{k} (X-\alpha_i)$ - -\remark The unique roots $\alpha_1,\ldots,\alpha_l$ form a Basis: -$$ - \text{span}(\S) = \{ x^je^{\alpha_i x} \sep i \leq l,\quad 0 \leq j \leq v_i \} -$$ -\subtext{$v_1,\ldots,v_k$ are the Multiplicities of $\alpha_1,\ldots,\alpha_k$} - -\remark If $\alpha_j = \beta + \gamma i \in \C$ is a root, $\bar{\alpha_j} = \beta - \gamma i$ is too.\\ -To get a real-valued solution, apply: -$$ -e^{\alpha_j x} = e^{\beta x}\left( \cos(\gamma x) + i \sin(\gamma x) \right) -$$ - -\begin{subbox}{Explicit Homogeneous Solution} - \smalltext{Using $\alpha_1,\ldots,\alpha_k$ from $P(X)$ s.t. $\alpha_i \neq \alpha_j$, $z_i \in \C$ arbitrary} - $$ - f(x) = \prod_{i=1}^{k} z_i \cdot e^{\alpha_i x} \quad\text{with}\quad f^{(j)(x)} = \prod_{i=1}^{k} z_i \cdot \alpha_i^j e^{\alpha_i x} - $$ - \smalltext{Multiple roots: same scheme, using the basis vectors of $\S$} -\end{subbox} -\subtext{Solutions exist $\forall Z = (z_1,\ldots,z_k)$ since that system's $\det(M_Z) \neq 0$.} - -\newpage -\subsubsection{Inhomogeneous Equations} - -\method \textbf{Undetermined Coefficients}: An educated guess. -\begin{enumerate} - \item $b(x) = cx^d \cdot e^{\alpha x} \implies f_p(x) = Q(x)e^{\alpha x}$\\ - \subtext{$\deg(Q) \leq d + v_\alpha$, where $v_\alpha$ is $\alpha$'s multiplicity in $P(X)$} - \item $\begin{rcases*} - b(x) = cx^d \cdot \cos(\alpha x) \\ - b(x) = cx^d \cdot \sin(\alpha x) - \end{rcases*} f_p = Q_1(x)\cos(\alpha x) + Q_2(x)\sin(\alpha x)$ - \subtext{$\deg(Q_{1,2}) \leq d + v_\alpha$, where $v_\alpha$ is $\alpha$'s multiplicity in $P(X)$} -\end{enumerate} - -\remark \textbf{Applying Linearity}\\ -If $b(x) = \sum_{i=1}^{n} b_i(x)$, A solution for $b(x)$ is $f(x) = \sum_{i=1}^{n} f_i(x)$\\ -\subtext{Sometimes called \textit{Superposition Principle} in this context} - -\subsection{Other Methods} - -\method \textbf{Change of Variable}\\ -If $f(x)$ is replaced by $h(y) = f(g(y))$, then $h$ is a sol. too.\\ -\subtext{Changes like $h(t) = f(e^t)$ may lead to useful properties.} - -\begin{subbox}{Separation of Variables} - Form: - $$ - y' = a(y)\cdot b(x) - $$ - Solve using: - $$ - \int \frac{1}{a(y)}\ \text{d}y = \int b(x) \dx + c - $$ -\end{subbox} -\subtext{Usually $\int 1/a(y)\ \text{d}y$ can be solved directly for $\ln|a(y)|+c$.} - -\subsection{Method Overview} - -\begin{center} - \begin{tabular}{l|l} - \textbf{Method} & \textbf{Use case} \\ - \hline - Variation of constants & LDE with $\ord(F)=1$ \\ - Characteristic Polynomial & Hom. LDE w/ const. coeff. \\ - Undetermined Coefficients & Inhom. LDE w/ const. coeff. \\ - Separation of Variables & ODE s.t. $y' = a(y)\cdot b(x)$ \\ - Change of Variables & e.g. $y' = f(ax + by + c)$ \\ - \end{tabular} -\end{center} \ No newline at end of file +Follows from: $D(f_1) + D(f_2) = b_1 + b_2$. \ No newline at end of file diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/03_diffeq_sol.