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-rw-r--r--FuVar.tex178
-rw-r--r--build/FuVar.pdfbin183310 -> 191614 bytes
-rw-r--r--tex/hsrstud.sty12
3 files changed, 184 insertions, 6 deletions
diff --git a/FuVar.tex b/FuVar.tex
index fbc8a06..4768341 100644
--- a/FuVar.tex
+++ b/FuVar.tex
@@ -3,7 +3,7 @@
% !TeX root = FuVar.tex
%% TODO: publish to CTAN
-\documentclass[twocolumn, margin=normal]{tex/hsrzf}
+\documentclass[twocolumn, margin=small]{tex/hsrzf}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Packages
@@ -498,7 +498,7 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
% and similarly the \emph{area moment of inertia} \(I\)
\end{application}
-\section{Parametric curves and line integrals}
+\section{Parametric curves, line and surface integrals}
\begin{definition}[Parametric curve]
A parametric curve is a vector function \(\mathcal{C} : \mathbb{R} \to W
@@ -620,10 +620,182 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
\end{align*}
\end{theorem}
-\section{Surface integrals}
+\begin{definition}[Parametric surface]
+ A parametric surface is a vector function \(\mathcal{S}: W \subseteq \mathbb{R}^2 \to
+ \mathbb{R}^3\).
+\end{definition}
+
+\begin{theorem}[Area of a parametric surface]
+ The area spanned by a parametric surface \(\vec{s}(u,v)\), with continuous
+ partial derivatives and that satisfy \(\partial_u \vec{s} \crossp \partial_v
+ \vec{s} \neq \vec{0}\), is given by
+ \[
+ A = \int_\mathcal{S} ds
+ = \iint |\partial_u \vec{s} \crossp \partial_v \vec{s}| \,dudv
+ \]
+\end{theorem}
+
+\begin{definition}[Scalar surface integral]
+ Let \(f: \mathbb{R}^3 \to \mathbb{R}\) be a function on a parametric surface
+ \(\vec{s}: W \subseteq \mathbb{R}^2 \to \mathbb{R}^3\). The surface integral
+ of \(f\) over \(\mathcal{S}\) is
+ \[
+ \int_\mathcal{S} f \,ds =
+ \iint_W f(\vec{s}(u,v)) \cdot
+ |\partial_u \vec{s} \crossp \partial_v \vec{s}| \,dudv
+ \]
+\end{definition}
\section{Vector analysis}
+\begin{definition}[Flux]
+ In a vector field \(\vec{F}: \mathbb{R}^m \to \mathbb{R}^n\) we define the
+ \emph{flux} through a parametric surface \(\mathcal{S}\) as
+ \[
+ \Phi = \int_\mathcal{S} \vec{F} \dotp d\vec{s}
+ = \int_\mathcal{S} \vec{F} \dotp \uvec{n} \,ds .
+ \]
+ If \(\mathcal{S}\) is a closed surface we write
+ \(
+ \mathring{\Phi} = \oint_\mathcal{S} \vec{F} \dotp d\vec{s}
+ \).
+\end{definition}
+
+If we now take the normalized flux on the surface of an arbitrarily small
+(limit) volume \(V\) we get the \emph{divergence}
+\[
+ \div \vec{F} = \lim_{V\to 0} \frac{1}{V} \oint_{\partial V} \vec{F}\dotp d\vec{s} .
+\]
+
+\begin{theorem}[Formula for divergence]
+ Let \(\vec{F}: \mathbb{R}^m \to \mathbb{R}^m\) be a vector field.
+ The divergence of \(\vec{F} = (F_{x_1},\ldots, F_{x_m})^t\) is
+ \[
+ \div\vec{F} = \sum_{i = 1}^m \partial_{x_i} F_{x_i} ,
+ \]
+ as suggested by the (ab)use of the dot product notation.
+\end{theorem}
+
+\begin{theorem}[Divergence theorem, Gauss's theorem]
+ Because the flux on the boundary \(\partial V\) of a volume \(V\) contains
+ information of the field inside of \(V\), it is possible relate the two with
+ \[
+ \int_V \div \vec{F} \,dv = \oint_{\partial V} \vec{F} \dotp d\vec{s} .
+ \]
+\end{theorem}
+
+\begin{definition}[Circulation, Vorticity]
+ The result of a closed line integral can be interpreted as a macroscopic
+ measure how much the field rotates around a given point, and is thus
+ sometimes called \emph{circulation} or \emph{vorticity}.
+\end{definition}
+
+As before, if we now make the area \(A\) enclosed by the parametric curve for
+the circulation arbitrarily small, normalize it, and use Gauss's theorem we get
+a local measure called \emph{curl}
+\[
+ \curl \vec{F} =
+ \lim_{A\to 0} \frac{\uvec{n}}{A} \oint_{\partial A} \vec{F} \dotp d\vec{s}
+\]
+Notice that the curl is a vector, normal to the enclosed surface \(A\).
+
+\begin{theorem}[Formula for curl]
+ Let \(\vec{F}\) be a vector field. In 2 dimensions
+ \[
+ \curl \vec{F} = \left(\partial_x F_y - \partial_y F_x\right)\uvec{z}.
+ \]
+ And in 3D
+ \[
+ \curl \vec{F} = \begin{pmatrix}
+ \partial_y F_z - \partial_z F_y \\
+ \partial_z F_x - \partial_x F_z \\
+ \partial_x F_y - \partial_y F_x
+ \end{pmatrix}
+ = \begin{vmatrix}
+ \uvec{x} & \uvec{y} & \uvec{z} \\
+ \partial_x & \partial_y & \partial_z \\
+ F_x & F_y & F_z
+ \end{vmatrix} .
