%% Bernd Berg apr 11 2016.
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{\bf\Large Typos, Errata and Additions} \bigskip

{\bf\Large Essential Graduate Electrodynamics} \bigskip

{\bf\Large Lecture Notes} \bigskip

{\bf\Large Bernd Berg, Fall 2016} \bigskip

{\bf p.11:} After equation (1.38): \smallskip

By reasons that will become clear in the later physical interpretation
the variable $\zeta$ is called {\it rapidity}\index{Rapidity}. For each
value of $d$ there is precisely one value of $\zeta$ and vice versa,
because the hyperbolic sine performs a one to one mapping. Such a
function is called {\it isomorphic}\index{Isomorphic} in calculus. 

...

Using finally $\begin{pmatrix}x^0\cr x^1\end{pmatrix}=
\begin{pmatrix}1\cr 1\end{pmatrix}$ yields 
\begin{eqnarray} \nonumber
  0  &=& [\cosh(\zeta)-\sinh(\eta)]^2-[-\sinh(\zeta)+\cosh(\eta)]^2 
  \\ &=& -2\,\cosh(\zeta)\,\sinh(\eta)+2\,\sinh(\zeta)\,\cosh(\eta) 
  \nonumber \\ &=& -2\,\tanh(\eta) + 2\,\tanh(\zeta)\ \Rightarrow\ 
  \zeta = \eta\, .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1.41)\nonumber
\end{eqnarray}


{\bf p.28:}

Consequently, we have
$$ \lambda' = \lambda\, \sqrt{1+\beta \over 1-\beta} =
  \lambda\,\sqrt{\cosh\zeta+\sinh\zeta\over\cosh\zeta-\sinh\zeta} =
  \lambda\, \exp(\zeta) \, . ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1.130) $$
\smallskip


{\bf p.70:} 

$$  Y_l^{\ -l}\ =\ \sqrt{\frac{(2l+1)\,(2l)!}{4\pi}}\,
    \frac{\sin^l\theta}{2^l\,l!}\,e^{-il\phi}\,.
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.154) $$
\smallskip


{\bf p.77:} 

The traceless {\it quadrupole} tensor is defined by
$$  Q^{ij} = Q^{ij}_S - Q^{ij}_D~~{\rm with}~~
    Q^{ij}_S = 3\int x'^i x'^j \rho(\vec{x}\,')\,d^3x'\,,~~
    Q^{ij}_D = \int r'^2\,\delta^{ij}\,\rho (\vec{x}\,')\,d^3x' $$
\rightline{(3.10)}
so that $\sum_{i=1}^3Q^{ii}_S=\sum_{i=1}^3Q^{ii}_D$ holds.


{\bf p.80:} 

$$\Phi(\vec{x}\,)={\rho(\vec{x}^{\,'})\over|\vec{x}-\vec{x}^{\,'}|} 
  + {\vec{p}(\vec{x}^{\,'})\cdot(\vec{x}-\vec{x}^{\,'})\over 
    |\vec{x}-\vec{x}^{\,'}|^3} \,. ~~~~~~~~~~~~~~~~~~~~~~~~~~~
  ~~~~~~~~~~~~~~~~(3.22)$$
\bigskip


{\bf p.101:}  \smallskip

{\it Faraday's Law} law of induction
$$ \epsilon_{\rm emf} = \oint_C \left(\vec{E} +
   \vec{\beta}\times\vec{B}\right)\cdot d\vec{l} =
    - \frac{1}{c} {d\ \over dt} \Phi_m~~{\rm with}~~
   \Phi_m = \int_S \vec{B}\cdot d\vec{a}\,,~~~~~~~~(4.4) $$
\bigskip


{\bf p.102:}  

$$ \nabla\cdot\vec{B}\ =\ 0\,,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(4.7)$$
which implies due to Gauss' theorem that the flux through a 
closed surface is zero.
\bigskip


{\bf p.183:} \smallskip

$$ v_g = {d\omega \over dk_q} = {c\over \sqrt{\mu\epsilon}}\,
  {d\ \over dk_q}\, \sqrt{k_q^2+\gamma_q^2} = {c\over 
  \sqrt{\mu\epsilon}}\, {k_q\over\sqrt{k_q^2+\gamma_q^2}}
  < \frac{c}{\sqrt{\mu\epsilon}}~~~~(6.48)  $$
\bigskip


{\bf p.190:} \smallskip

$$ \vec{E}_t = {1\over 2}\, (\vec{E}^+_t + \vec{E}^-_t)\ 
   =\ - \sin(kz) {k\over \gamma^2}\, \nabla_t \psi (x,y) 
   ~~~~(6.70) $$
\bigskip


