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-rw-r--r--FuVar.tex188
-rw-r--r--build/FuVar.pdfbin67594 -> 161858 bytes
-rw-r--r--img/double-integral.pdfbin0 -> 46507 bytes
-rw-r--r--img/double-integral.svg306
-rw-r--r--img/hsrlogo-bw.epsbin413070 -> 0 bytes
-rw-r--r--img/hsrlogo-color.epsbin413438 -> 0 bytes
-rw-r--r--img/lagrange-multipliers.pdfbin0 -> 34818 bytes
-rw-r--r--img/lagrange-multipliers.svg446
8 files changed, 896 insertions, 44 deletions
diff --git a/FuVar.tex b/FuVar.tex
index 13b0d2d..62d71f2 100644
--- a/FuVar.tex
+++ b/FuVar.tex
@@ -30,6 +30,7 @@
%% Layout
\usepackage{enumitem}
+\usepackage{booktabs}
%% Nice drwaings
\usepackage{tikz}
@@ -63,11 +64,11 @@
\theoremstyle{fuvarzf}
\newtheorem{theorem}{Theorem}
-\newtheorem{proposition}{Proposition}
\newtheorem{method}{Method}
+\newtheorem{application}{Application}
\newtheorem{definition}{Definition}
-\newtheorem{lemma}{Lemma}
\newtheorem{remark}{Remark}
+\newtheorem{note}{Note}
\DeclareMathOperator{\tr}{\mathrm{tr}}
@@ -90,7 +91,7 @@ These are just my personal notes of the \themodule{} course, and definitively
not a rigorously constructed mathematical text. The good looking \LaTeX{}
typesetting may trick you into thinking it is rigorous, but really, it is not.
-\section{Derivatives of vector valued scalar functions}
+\section{Derivatives of vector valued functions}
\begin{definition}[Partial derivative]
A vector values function \(f: \mathbb{R}^m\to\mathbb{R}\), with
@@ -103,14 +104,14 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
\]
\end{definition}
-\begin{proposition}
+\begin{theorem}(Schwarz's theorem, symmetry of partial derivatives)
Under some generally satisfied conditions (continuity of \(n\)-th order
partial derivatives) Schwarz's theorem states that it is possible to swap
the order of differentiation.
\[
\partial_x \partial_y f(x,y) = \partial_y \partial_x f(x,y)
\]
-\end{proposition}
+\end{theorem}
\begin{definition}[Linearization]
A function \(f: \mathbb{R}^m\to\mathbb{R}\) has a linearization \(g\) at
@@ -135,7 +136,10 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
\begin{definition}[Gradient vector]
The \emph{gradient} of a function \(f(\vec{x}), \vec{x}\in\mathbb{R}^m\) is a
- vector containing the derivatives in each direction.
+ column vector\footnote{In matrix notation it is also often defined as row
+ vector to avoid having to do some transpositions in the Jacobian matrix and
+ dot products in directional derivatives} containing the derivatives in each
+ direction.
\[
\grad f (\vec{x}) = \sum_{i=1}^m \partial_{x_i} f(\vec{x}) \vec{e}_i
= \begin{pmatrix}
@@ -151,7 +155,7 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
\(\vec{r}\) (with \(|\vec{r}| = 1\)) given by
\[
\frac{\partial f}{\partial\vec{r}}
- = \nabla_\vec{r} f = \vec{r} \dotp \grad f
+ = \nabla\vec{r} f = \vec{r} \dotp \grad f
\]
\end{definition}
@@ -160,6 +164,52 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
ascent}.
\end{theorem}
+\begin{definition}[Jacobian Matrix]
+ The \emph{Jacobian} \(\mx{J}_f\) (sometimes written as
+ \(\frac{\partial(f_1,\ldots f_m)}{\partial(x_1,\ldots,x_n)}\)) of a function
+ \(\vec{f}: \mathbb{R}^n \to \mathbb{R}^m\) is a matrix
+ \(\in\mathbb{R}^{n\times m}\) whose entry at the \(i\)-th row and \(j\)-th
+ column is given by \((\mx{J}_f)_{i,j} = \partial_{x_j} f_i\), so
+ \[
+ \mx{J}_f = \begin{pmatrix}
+ \partial_{x_1} f_1 & \cdots & \partial_{x_n} f_1 \\
+ \vdots & \ddots & \vdots \\
+ \partial_{x_1} f_m & \cdots & \partial_{x_n} f_m \\
+ \end{pmatrix}
+ = \begin{pmatrix}
+ (\grad f_1)^t \\
+ \vdots \\
+ (\grad f_m)^t \\
+ \end{pmatrix}
+ \]
+\end{definition}
+
+\begin{remark}
+ In the scalar case (\(m = 1\)) the Jacobian matrix is the transpose of the
+ gradient vector.
