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cleaned up math in docs

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This commit is contained in:
Lindsey Heagy
2015-07-06 23:59:21 -05:00
parent 037155e3f3
commit cb73c38e8e
3 changed files with 61 additions and 46 deletions
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docs/api_FDEM.rst
+7 -6
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@@ -44,12 +44,13 @@ In the frequency domain, Maxwell's equations are given by
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
- \\(\\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{S}\\)
docs/conf.py
+2 -2
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@@ -43,7 +43,7 @@ master_doc = 'index'
# General information about the project.
project = u'SimPEG'
copyright = u'2013, SimPEG Developers'
copyright = u'2013-2015, SimPEG Developers'
# The version info for the project you're documenting, acts as replacement for
# |version| and |release|, also used in various other places throughout the
@@ -93,7 +93,7 @@ pygments_style = 'sphinx'
# The theme to use for HTML and HTML Help pages. See the documentation for
# a list of builtin themes.
html_theme = 'default'
html_theme = 'sphinx_rtd_theme'
# Theme options are theme-specific and customize the look and feel of a theme
# further. For a list of options available for each theme, see the
simpegEM/FDEM/FDEM.py
+52 -38
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@@ -8,23 +8,27 @@ from simpegEM.Utils.EMUtils import omega
class BaseFDEMProblem(BaseEMProblem):
"""
We start by looking at Maxwell's equations in the electric field \\(\\mathbf{e}\\) and the magnetic flux density \\(\\mathbf{b}\\):
We start by looking at Maxwell's equations in the electric
field \\\(\\\mathbf{e}\\\) and the magnetic flux
density \\\(\\\mathbf{b}\\\)
.. math ::
\\mathbf{C} \\mathbf{e} + i \\omega \\mathbf{b} = \\mathbf{s_m} \\\\
\\mathbf{C}^T \\mathbf{M^f_{\\mu^{-1}} \\mathbf{b} - \\mathbf{M^e_{\\sigma}} \\mathbf{e} = \\mathbf{s_e}
if using the E-B formulation (:code:`ProblemFDEM_e` or :code:`ProblemFDEM_b`) or the magnetic field \\(\\mathbf{h}\\) and current density \\(\\mathbf{j}\\)
\mathbf{C} \mathbf{e} + i \omega \mathbf{b} = \mathbf{s_m} \\\\
{\mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{b} - \mathbf{M_{\sigma}^e} \mathbf{e} = \mathbf{s_e}}
if using the E-B formulation (:code:`ProblemFDEM_e`
or :code:`ProblemFDEM_b`) or the magnetic field
\\\(\\\mathbf{h}\\\) and current density \\\(\\\mathbf{j}\\\)
.. math ::
\\mathbf{C}^T \\mathbf{M^f_{\\rho}} \\mathbf{j} + i \\omega \\mathbf{M^e_{\\mu}} \\mathbf{h} = \\mathbf{s_m} \\\\
\\mathbf{C} \\mathbf{h} - \\mathbf{j} = \\mathbf{s_e}
\mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{j} + i \omega \mathbf{M_{\mu}^e} \mathbf{h} = \mathbf{s_m} \\\\
\mathbf{C} \mathbf{h} - \mathbf{j} = \mathbf{s_e}
if using the H-J formulation (:code:`ProblemFDEM_j` or :code:`ProblemFDEM_h`).
The problem performs the elimination so that we are solving the system for \\(\\mathbf{e},\\mathbf{b},\\mathbf{j} or \\mathbf{h}\\)
The problem performs the elimination so that we are solving the system for \\\(\\\mathbf{e},\\\mathbf{b},\\\mathbf{j} \\\) or \\\(\\\mathbf{h}\\\)
"""
surveyPair = SurveyFDEM
@@ -187,17 +191,16 @@ class ProblemFDEM_e(BaseFDEMProblem):
.. math ::
\\mathbf{b} = \\frac{1}{i \\omega}\\left(-\\mathbf{C} \\mathbf{e} + \\mathbf{s_m}\\right) \\\\
\mathbf{b} = \\frac{1}{i \omega}\\left(-\mathbf{C} \mathbf{e} + \mathbf{s_m}\\right)
we can write Maxwell's equations as a second order system in \\(\\mathbf{e}\\) only:
we can write Maxwell's equations as a second order system in \\\(\\\mathbf{e}\\\) only:
.. math ::
\\left(\\mathbf{C}^T \\mathbf{M^f_{\\mu^{-1}} \\mathbf{C} + i \\omega \\mathbf{M^e_{\\sigma}} \\right)\\mathbf{e}
= \\mathbf{C}^T \\mathbf{M^f_{\\mu^{-1}}\\mathbf{s_m} -i\\omega\\mathbf{s_e} \\\\
\\left(\mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{C}+ i \omega \mathbf{M^e_{\sigma}} \\right)\mathbf{e} = \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f}\mathbf{s_m} -i\omega\mathbf{s_e}
which we solve for \\(\\mathbf{e}\\).
which we solve for \\\(\\\mathbf{e}\\\).