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/03_diffeq_sol.tex new file mode 100644 index 0000000..b66a881 --- /dev/null +++ b/semester3/analysis-ii/cheat-sheet-rb/parts/03_diffeq_sol.tex @@ -0,0 +1,133 @@ + + +\subsection{Linear Solutions: First Order} +\textbf{Form:}$\quad y' + ay = b\quad $ \subtext{ $I \subset \R, \quad a,b: I \to \R$ } + +\textbf{Approach:} +\begin{enumerate} + \item Hom. Solution $f_1$ for: $y' + ay = 0$\\ + \subtext{Note that $\S$ has $\dim(\S) = 1$, so $f_1 \neq 0$ is a Basis for $\S$} + \item Part. Solution $f_0$ for $y' + ay = b$ +\end{enumerate} +\textbf{Solutions:} $ f_0 + zf_1 \quad \text{ for } z \in \C $ + +\begin{subbox}{Explicit Homogeneous Solution} + \smalltext{$A(x)$ is a primitive of $a$, $f(x_0) = y_0$} + \begin{align*} + f_1(x) &= z \cdot \exp(-A(x)) \\ + f_1(x) &= y_0 \cdot \exp(A(x_0) - a(x)) + \end{align*} +\end{subbox} + +\method \textbf{Variation of Constants}: Treating $z$ as $z(x)$ yields: + +\begin{subbox}{Explicit Inhomogeneous Solution} + \smalltext{$A(x)$ is a primitive of $a$} + $$ + f_0(x) = \underbrace{\left(\int b(x)\cdot\exp(A(x)) \right)}_{z(x)} \cdot \exp\left(-A(x)\right) + $$ +\end{subbox} + +\method \textbf{Educated Guess}\\ +Usually, $y$ has a similar form to $b$: + +\begin{tabular}{ll} + \hline + $b(x)$ & \text{Guess} \\ + \hline + $a \cdot e^{\alpha x}$ & $b \cdot e^{\alpha x}$ \\ + $a \cdot \sin(\beta x)$ & $c\sin(\beta x) + d\cos(\beta x)$\\ + $b \cdot \cos(\beta x)$ & $c\sin(\beta x) + d\cos(\beta x)$\\ + $ae^{\alpha x} \cdot \sin(\beta x)$ & $e^{\alpha x}\left(c\sin(\beta x) + d\cos(\beta x)\right)$\\ + $be^{\alpha x} \cdot \cos(\beta x)$ & $e^{\alpha x}\left(c\sin(\beta x) + d\cos(\beta x)\right)$\\ + $P_n(x) \cdot e^{\alpha x}$ & $R_n(x) \cdot e^{\alpha x}$\\ + $P_n(x) \cdot e^{\alpha x}\sin(\beta x)$ & $e^{\alpha x}\left( R_n(x) \sin(\beta x) + S_n(x) \cos(\beta x) \right)$\\ + $P_n(x) \cdot e^{\alpha x}\cos(\beta x)$ & $e^{\alpha x}\left( R_n(x) \sin(\beta x) + S_n(x) \cos(\beta x) \right)$\\ + \hline +\end{tabular} + +\remark If $\alpha, \beta$ are roots of $P(X)$ with multiplicity $j$, multiply guess with a $P_j(x)$. + +\subsection{Linear Solutions: Constant Coefficients} +\textbf{Form:} +$$ + y^{(k)} + a_{k-1}y^{(k-1)} + \ldots + a_1y' + a_0y = b +$$ +\subtext{Where $a_0, \ldots, a_{k-1} \in \C$ are constants, $b(x)$ is continuous.} + +\subsubsection{Homogeneous Equations} + +The idea is to find a Basis of $\S$: + +\definition \textbf{Characteristic Polynomial} $P(X) = \prod_{i=1}^{k} (X-\alpha_i)$ + +\remark The unique roots $\alpha_1,\ldots,\alpha_l$ form a Basis: +$$ + \text{span}(\S) = \{ x^je^{\alpha_i x} \sep i \leq l,\quad 0 \leq j \leq v_i \} +$$ +\subtext{$v_1,\ldots,v_k$ are the Multiplicities of $\alpha_1,\ldots,\alpha_k$} + +\remark If $\alpha_j = \beta + \gamma i \in \C$ is a root, $\bar{\alpha_j} = \beta - \gamma i$ is too.\\ +To get a real-valued solution, apply: +$$ +e^{\alpha_j x} = e^{\beta x}\left( \cos(\gamma x) + i \sin(\gamma x) \right) +$$ + +\begin{subbox}{Explicit Homogeneous Solution} + \smalltext{Using $\alpha_1,\ldots,\alpha_k$ from $P(X)$ s.t. $\alpha_i \neq \alpha_j$, $z_i \in \C$ arbitrary} + $$ + f(x) = \prod_{i=1}^{k} z_i \cdot e^{\alpha_i x} \quad\text{with}\quad f^{(j)(x)} = \prod_{i=1}^{k} z_i \cdot \alpha_i^j e^{\alpha_i x} + $$ + \smalltext{Multiple roots: same scheme, using the basis vectors of $\S$} +\end{subbox} +\subtext{Solutions exist $\forall Z = (z_1,\ldots,z_k)$ since that system's $\det(M_Z) \neq 0$.