{\bf p.212:} \smallskip

$$ \vec{E}_{\rm in}\ =\ \hat{\epsilon}_{\rm in}\,E_{\rm in}\,
  e^{ik_{\rm in} \hat{k}_{\rm in}\cdot\vec{x}}\ ,\qquad
  \vec{B}_{\rm in}\ =\ \hat{k}_{\rm in} \times \vec{E}_{\rm in}
  ~~~~~~(7.87) $$
\bigskip


{\bf p.242:} \smallskip

The equation \begin{eqnarray} \nonumber
  0 &=& L_-\, Y_l^{~-l} \,=\, L_-\,P_l^{~-l}\,e^{-i\,l\,\phi}
  \\ \nonumber &=& e^{-i\,\phi}\,\left(-{\partial\ \over\partial\theta}
     + i\,\cot\theta\, {\partial\ \over\partial\phi}\right)\,
     P_l^{~-l}\,e^{-i\,l\,\phi}\ ,
\end{eqnarray}

% implies
\bigskip


{\bf p.256:} \smallskip

(3) Show that 
\begin{eqnarray} \nonumber
   \epsilon_{\alpha\beta_1\gamma_1\delta_1} 
   \epsilon^{\alpha\beta_2\gamma_2\delta_2}\, &=&
  - \delta_{\beta_1}^{~\beta_2}\delta_{\gamma_1}^{~\gamma_2}
    \delta_{\delta_1}^{~\delta_2}
  - \delta_{\beta_1}^{~\gamma_2}\delta_{\gamma1}^{~\delta_2}
    \delta_{\delta_1}^{~\beta_2}
  - \delta_{\beta_1}^{~\delta_2}\delta_{\gamma_1}^{~\beta_2}
    \delta_{\delta_1}^{~\gamma_2}
    \\ \nonumber &+&
     \delta_{\beta_1}^{~\beta_2}\delta_{\gamma_1}^{~\delta_2}
     \delta_{\delta_1}^{~\gamma_2}
   + \delta_{\beta_1}^{~\delta_2}\delta_{\gamma_1}^{~\gamma_2}
     \delta_{\delta_1}^{~\beta_2}
   + \delta_{\beta_1}^{~\gamma_2}\delta_{\gamma_1}^{~\beta_2}
     \delta_{\delta_1}^{~\delta_2} \,.
\end{eqnarray}
\bigskip

{\bf p.285:} \smallskip

(4) Use the limit of figure~E.8 to calculate the principal
value integral defined by
\begin{eqnarray} \nonumber 
  \frac{1}{2\pi i}\,P\int_{-\infty}^{\infty}dx\,\frac{f(x)}{x-\omega_R} 
  = {1\over 2\pi\,i}\,\lim_{\eta\to 0^+}\left[
  \int_{-\infty}^{\omega_R-\eta} \frac{f(x)\,dx}{x-\omega_R} +
  \int_{\omega_R+\eta}^{+\infty} \frac{f(x)\,dx}{x-\omega_R} \right]
  &~&\\ \nonumber =\ {1\over 2\pi\,i} \lim_{\eta\to 0^+}\left[ 
  \int_{-\infty}^{\omega_R+i\eta-\eta} \frac{f(z)\,dz}{z-\omega_R-i\eta} 
  +\int_{\omega_R+i\eta+\eta}^{+\infty}\frac{f(z)\,dz}{z-\omega_R-i\eta} 
  \right] &~&
\end{eqnarray}

\vfill
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