+\end{remark}
+
+\begin{definition}[Hessian matrix]
+ Given a function \(f: \mathbb{R}^m \to \mathbb{R}\), the square matrix whose
+ entry at the \(i\)-th row and \(j\)-th column is the second derivative of
+ \(f\) first with respect to \(x_j\) and then to \(x_i\) is know as the
+ \emph{Hessian} matrix.
+ \(
+ \left(\mx{H}_f\right)_{i,j} = \partial_{x_i}\partial_{x_j} f
+ \)
+ or
+ \[
+ \mx{H}_f = \begin{pmatrix}
+ \partial_{x_1}\partial_{x_1} f & \cdots & \partial_{x_1}\partial_{x_m} f \\
+ \vdots & \ddots & \vdots \\
+ \partial_{x_m}\partial_{x_1} f & \cdots & \partial_{x_m}\partial_{x_m} f \\
+ \end{pmatrix}
+ \]
+ Because (almost always) the order of differentiation
+ does not matter, it is a symmetric matrix.
+\end{definition}
+
+
\section{Methods for maximization and minimization problems}
\begin{method}[Find stationary points]
@@ -196,30 +246,10 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
vector \(\vec{r} = \vec{e}_1\cos(\alpha) + \vec{e}_2\sin(\alpha)\)
\end{remark}
-\begin{definition}[Hessian matrix]
- Given a function \(f: \mathbb{R}^m \to \mathbb{R}\), the square matrix whose
- entry at the \(i\)-th row and \(j\)-th column is the second derivative of
- \(f\) first with respect to \(x_j\) and then to \(x_i\) is know as the
- \emph{Hessian} matrix.
- \(
- \left(\mtx{H}_f\right)_{i,j} = \partial_{x_i}\partial_{x_j} f
- \)
- or
- \[
- \mtx{H}_f = \begin{pmatrix}
- \partial_{x_1}\partial_{x_1} f & \cdots & \partial_{x_1}\partial_{x_m} f \\
- \vdots & \ddots & \vdots \\
- \partial_{x_m}\partial_{x_1} f & \cdots & \partial_{x_m}\partial_{x_m} f \\
- \end{pmatrix}
- \]
- Because (almost always) the order of differentiation
- does not matter, it is a symmetric matrix.
-\end{definition}
-
\begin{method}[Determine the type of stationary point in higher dimensions]
Given a scalar function of two variables \(f(x,y)\) and a stationary point
\(\vec{x}_s\) (where \(\grad f(\vec{x}_s) = \vec{0}\)), we compute the
- Hessian matrix \(\mtx{H}_f(\vec{x}_s)\). Then we compute its eigenvalues
+ Hessian matrix \(\mx{H}_f(\vec{x}_s)\). Then we compute its eigenvalues
\(\lambda_1, \ldots, \lambda_m\) and
\begin{itemize}
\item if all \(\lambda_i > 0\), the point is a minimum;
@@ -233,23 +263,20 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
\begin{remark}
Recall that to compute the eigenvalues of a matrix, one must solve the
- equation \((\mtx{H} - \lambda\mtx{I})\vec{x} = \vec{0}\). Which can be done
- by solving the characteristic polynomial \(\det\left(\mtx{H} -
- \lambda\mtx{I}\right) = 0\) to obtain \(\dim(\mtx{H})\) \(\lambda_i\), which
+ equation \((\mx{H} - \lambda\mx{I})\vec{x} = \vec{0}\). Which can be done
+ by solving the characteristic polynomial \(\det\left(\mx{H} -
+ \lambda\mx{I}\right) = 0\) to obtain \(\dim(\mx{H})\) \(\lambda_i\), which
when plugged back in result in a overdetermined system of equations.
\end{remark}
\begin{method}[Quickly find the eigenvalues of a \(2\times 2\) matrix]
Let
\[
- m = \frac{1}{2}\tr \mtx{H} = \frac{a + d}{2}
- \text{ and }
- p = \det\mtx{H} = ad - bc ,
- \]
- then
- \[
- \lambda = m \pm \sqrt{m^2 - p} .
+ m = \frac{1}{2}\tr \mx{H} = \frac{a + d}{2} ,
+ \qquad
+ p = \det\mx{H} = ad - bc ,
\]
+ then \(\lambda = m \pm \sqrt{m^2 - p}\).
\end{method}
\begin{method}[Search for a constrained extremum in 2 dimensions]
@@ -273,6 +300,15 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
\end{itemize}
\end{method}
+\begin{figure}
+ \centering
+ \includegraphics{img/lagrange-multipliers}
+ \caption{
+ Intuition for the method of Lagrange multipliers. Extrema of a constrained
+ function are where \(\grad f\) is proportional to \(\grad n\).