+ \]
+\end{theorem}
+
+\begin{theorem}[Stokes' theorem]
+ \[
+ \int_\mathcal{S} \curl \vec{F} \dotp d\vec{s}
+ = \oint_{\partial\mathcal{S}} \vec{F} \dotp d\vec{r}
+ \]
+\end{theorem}
+
+\begin{theorem}[Green's theorem]
+ The special case of Stokes' theorem in 2D is knowns as Green's theorem.
+ \[
+ \int_\mathcal{S} \partial_x F_y - \partial_y F_x \,ds
+ = \oint_{\partial\mathcal{S}} \vec{F} \dotp d\vec{r}
+ \]
+\end{theorem}
+
+\begin{definition}[Laplacian operator]
+ A second vector derivative is so important that it has a special name. For a
+ scalar function \(f: \mathbb{R}^m \to \mathbb{R}\) the divergence of the
+ gradient
+ \[
+ \laplacian = \div (\grad f) = \sum_{i=1}^m \partial_{x_i}^2 f_{x_i}
+ \]
+ is called the \emph{Laplacian operator}.
+\end{definition}
+
+\begin{definition}[Vector Laplacian]
+ The Laplacian operator can be extended on a vector field \(\vec{F}\) to the
+ \emph{Laplacian vector} by applying the Laplacian to each component:
+ \[
+ \vlaplacian \vec{F}
+ = (\laplacian F_x)\uvec{x}
+ + (\laplacian F_y)\uvec{y}
+ + (\laplacian F_z)\uvec{z} .
+ \]
+ The vector laplacian can also be defined as
+ \[
+ \vlaplacian \vec{F} = \grad (\div \vec{F}) - \curl (\curl \vec{F}).
+ \]
+\end{definition}
+
+\begin{theorem}[Product rules and second derivatives]
+ Let \(f,g\) be sufficiently differentiable scalar functions \(D
+ \subseteq\mathbb{R}^m \to \mathbb{R}\) and \(\vec{A}, \vec{B}\) be
+ sufficiently differentiable vector fields in \(\mathbb{R}^m\) (with \(m = 2\)
+ or 3 for equations with the curl).
+ \begin{itemize}
+ \item Rules with the gradient
+ \begin{align*}
+ \grad (\div \vec{A}) &= \curl \curl \vec{A} + \vlaplacian \vec{A} \\
+ \grad (f\cdot g) &= (\grad f)\cdot g + f\cdot \grad g \\
+ \grad (\vec{A} \dotp \vec{B}) &=
+ (\vec{A} \dotp \grad) \vec{B}
+ + (\vec{B} \dotp \grad) \vec{A} \\
+ & + \vec{A} \crossp (\curl \vec{B})
+ + \vec{B} \crossp (\curl \vec{A})
+ \end{align*}
+ \item Rules with the divergence
+ \begin{align*}
+ \div (\grad f) &= \laplacian f \\
+ \div (\curl \vec{A}) &= 0 \\
+ \div (f\cdot \vec{A}) &= (\grad f) \dotp \vec{A} + f\cdot (\div \vec{A}) \\
+ \div (\vec{A}\crossp\vec{B}) &= (\curl \vec{A})\dotp \vec{B}
+ - \vec{A} \cdot (\curl\vec{B})
+ \end{align*}
+ \item Rules with the curl
+ \begin{align*}
+ \curl (\grad f) &= \vec{0} \\
+ \curl (\curl \vec{A}) &= \grad (\div \vec{A}) - \vlaplacian \vec{A} \\
+ \curl (\vlaplacian \vec{A}) &= \vlaplacian (\curl \vec{A}) \\
+ \curl (f\cdot \vec{A}) &= (\grad f)\crossp \vec{A} + f\cdot \curl \vec{A} \\
+ \curl (\vec{A}\crossp\vec{B}) &=
+ (\vec{B} \dotp \grad) \vec{A} - (\vec{A} \dotp \grad) \vec{B} \\
+ &+ \vec{A} \dotp (\div \vec{B}) - \vec{B} \dotp (\div \vec{A})
+ \end{align*}
+ \end{itemize}
+\end{theorem}
+
\section*{License}
\doclicenseText
diff --git a/build/FuVar.pdf b/build/FuVar.pdf
index 06ab09e..d042a4e 100644
--- a/build/FuVar.pdf
+++ b/build/FuVar.pdf
Binary files differ
diff --git a/tex/hsrstud.sty b/tex/hsrstud.sty
index 97c148f..a65cfa4 100644
--- a/tex/hsrstud.sty
+++ b/tex/hsrstud.sty
@@ -176,7 +176,7 @@
%% Products ((
\newcommand{\dotp}{\boldsymbol\cdot}
-\newcommand{\crossp}{\boldsymbol\crossp}
+\newcommand{\crossp}{\boldsymbol\times}
\newcommand{\cross}{\crossp}
%%))
@@ -222,9 +222,15 @@
%% laplacian ((
\ifhsr@textvecdiff
- \DeclareMathOperator{\laplace}{div grad}
+ \DeclareMathOperator{\laplacian}{div grad}
\else
- \DeclareMathOperator{\laplace}{\nabla^2}
+ \DeclareMathOperator{\laplacian}{\nabla^2}
+\fi
+
+\ifhsr@textvecdiff
+ \DeclareMathOperator{\vlaplacian}{div grad}
+\else
+ \DeclareMathOperator{\vlaplacian}{\vec{\nabla}^2}
\fi
%% ))