From fa657eb7d323e5bf4d791a690d4b7f02ea08ded0 Mon Sep 17 00:00:00 2001
From: Lindsey Heagy <lindseyheagy@gmail.com>
Date: Sat, 5 Apr 2014 11:21:27 -0700
Subject: [PATCH] start of documentation for FDEM

---
 docs/api_FDEM.rst     | 77 +++++++++++++++++++++++++++++++++++++++++--
 simpegEM/FDEM/FDEM.py | 17 ++++++----
 2 files changed, 85 insertions(+), 9 deletions(-)

diff --git a/docs/api_FDEM.rst b/docs/api_FDEM.rst
index 9fdf123c..b9547a40 100644
--- a/docs/api_FDEM.rst
+++ b/docs/api_FDEM.rst
@@ -1,9 +1,82 @@
 .. _api_FDEM:
 
+.. math::
+    \renewcommand{\div}{\nabla\cdot\,}
+    \newcommand{\grad}{\vec \nabla}
+    \newcommand{\curl}{{\vec \nabla}\times\,}
 
-The API
-=======
 
+Frequency Domain Electromagnetics
+*********************************
+
+Intro Here
+
+Background
+==========
+Electromagnetic geophysical methods are used
+
+Conventions
+-----------
+In order to examine Maxwell's equations in the frequency domain, we must first define our choice of harmonic time-dependence by choosing a Fourier transform convention. We use the \\e^{i\omega t}\\ convention, so we define our Fourier Transform pair as
+.. math ::
+	F(\omega) = \int_{-\infty}^{\infty} f(t) e^{- i \omega t} dt
+.. math ::
+	f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega) e^{i \omega t} d\omega
+where \\ \omega\\ is angular frequency, \\t\\ is time, \\ F(\omega) \\ is the function defined in the frequency domain and \\ f(t) \\ is the function defined in the time domain.
+
+
+Maxwell's Equations
+===================
+In the frequency domain, Maxwell's equations are given by
+.. math::
+	\curl \vec{E} = - i \omega \vec{B} \\
+	\curl \vec{H} = \vec{J} + i \omega \vec{D} + \vec{J_s} \\
+	\div \vec{B} = 0 \\
+	\div \vec{D} = \rho_f
+
+where:
+- \\ \vec{E} \\ : electric field (\\V/m\\)
+- \\ \vec{H} \\ : magnetic field (\\A/m\\)
+- \\ \vec{B} \\ : magnetic flux density (\\Wb/m^2\\)
+- \\ \vec{D} \\ : electric displacement / electric flux density (\\C/m^2\\)
+- \\ \vec{J} \\ : electric current density (\\A/m^2\\)
+- \\ \rho_f \\ : free charge density
+The source term is \\ \vec{J_s} \\
+
+
+Constitutive Relations
+----------------------
+The fields and fluxes are related through the constitutive relations. At each frequency, they are given by
+.. math::
+	\vec{J} = \sigma \vec{E} \\
+
+	\vec{B} = \mu \vec{H} \\
+
+	\vec{D} = \varepsilon \vec{E}
+where
+- \\ \sigma \\ : electrical conductivity (S/m)
+- \\ \mu \\ : magnetic permeability (H/m)
+- \\ \varepsilon \\ : dielectric permittivity (F/m)
+
+\\ \sigma \\, \\ \mu \\, \\ \varepsilon \\ are physical properties which depend on the material. \\ \sigma \\ describes how easily electric current passes through a material, \\ \mu \\ describes how easily a material is magnetized, and \\ \varepsilon \\ describes how easily a material is electrically polarized. In most geophysical applications of EM, \\ \sigma \\ is the the primary physical property of interest, and \\ \mu \\, \\ \varepsilon \\ are assumed to have their free-space values \\ \mu_0 = 4\pi \times 10^{-7} H/m\\, \\ \varepsilon_0 = 8.85\times 10^{-12} F/m\\
+For a more complete discussion of physical properties see `GPG <http://www.eos.ubc.ca/courses/eosc350/content/index.htm>`_
+
+
+Quasi-static Approximation
+--------------------------
+
+
+Fields from a Dipole
+--------------------
+
+Forward Problem
+===============
+
+Inverse Problem
+===============
+
+API
+===
 .. automodule:: simpegEM.FDEM.FDEM
     :show-inheritance:
     :members:
diff --git a/simpegEM/FDEM/FDEM.py b/simpegEM/FDEM/FDEM.py
index e9c233df..8074fdba 100644
--- a/simpegEM/FDEM/FDEM.py
+++ b/simpegEM/FDEM/FDEM.py
@@ -9,13 +9,16 @@ def omega(freq):
 
 class BaseProblemFDEM(Problem.BaseProblem):
     """
-        Frequency-Domain EM problem - E-formulation
-
-
-        .. math::
-
-            \dcurl E + i \omega B = 0 \\\\
-            \dcurl^\\top \MfMui B - \MeSig E = \Me \j_s
+        We start with the E-formulation Maxwell's equations in the frequency domain:
+        .. math ::
+            \\nabla \\times \\vec{E} + i \\omega \\vec{B} = 0 \\\\
+            \\nabla \\times \\mu^{-1} \\vec{B} - \\sigma \\vec{E} = \\vec{J_s}
+        By eliminating the magnetic flux density using
+        .. math ::
+            \\vec{B} = \\frac{-1}{i\\omega}\\nabla\\times\\vec{E},
+        we can write Maxwell's equations as a second order system in \\ \\vec{E} \\ only:
+        .. math ::
+            \\nabla \\times \\mu^{-1} \\nabla \\times \\vec{E} + i \\omega \\sigma \\vec{E} = \\vec{J_s}
     """
     def __init__(self, model, **kwargs):
         Problem.BaseProblem.__init__(self, model, **kwargs)