+ }
+\end{figure}
+
\begin{method}[%
Search for a constrained extremum in higher dimensions,
method of Lagrange multipliers]
@@ -305,16 +341,80 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
\mathcal{L}(\vec{u}, \vec{\lambda})
= f(\vec{u}) - \sum_{i = 0}^k \lambda_i n_i(\vec{u})
\]
- where \(\vec{\lambda} = \lambda_1, \ldots, \lambda_k\) and then
- evaluating \(\grad \mathcal{L}(\vec{u}, \vec{\lambda}) = \vec{0}\).
+ where \(\vec{\lambda} = \lambda_1, \ldots, \lambda_k\) and then solving
+ \(\grad \mathcal{L}(\vec{u}, \vec{\lambda}) = \vec{0}\). This is
+ generally used in numerical computations and not very useful by hand.
\end{itemize}
\end{method}
-\section{Integration}
-\begin{remark}
-
-\end{remark}
+\section{Integration of vector values scalar functions}
+
+\begin{figure}
+ \centering
+ \includegraphics{img/double-integral}
+ \caption{
+ Double integral.
+ \label{fig:double-integral}
+ }
+\end{figure}
+
+\begin{theorem}[Change the order of integration for double integrals] For a
+ double integral over a region \(S\) (see Fig. \ref{fig:double-integral}) we
+ need to compute
+ \[
+ \iint_S f(x,y) \,ds =
+ \int\limits_{x_1}^{x_2} \int\limits_{y_1(x)}^{y_2(x)} f(x,y) \,dydx .
+ \]
+ If \(y_1(x)\) and \(y_2(x)\) are bijective we can swap the order of
+ integration by finding the inverse functions \(x_1(y)\) and \(x_2(y)\). If
+ they are not bijective (like in Fig. \ref{fig:double-integral}), the region
+ must be split into smaller parts. If the region is a rectangle it is always
+ possible to change the order of integration.
+\end{theorem}
+
+\begin{theorem}[Transformation of coordinates in 2 dimensions]
+ \label{thm:transform-coords}
+ Given two ``nice'' functions \(x(u,v)\) and \(y(u,v)\), that means are a
+ bijection from \(S\) to \(S'\) with continuous partial derivatives and
+ nonzero Jacobian determinant \(|\mx{J}_f| = \partial_u x \partial_v y -
+ \partial_v x \partial_u y\), which transform the coordinate system. Then
+ \[
+ \iint_S f(x,y) \,ds = \iint_{S'} f(x(u,v), y(u,v)) |\mx{J}_f| \,ds
+ \]
+\end{theorem}
+
+\begin{theorem}[Transformation of coordinates]
+ The generalization of theorem \ref{thm:transform-coords} is quite simple.
+ For an \(n\)-integral of a function \(f:\mathbb{R}^m\to\mathbb{R}\) over a
+ region \(B\), we let \(\vec{x}(\vec{u})\) be ``nice'' functions that
+ transform the coordinate system. Then as before
+ \[
+ \int_B f(\vec{x}) \,ds = \int_{B'} f(\vec{x}(\vec{u})) |\mx{J}_f| \,ds
+ \]
+\end{theorem}
+\begin{table}
+ \centering
+ \begin{tabular}{l >{\(}l<{\)} >{\(}l<{\)}}
+ \toprule
+ & \text{Volume } dv & \text{Surface } d\vec{s}\\
+ \midrule
+ Cartesian & - & dx\,dy \\
+ Polar & - & rd\,rd\phi \\
+ Curvilinear & - & |\mx{J}_f|\,du\,dv \\
+ \midrule
+ Cartesian & dx\,dy\,dz & \uvec{z}\,dx\,dy \\
+ Cylindrical & r\,dr\,d\phi\,dz & \uvec{z}r\,dr\,d\phi \\
+ & & \uvec{\phi}\,dr\,dz \\
+ & & \uvec{r}r\,d\phi\,dz \\
+ Spherical & r^2\sin\theta\, dr\,d\theta\,d\phi & \uvec{r}r^2\sin\theta\,d\theta\,d\phi \\
+ Curvilinear & |\mx{J}_f|\,du\,dv\,dw & - \\
+ \bottomrule
+ \end{tabular}
+ \caption{Differential elements for integration.}
+\end{table}
+
+\section{Derivatives of curves}
\section*{License}
\doclicenseText
diff --git a/build/FuVar.pdf b/build/FuVar.pdf
index 6271ac5..000648d 100644
--- a/build/FuVar.pdf
+++ b/build/FuVar.pdf
Binary files differ
diff --git a/img/double-integral.pdf b/img/double-integral.pdf
new file mode 100644
index 0000000..ba8b808
--- /dev/null
+++ b/img/double-integral.pdf
Binary files differ
diff --git a/img/double-integral.svg b/img/double-integral.svg
new file mode 100644
index 0000000..61dbcc3
--- /dev/null
+++ b/img/double-integral.svg
@@ -0,0 +1,306 @@
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