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diff --git a/FuVar.tex b/FuVar.tex
index 0e32511..b607802 100644
--- a/FuVar.tex
+++ b/FuVar.tex
@@ -90,8 +90,7 @@
\section{Preface}
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.
+not a rigorously constructed mathematical text.
\section{Derivatives of vector valued scalar functions}
@@ -185,11 +184,11 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
\begin{definition}[Directional derivative]
A function \(f(\vec{x})\) has a directional derivative in direction
- \(\vec{r}\) (with \(|\vec{r}|=1\)) of
+ \(\vec{v}\) (with \(|\vec{v}|=1\)) of
\[
- \frac{\partial f}{\partial\vec{r}}
- = \nabla_\vec{r} f = \vec{r} \dotp \grad f
- = \sum_{i=1}^m r_i \partial_{x_i} f
+ \frac{\partial f}{\partial\vec{v}}
+ = \nabla_\vec{v} f = \vec{v} \dotp \grad f
+ = \sum_{i=1}^m v_i \partial_{x_i} f
\]
\end{definition}
@@ -274,8 +273,8 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
\begin{remark}
The previous method is obtained by studying the second directional derivative
- \(\nabla_\vec{r}\nabla_\vec{r} f\) at the stationary point in direction of a
- vector \(\vec{r} = \vec{e}_1\cos(\alpha) + \vec{e}_2\sin(\alpha)\).
+ \(\nabla_\vec{v}\nabla_\vec{v} f\) at the stationary point in direction of a
+ vector \(\vec{v} = \vec{e}_1\cos(\alpha) + \vec{e}_2\sin(\alpha)\).
\end{remark}
\begin{method}[Determine the type of stationary point in higher dimensions]
@@ -447,51 +446,29 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
\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 -
+ nonzero Jacobian determinant \(|\mx{J}| = \partial_u x \partial_v y -
\partial_v x \partial_u y\), which transform the coordinate system. Then
\[
- \iint_S f(x,y) \,ds = \iint_{S'} f(x(u,v), y(u,v)) |\mx{J}_f| \,ds .
+ \iint_S f(x,y) \,ds = \iint_{S'} f(x(u,v), y(u,v)) |\mx{J}| \,ds .
\]
\end{theorem}
\begin{theorem}[Transformation of coordinates]
The generalization of theorem \ref{thm:transform-coords} is quite simple.
For an \(m\)-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
+ region \(B\), we let \(\vec{g}(\vec{u})\) be ``nice'' functions that
transform the coordinate system. Then as before
\[
- \int_B f(\vec{x}) \,ds = \int_{B'} f(\vec{x}(\vec{u})) |\mx{J}_f| \,ds .
+ \int_B f(\vec{r}) \,ds = \int_{B'} f(\vec{g}(\vec{u})) |\mx{J}_\vec{g}| \,ds .
\]
\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}
-
\begin{application}[Physics]
Given the mass \(m\) and density function \(\rho\) of an object,
its \emph{center of mass} is calculated with
\[
- \vec{x}_c = \frac{1}{m}\int_V \vec{x}\rho(\vec{x}) \,dv
- \stackrel{\rho\text{ const.}}{=} \frac{1}{V} \int_V \vec{x}\,dv .
+ \vec{x}_c = \frac{1}{m}\int_V \rho(\vec{r})\vec{r} \,dv
+ \stackrel{\rho\text{ const.}}{=} \frac{1}{V} \int_V \vec{r}\,dv .
\]
The (scalar) \emph{moment of inertia} \(J\) of an object is given by
\[
@@ -517,6 +494,14 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
\end{align*}
\end{theorem}
+% \begin{figure}
+% \centering
+% \includegraphics{img/multivariable-chain-rule}
+% \caption{
+% Multivariable chain rule.
+% }
+% \end{figure}
+
\begin{theorem}[Multivariable chain rule]
Let \(\vec{x}: \mathbb{R} \to \mathbb{R}^m\) and \(f: \mathbb{R}^m \to
\mathbb{R}\), so that \(f\circ\vec{x}: \mathbb{R} \to \mathbb{R}\), then
@@ -537,31 +522,39 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
\end{theorem}
\begin{definition}[Line integral in a scalar field]
- Let \(\mathcal{C}:[a,b]\to\mathbb{R}^n, t \mapsto \vec{x}(t)\) be a
- parametric curve. The \emph{line integral} in a field \(f(\vec{x})\) is the
+ Let \(\mathcal{C}:[a,b]\to\mathbb{R}^n, t \mapsto \vec{r}(t)\) be a
+ parametric curve. The \emph{line integral} in a field \(f(\vec{r})\) is the
integral of the signed area under the curve traced in \(\mathbb{R}^n\), and
is computed with
\[
- \int_\mathcal{C} f(\vec{x}) \,d\ell
- = \int_\mathcal{C} f(\vec{x}) \,|d\vec{x}|
- = \int_a^b f(\vec{x}(t)) |\vec{x}'(t)| \, dt .
+ \int_\mathcal{C} f(\vec{r}) \,d\ell
+ = \int_\mathcal{C} f(\vec{r}) \,|d\vec{r}|
+ = \int_a^b f(\vec{r}(t)) |\vec{r}'(t)| \, dt .
\]
\end{definition}
\begin{application}[Length of a parametric curve]
- By computing the line integral of the function \(1(\vec{x})\) we get the
+ By computing the line integral of the function \(1(\vec{r})\) we get the
length of the parametric curve \(\mathcal{C}:[a,b]\to\mathbb{R}^n\).
\[
\int_\mathcal{C}d\ell
- = \int_\mathcal{C} |d\vec{x}|
+ = \int_\mathcal{C} |d\vec{r}|
= \int_a^b \sqrt{\sum_{i=1}^n x'_i(t)^2} \,dt
\]
The special case with the scalar function \(f(x)\) results in
\(\int_a^b\sqrt{1+f'(x)^2}\,dx\).
\end{application}
+\begin{figure}
+ \centering
+ \includegraphics{img/line-integral}
+ \caption{
+ Line integral in a vector field.
+ }
+\end{figure}
+
\begin{definition}[Line integral in a vector field]
- The line integral in a vector field \(\vec{F}(\vec{x})\) is the ``sum'' of
+ The line integral in a vector field \(\vec{F}(\vec{r})\) is the ``sum'' of
the projections of the field's vectors on the tangent of the parametric curve
\(\mathcal{C}\).
\[
@@ -587,7 +580,7 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
\end{definition}
\begin{theorem}
- For a twice partially differentiable vector field \(\vec{F}(\vec{x})\) in
+ For a twice partially differentiable vector field \(\vec{F}\) in
\(n\) dimensions without ``holes'', i.e. in which each closed curve can be
contracted to a point (simply connected open set), the following statements
are equivalent:
@@ -627,6 +620,14 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
\mathbb{R}^3\).
\end{definition}
+\begin{figure}
+ \centering
+ \includegraphics{img/surface-integral}
+ \caption{
+ Surface integral.
+ }
+\end{figure}
+
\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
@@ -648,6 +649,28 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
\]
\end{definition}
+\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{Vector analysis}
\begin{definition}[Flux]
@@ -729,7 +752,7 @@ Notice that the curl is a vector, normal to the enclosed surface \(A\).
\end{theorem}
\begin{theorem}[Green's theorem]
- The special case of Stokes' theorem in 2D is knowns as Green's theorem.
+ The special case of Stokes' theorem in 2D is known 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}