"""
_fieldType = 'e'
@@ -210,7 +213,7 @@ class ProblemFDEM_e(BaseFDEMProblem):
def getA(self, freq):
"""
.. math ::
\\mathbf{A} = \\mathbf{C}^T \\mathbf{M^f_{\\mu^{-1}} \\mathbf{C} + i \\omega \\mathbf{M^e_{\\sigma}}
\mathbf{A} = \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{C} + i \omega \mathbf{M^e_{\sigma}}
:param float freq: Frequency
:rtype: scipy.sparse.csr_matrix
@@ -235,7 +238,7 @@ class ProblemFDEM_e(BaseFDEMProblem):
def getRHS(self, freq):
"""
.. math ::
\\mathbf{RHS} = \\mathbf{C}^T \\mathbf{M^f_{\\mu^{-1}}\\mathbf{s_m} -i\\omega\\mathbf{s_e}
\mathbf{RHS} = \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f}\mathbf{s_m} -i\omega\mathbf{s_e}
:param float freq: Frequency
:rtype: numpy.ndarray (nE, nSrc)
@@ -282,15 +285,17 @@ class ProblemFDEM_e(BaseFDEMProblem):
class ProblemFDEM_b(BaseFDEMProblem):
"""
We eliminate \\(\\mathbf{e}\\) using
We eliminate \\\(\\\mathbf{e}\\\) using
.. math ::
\\mathbf{e} = \\mathbf{M^e_{\\sigma}}^{-1} \\left(\\mathbf{C}^T \\mathbf{M^f_{\\mu^{-1}} \\mathbf{b} - \\mathbf{s_e}\\right)
\mathbf{e} = \mathbf{M^e_{\sigma}}^{-1} \\left(\mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{b} - \mathbf{s_e}\\right)
and solve for \\(\\mathbf{b}\\) using:
and solve for \\\(\\\mathbf{b}\\\) using:
.. math ::
\\left(\\mathbf{C} \\mathbf{M^e_{\\sigma}}^{-1} \\mathbf{C}^T \\mathbf{M^f_{\\mu^{-1}} + i \\omega \\right)\\mathbf{b} = \\mathbf{s_m} + \\mathbf{M^e_{\\sigma}}^{-1}\\mathbf{s_e}
\\left(\mathbf{C} \mathbf{M^e_{\sigma}}^{-1} \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} + i \omega \\right)\mathbf{b} = \mathbf{s_m} + \mathbf{M^e_{\sigma}}^{-1}\mathbf{s_e}
.. note ::
The inverse problem will not work with full anisotropy
@@ -306,7 +311,7 @@ class ProblemFDEM_b(BaseFDEMProblem):
def getA(self, freq):
"""
.. math ::
\\mathbf{A} = \\mathbf{C} \\mathbf{M^e_{\\sigma}}^{-1} \\mathbf{C}^T \\mathbf{M^f_{\\mu^{-1}} + i \\omega
\mathbf{A} = \mathbf{C} \mathbf{M^e_{\sigma}}^{-1} \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} + i \omega
:param float freq: Frequency
:rtype: scipy.sparse.csr_matrix
@@ -346,7 +351,7 @@ class ProblemFDEM_b(BaseFDEMProblem):
def getRHS(self, freq):
"""
.. math ::
\\mathbf{RHS} = \\mathbf{s_m} + \\mathbf{M^e_{\\sigma}}^{-1}\\mathbf{s_e}
\mathbf{RHS} = \mathbf{s_m} + \mathbf{M^e_{\sigma}}^{-1}\mathbf{s_e}
:param float freq: Frequency
:rtype: numpy.ndarray (nE, nSrc)
@@ -419,16 +424,17 @@ class ProblemFDEM_b(BaseFDEMProblem):
class ProblemFDEM_j(BaseFDEMProblem):
"""
We eliminate \\(\\mathbf{h}\\) using
We eliminate \\\(\\\mathbf{h}\\\) using
.. math ::
\\mathbf{h} = \\frac{1}{i \\omega} \\mathbf{M^e_{\\mu}}^{-1} \\left(- \\mathbf{C}^T \\mathbf{M^f_{\\rho}} \\mathbf{j} + \\mathbf{s_m} \\right)
\mathbf{h} = \\frac{1}{i \omega} \mathbf{M_{\mu}^e}^{-1} \\left(-\mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{j} + \mathbf{s_m} \\right)
and solve for \\(\\mathbf{j}\\) using
and solve for \\\(\\\mathbf{j}\\\) using
.. math ::
\\left(\\mathbf{C} \\mathbf{M^e_{\\mu}}^{-1} \\mathbf{C}^T \\mathbf{M^f_{\\rho}} + i \\omega\\right))\\mathbf{j} = \\mathbf{C} \\mathbf{M^e_{\\mu}}^{-1}\\mathbf{s_m} -i\\omega\\mathbf{s_e}
.. math ::
\\left(\mathbf{C} \mathbf{M_{\mu}^e}^{-1} \mathbf{C}^T \mathbf{M_{\\rho}^f} + i \omega\\right)\mathbf{j} = \mathbf{C} \mathbf{M_{\mu}^e}^{-1}\mathbf{s_m} -i\omega\mathbf{s_e}
.. note::
This implementation does not yet work with full anisotropy!!