} + +\newpage +\subsubsection{Inhomogeneous Equations} + +\method \textbf{Undetermined Coefficients}: An educated guess. +\begin{enumerate} + \item $b(x) = cx^d \cdot e^{\alpha x} \implies f_p(x) = Q(x)e^{\alpha x}$\\ + \subtext{$\deg(Q) \leq d + v_\alpha$, where $v_\alpha$ is $\alpha$'s multiplicity in $P(X)$} + \item $\begin{rcases*} + b(x) = cx^d \cdot \cos(\alpha x) \\ + b(x) = cx^d \cdot \sin(\alpha x) + \end{rcases*} f_p = Q_1(x)\cos(\alpha x) + Q_2(x)\sin(\alpha x)$ + \subtext{$\deg(Q_{1,2}) \leq d + v_\alpha$, where $v_\alpha$ is $\alpha$'s multiplicity in $P(X)$} +\end{enumerate} + +\remark \textbf{Applying Linearity}\\ +If $b(x) = \sum_{i=1}^{n} b_i(x)$, A solution for $b(x)$ is $f(x) = \sum_{i=1}^{n} f_i(x)$\\ +\subtext{Sometimes called \textit{Superposition Principle} in this context} + +\subsection{Other Methods} + +\method \textbf{Change of Variable}\\ +If $f(x)$ is replaced by $h(y) = f(g(y))$, then $h$ is a sol. too.\\ +\subtext{Changes like $h(t) = f(e^t)$ may lead to useful properties.} + +\begin{subbox}{Separation of Variables} + Form: + $$ + y' = a(y)\cdot b(x) + $$ + Solve using: + $$ + \int \frac{1}{a(y)}\ \text{d}y = \int b(x) \dx + c + $$ +\end{subbox} +\subtext{Usually $\int 1/a(y)\ \text{d}y$ can be solved directly for $\ln|a(y)|+c$.} + +\subsection{Method Overview} + +\begin{center} + \begin{tabular}{l|l} + \textbf{Method} & \textbf{Use case} \\ + \hline + Variation of constants & LDE with $\ord(F)=1$ \\ + Characteristic Polynomial & Hom. LDE w/ const. coeff. \\ + Undetermined Coefficients & Inhom. LDE w/ const. coeff. \\ + Separation of Variables & ODE s.t. $y' = a(y)\cdot b(x)$ \\ + Change of Variables & e.g. $y' = f(ax + by + c)$ \\ + \end{tabular} +\end{center} \ No newline at end of file diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/03_cont.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/04_cont.tex similarity index 100% rename from semester3/analysis-ii/cheat-sheet-rb/parts/03_cont.tex rename to semester3/analysis-ii/cheat-sheet-rb/parts/04_cont.tex diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/04_diff.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/05_diff.tex similarity index 97% rename from semester3/analysis-ii/cheat-sheet-rb/parts/04_diff.tex rename to semester3/analysis-ii/cheat-sheet-rb/parts/05_diff.tex index 5793715..5f633ae 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/parts/04_diff.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/parts/05_diff.tex @@ -50,7 +50,7 @@ $\\ Partial derivatives don't provide a good approx. of $f$, unlike in the $1$-dimensional case. The \textit{differential} is a linear map which replicates this purpose in $\R^n$. } -\begin{subbox}{Differentiability in $\R^n$} +\begin{subbox}{Differentiability in $\R^n$ \& the Differential} \smalltext{$X \subset \R^n \text{ open},\quad f: X \to \R^n,\quad u: \R^n \to \R^m \text{ linear map}$} $$ df(x_0) := u