adding basic kernels info

docs/conf.py

@@ -17,21 +17,17 @@
 # add these directories to sys.path here. If the directory is relative to the
 # documentation root, use os.path.abspath to make it absolute, like shown here.
 #
-# import os
-# import sys
-# sys.path.insert(0, os.path.abspath('.'))
-
 
 # -- General configuration ------------------------------------------------
 
 # If your documentation needs a minimal Sphinx version, state it here.
 #
-# needs_sphinx = '1.0'
+needs_sphinx = '1.6'
 
 # Add any Sphinx extension module names here, as strings. They can be
 # extensions coming with Sphinx (named 'sphinx.ext.*') or your custom
 # ones.
-extensions = ['sphinx.ext.mathjax']
+extensions = ['sphinx.ext.mathjax', 'sphinx_math_dollar']
 
 # Add any paths that contain templates here, relative to this directory.
 templates_path = ['_templates']

docs/index.rst

@@ -139,5 +139,6 @@ license.
    getchunkie
    guide
    gallery
+   kerns
 
 

docs/kerns.rst

@@ -0,0 +1,60 @@
+.. role:: matlab(code)
+   :language: matlab  
+
+The kernel class
+==================
+The kernel class is an easy way of specifying integral operators
+in classical potential theory. 
+The kernel object has several attributes including parameters required
+to define the kernel, the type of singularity for determining the quadrature
+to be used, function handles for evaluating the kernel, 
+and fmm acceleration routines if available. 
+
+The constructor for the kernel objects take the form::
+
+    kernel('PDE name', 'kernel name', 'extra params')
+
+Chunkie includes pre-edefined kernels that arise in the solution of the
+following PDEs.
+
+- :ref:`lap`
+- :ref:`helm`
+- :ref:`stokes`
+- :ref:`elasticity`
+
+.. _lap:
+
+Laplace kernels
+----------------
+The free-space Green's function for Laplace's equation in two dimensions,
+denoted by $G_{0}(x, y)$, is given by
+
+.. math::
+
+   G_{0}(x,y) = -\frac{1}{2\pi} \log{(\sqrt{(x_{1} - y_{1})^2 + (x_{2} -  y_{2})^2)}} \,,
+
+where $x=(x_{1},x_{2})$, and $y=(y_{1},y_{2})$.
+
+The PDE keywords for using any Laplace kernels are 'Laplace', or 'l'. 
+In the following, the variable $y$ will be referred to as the source, and the
+variable x will be the target.
+
+The following layer potentials
+defined by their associated kernels are supported:
+
+- 'single' or 's': Laplace single layer potential, $G_{0}(x,y)$
+- 'double' or 'd': Laplace double layer potential, $n(y) \cdot \partial_{y} G_{0}(x,y)$
+- 'combined' or 'c': Laplace combined layer potential, $c_{1} n(y) \cdot
+  \partial_{y} G_{0}(x,y) + c_{2} G_{0}(x,y)$, where the coefficients
+  $c_{1},c_{2}$ are passed as an extra array of length 2, with default values of
+  $(c_{1},c_{2}) = (1,1)$.
+- 'sprime' or 'sp': Normal derivative of Laplace single layer
+   potential, $n(x) \cdot \partial_{x} G_{0}(x,y)$
+- 'stau' or 'st': Tangential derivative of Laplace single layer potential,
+   $\tau(x) \cdot \partial_{x} G_{0}(x,y)$
+- 'dprime' or 'dp': Normal derivative of Laplace double layer potential, 
+   $n(x) \cdot \partial_{x} n(y) \cdot \partial_{y} G_{0}(x,y)$
+- 'sgrad' or 'sg': Gradient of Laplace single layer potential, 
+   $\partial_{x} G_{0}(x,y)$
+- 'dgrad' or 'dg': Gradient of Laplace double layer potential,
+   $\partial_{x} n_{y} \cdot \partial_{y} G_{0}(x,y)$