@@ -466,9 +472,11 @@ class ProblemFDEM_j(BaseFDEMProblem):
def getADeriv_m(self, freq, u, v, adjoint=False):
"""
In this case, we assume that electrical conductivity, \(\\sigma\) is the physical property of interest (i.e. \(\sigma\) = model.transform). Then we want
In this case, we assume that electrical conductivity, \\\(\\\sigma\\\) is the physical property of interest (i.e. \\\(\\\sigma\\\) = model.transform). Then we want
.. math ::
\\frac{\mathbf{A(\\sigma)} \mathbf{v}}{d \\mathbf{m}} &= \\mathbf{C} \\mathbf{M^e_{mu^{-1}}} \\mathbf{C^T} \\frac{d \\mathbf{M^f_{\\sigma^{-1}}}}{d \\mathbf{m}}
\\frac{\mathbf{A(\sigma)} \mathbf{v}}{d \\mathbf{m}} &= \\mathbf{C} \\mathbf{M^e_{mu^{-1}}} \\mathbf{C^T} \\frac{d \\mathbf{M^f_{\\sigma^{-1}}}}{d \\mathbf{m}}
&= \\mathbf{C} \\mathbf{M^e_{mu}^{-1}} \\mathbf{C^T} \\frac{d \\mathbf{M^f_{\\sigma^{-1}}}}{d \\mathbf{\\sigma^{-1}}} \\frac{d \\mathbf{\\sigma^{-1}}}{d \\mathbf{\\sigma}} \\frac{d \\mathbf{\\sigma}}{d \\mathbf{m}}
"""
@@ -490,7 +498,8 @@ class ProblemFDEM_j(BaseFDEMProblem):
def getRHS(self, freq):
"""
.. math ::
\\mathbf{RHS} = \\mathbf{C} \\mathbf{M^e_{\\mu}}^{-1}\\mathbf{s_m} -i\\omega\\mathbf{s_e}
\mathbf{RHS} = \mathbf{C} \mathbf{M_{\mu}^e}^{-1}\mathbf{s_m} -i\omega \mathbf{s_e}
:param float freq: Frequency
:rtype: numpy.ndarray (nE, nSrc)
:return: RHS
@@ -549,15 +558,17 @@ class ProblemFDEM_j(BaseFDEMProblem):
class ProblemFDEM_h(BaseFDEMProblem):
"""
We eliminate \\(\\mathbf{j}\\) using
.. math ::
\\mathbf{j} = \\mathbf{C} \\mathbf{h} - \\mathbf{s_e}
and solve for \\(\\mathbf{h}\\) using
We eliminate \\\(\\\mathbf{j}\\\) using
.. math ::
\\left(\\mathbf{C}^T \\mathbf{M^f_{\\rho}} \\mathbf{C} + i \\omega \\mathbf{M^e_{\\mu}}\\right) \\mathbf{h} = \\mathbf{s_m} + \\mathbf{C}^T \\mathbf{M^f_{\\rho}} \\mathbf{s_e}
\mathbf{j} = \mathbf{C} \mathbf{h} - \mathbf{s_e}
and solve for \\\(\\\mathbf{h}\\\) using
.. math ::
\\left(\mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{C} + i \omega \mathbf{M_{\mu}^e}\\right) \mathbf{h} = \mathbf{s_m} + \mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{s_e}
"""
@@ -571,7 +582,9 @@ class ProblemFDEM_h(BaseFDEMProblem):
def getA(self, freq):
"""
.. math ::
\\mathbf{A} = \\mathbf{C}^T \\mathbf{M^f_{\\rho}} \\mathbf{C} + i \\omega \\mathbf{M^e_{\\mu}}
\mathbf{A} = \mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{C} + i \omega \mathbf{M_{\mu}^e}
:param float freq: Frequency
:rtype: scipy.sparse.csr_matrix
:return: A
@@ -596,7 +609,8 @@ class ProblemFDEM_h(BaseFDEMProblem):
def getRHS(self, freq):
"""
.. math ::
\\mathbf{RHS} = \\mathbf{s_m} + \\mathbf{C}^T \\mathbf{M^f_{\\rho}} \\mathbf{s_e}
\mathbf{RHS} = \mathbf{s_m} + \mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{s_e}
:param float freq: Frequency
:rtype: numpy.ndarray (nE, nSrc)