fixed math and added a small intro (that could use a bit of work)

docs/api_FDEM.rst

@@ -13,57 +13,70 @@ Intro Here
 
 Background
 ==========
-Electromagnetic geophysical methods are used
+Electromagnetic (EM) geophysical methods are used in a variety of applications from resource exploration, including for hydrocarbons and minerals, to environmental applications, such as groundwater monitoring.
+
+Fourier Transform Convention
+----------------------------
+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
 
-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(\omega) = \int_{-\infty}^{\infty} f(t) e^{- i \omega t} dt \\
+
 	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} \\
+- \\(\\ \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\\
+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
 --------------------------
+For the frequency range typical of most geophysical surveys, the contribution of the electric displacement is negligible compared to the electric current density. In this case, we use the \emph{Quasi-static approximation} and assume that this term can be neglected, giving
+
+.. math::
+	\nabla \times \vec{E} = -i \omega \vec{B} \\
+	\nabla \times \vec{H} = \vec{J} + \vec{J_s}
 
 
 Fields from a Dipole