\section{Appendix - Fundamentele constanten}\label{appendix_c}

$c = 2.998 \times 10^8$ m$/s$
\hfill
lichtsnelheid

\smallskip
\noindent
$h = 6.626 \times 10^{-34}$ Js
\hfill
constante van Planck

\smallskip
\noindent
$e = 1.602 \times 10^{-19}$ C
\hfill
lading van het elektron

\smallskip
\noindent
$m_e = 9.109 \times 10^{-31}$ kg
\hfill
massa van het elektron

\smallskip
\noindent
$m_p = 1.672 \times 10^{-27}$ kg
\hfill
massa van het proton

\smallskip
\noindent
$\epsilon_0 = 8.854 \times 10^{-12}$ C$^2$/Nm$^2$
\hfill
permittiviteit van het vacuum

\smallskip
\noindent
$\mu_0 = 4 \pi \times 10^{-7}$ N/A$^2$
\hfill
permeabiliteit van het vacuum

\smallskip
\noindent
$N_A = 6.022 \times 10^{23}$ 1/mol
\hfill
constante van Avogadro

\smallskip
\noindent
$k =  1.381 \times 10^{-23}$ J/K
\hfill
constante van Boltzmann


\vfill
\eject

\section{Appendix - Co\"ordinatensystemen}\label{appendix_d}

\subsection{Cartesiaanse co\"ordinaten}

\medskip
$d{\bf l} = dx ~ \hat {\bf x} + dy ~ \hat {\bf y} + dz ~ \hat {\bf z}$
\hfill
lijn-element

\smallskip
$d\tau = dxdydz$
\hfill
volume-element

\smallskip
$\nabla t = {\partial t \over \partial x} \hat {\bf x} +
{\partial t \over \partial y} \hat {\bf y} +
{\partial t \over \partial z} \hat {\bf z}$
\hfill
gradi\" ent

\smallskip
$\nabla \cdot {\bf v} = {\partial v_x \over \partial x} +
{\partial v_y \over \partial y} +
{\partial v_z \over \partial z}$
\hfill
divergentie

\smallskip
$\nabla \times {\bf v} =
\left( {\partial v_z \over \partial y} -
{\partial v_y \over \partial z} \right) \hat {\bf x} +
\left( {\partial v_x \over \partial z} -
{\partial v_z \over \partial x} \right) \hat {\bf y} +
\left( {\partial v_y \over \partial x} -
{\partial v_x \over \partial y} \right) \hat {\bf z}$
\hfill
rotatie

\smallskip
$\Delta t = \nabla^2 t = {\partial^2 t \over \partial x^2} +
{\partial^2 t \over \partial y^2} +{\partial^2 t \over \partial z^2}$
\hfill
Laplace operator

\bigskip
\noindent
\subsection{Sferische co\"ordinaten}

\medskip
$d{\bf l} = dr ~ \hat {\bf r} + r d\theta ~ \hat {\bf \theta} +
r \sin {\theta}d\phi ~ \hat {\bf \phi}$
\hfill
lijn-element

\smallskip
$d\tau = r^2 \sin{\theta} dr d\theta d\phi$
\hfill
volume-element

\smallskip
$\nabla t = {\partial t \over \partial r} \hat {\bf r} +
{1 \over r}{\partial t \over \partial \theta} \hat {\bf \theta} +
{1 \over r \sin{\theta}}{\partial t \over \partial \phi} \hat {\bf \phi}$
\hfill
gradi\" ent

\smallskip
$\nabla \cdot {\bf v} = {1 \over r^2}{\partial \over \partial r}(r^2 v_r) +
{1 \over r \sin{\theta}}{\partial \over \partial \theta}(\sin{\theta}v_\theta ) +
{1 \over r \sin{\theta}}{\partial v_\phi \over \partial \phi}$
\hfill
divergentie

\smallskip
$\nabla \times {\bf v} =
{1 \over r \sin{\theta}} \left[ {\partial \over \partial \theta}
(\sin{\theta} v_\phi ) - {\partial v_\theta \over \partial \phi} \right]
\hat {\bf r} +
{1 \over r} \left[ {1 \over \sin{\theta}} {\partial v_r \over \partial \phi}
- {\partial \over \partial r} (rv_\phi ) \right] \hat {\bf \theta} +
{1 \over r} \left[ {\partial \over \partial r} (r v_\theta )
- {\partial v_r \over \partial \theta }\right] \hat {\bf \phi}$
\hfill
rotatie

\smallskip
$\Delta t = \nabla^2 t = {1 \over r^2}{\partial \over \partial r}
\left( r^2 {\partial t \over \partial r} \right)
+ {1 \over r^2 \sin{\theta}}{\partial \over \partial \theta}
\left( \sin{\theta}{\partial t \over \partial \theta} \right) +
{1 \over r^2 \sin^2{\theta} }{\partial^2 t \over \partial \phi^2}$
\hfill
Laplace operator

\bigskip
\noindent
\subsection{Cilindirische co\"ordinaten}

\medskip
$d{\bf l} = ds ~ \hat {\bf s} + s d\phi ~ \hat {\bf \phi} + dz ~ \hat {\bf z}$
\hfill
lijn-element

\smallskip
$d\tau = sds d\phi dz$
\hfill
volume-element

\smallskip
$\nabla t = {\partial t \over \partial s} \hat {\bf s} +
{1 \over s}{\partial t \over \partial \phi} \hat {\bf \phi} +
{\partial t \over \partial z} \hat {\bf z}$
\hfill
gradi\" ent

\smallskip
$\nabla \cdot {\bf v} = {1 \over s}{\partial \over \partial s}(s v_s) +
{1 \over s}{\partial v_\phi \over \partial \phi} +
{\partial v_z \over \partial z}$
\hfill
divergentie

\smallskip
$\nabla \times {\bf v} =
\left[ {1 \over s}{\partial v_z \over \partial \phi} -
{\partial v_\phi \over \partial z} \right] \hat {\bf s} +
\left[ {\partial v_s \over \partial z} -
{\partial v_z \over \partial s} \right] \hat {\bf \phi} +
{1 \over s}\left[ {\partial \over \partial s}(sv_\phi ) -
{\partial v_s \over \partial \phi} \right] \hat {\bf z}$
\hfill
rotatie

\smallskip
$\Delta t = \nabla^2 t = {1 \over s}{\partial \over \partial s}
\left( s {\partial t \over \partial s} \right)
+ {1 \over s^2}{\partial^2 t \over \partial \phi^2} +
{\partial^2 t \over \partial z^2}$
\hfill
Laplace operator

\bigskip
\noindent
{\bf FUNDAMENTELE THEOREMAS}

\medskip
$\int_{\bf a}^{\bf b} (\nabla f) \cdot d{\bf l} = f({\bf b}) - f({\bf a})$
\hfill
Gradi\" ent theorema

\smallskip
$\int (\nabla \cdot {\bf A} ) d\tau = \oint {\bf A} \cdot d{\bf a}$
\hfill
Divergentie theorema (stelling van Gauss)

\smallskip
$\int (\nabla \times {\bf A} ) \cdot d{\bf a} = \oint {\bf A} \cdot d{\bf l}$
\hfill
Rotatie theorema (stelling van Stokes)

\vfill
\eject


