ylm::Tags Namespace Reference

Holds tags and ComputeItems associated with a ylm::Strahlkorper. More...

Namespaces
namespace	aliases
	Defines type aliases used in Strahlkorper-related Tags.

Classes
struct	CartesianCoords
	CartesianCoords(i) is $x_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^i$, the vector of $(x,y,z)$ coordinates of each point on the surface. More...
struct	CartesianCoordsCompute
struct	D2xRadius
	D2xRadius(i,j) is ${\partial }^2{r}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}/\partial x^i\partial x^j$. Here $r_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}=r_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}(\theta ,\varphi )$ is the function describing the surface, which is considered a function of Cartesian coordinates $r_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}=r_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}(\theta (x,y,z),\varphi (x,y,z))$ for this operation. More...
struct	D2xRadiusCompute
struct	DxRadius
	DxRadius(i) is $\partial r_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}/\partial x^i$. Here $r_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}=r_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}(\theta ,\varphi )$ is the function describing the surface, which is considered a function of Cartesian coordinates $r_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}=r_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}(\theta (x,y,z),\varphi (x,y,z))$ for this operation. More...
struct	DxRadiusCompute
struct	EuclideanAreaElement
	Computes the Euclidean area element on a Strahlkorper. Useful for flat space integrals. More...
struct	EuclideanAreaElementCompute
struct	EuclideanSurfaceIntegral
	Computes the flat-space integral of a scalar over a Strahlkorper. More...
struct	EuclideanSurfaceIntegralCompute
struct	EuclideanSurfaceIntegralVector
	Computes the Euclidean-space integral of a vector over a Strahlkorper, $\oint V^i{s}_i(s_j{s}_k{\delta }^{jk}{)}^{-1/2}{d}^{2}S$, where $s_i$ is the Strahlkorper surface unit normal and ${\delta }^{ij}$ is the Kronecker delta. Note that $s_i$ is not assumed to be normalized; the denominator of the integrand effectively normalizes it using the Euclidean metric. More...
struct	EuclideanSurfaceIntegralVectorCompute
struct	ExtrinsicCurvature
	Extrinsic curvature of a 2D Strahlkorper embedded in a 3D space. More...
struct	ExtrinsicCurvatureCompute
	Calculates the Extrinsic curvature of a 2D Strahlkorper embedded in a 3D space. More...
struct	GradUnitNormalOneForm
	The 3-covariant gradient $D_i{S}_j$ of a Strahlkorper's normal. More...
struct	GradUnitNormalOneFormCompute
	Computes 3-covariant gradient $D_i{S}_j$ of a Strahlkorper's normal. More...
struct	InvHessian
	InvHessian(k,i,j) is $\partial (J^{-1}){}_j^k/\partial x^i$, where $(J^{-1}){}_j^k$ is the inverse Jacobian. InvHessian is not symmetric because the Jacobians are Pfaffian. InvHessian doesn't depend on the shape of the surface. More...
struct	InvHessianCompute
struct	InvJacobian
	InvJacobian(0,i) is $r\partial \theta /\partial x^i$, and InvJacobian(1,i) is $r\sin \theta \partial \varphi /\partial x^i$. Here $r$ means $\sqrt{x^2+y^2+z^2}$. InvJacobian doesn't depend on the shape of the surface. More...
struct	InvJacobianCompute
struct	Jacobian
	Jacobian(i,0) is $\frac{1}{r}\partial x^i/\partial \theta$, and Jacobian(i,1) is $\frac{1}{r\sin \theta }\partial x^i/\partial \varphi$. Here $r$ means $\sqrt{x^2+y^2+z^2}$. Jacobian doesn't depend on the shape of the surface. More...
struct	JacobianCompute
struct	LaplacianRadius
	${\mathrm{\nabla }}^2{r}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$, the flat Laplacian of the surface. This is ${\eta }^{ij}{\partial }^2{r}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}/\partial x^i\partial x^j$, where $r_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}=r_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}(\theta (x,y,z),\varphi (x,y,z))$. More...
struct	LaplacianRadiusCompute
struct	MaxRicciScalar
	The pointwise maximum of the Strahlkorper's intrinsic Ricci scalar curvature. More...
struct	MaxRicciScalarCompute
	Computes the pointwise maximum of the Strahlkorper's intrinsic Ricci scalar curvature. More...
struct	MinRicciScalar
	The pointwise minimum of the Strahlkorper’s intrinsic Ricci scalar curvature. More...
struct	MinRicciScalarCompute
	Computes the pointwise minimum of the Strahlkorper’s intrinsic Ricci scalar curvature. More...
struct	NormalOneForm
	NormalOneForm(i) is $s_i$, the (unnormalized) normal one-form to the surface, expressed in Cartesian components. This is computed by $x_i/r-\partial r_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}/\partial x^i$, where $x_i/r$ is Rhat and $\partial r_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}/\partial x^i$ is DxRadius. See Eq. (8) of [15]. Note on the word "normal": $s_i$ points in the correct direction (it is "normal" to the surface), but it does not have unit length (it is not "normalized"; normalization requires a metric). More...
struct	NormalOneFormCompute
struct	OneOverOneFormMagnitude
	The OneOverOneFormMagnitude is the reciprocal of the magnitude of the one-form perpendicular to the horizon. More...
struct	OneOverOneFormMagnitudeCompute
	Computes the reciprocal of the magnitude of the one form perpendicular to the horizon. More...
struct	PhysicalCenter
	The geometrical center of the surface. Uses ylm::Strahlkorper::physical_center. More...
struct	PhysicalCenterCompute
struct	PreviousStrahlkorpers
	Tag for holding the previously-found values of a Strahlkorper, which are saved for extrapolation for future initial guesses and for computing the time deriv of a Strahlkorper. More...
struct	Radius
	(Euclidean) distance $r_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}(\theta ,\varphi )$ from the center to each point of the surface. More...
struct	RadiusCompute
struct	Rhat
	Rhat(i) is ${\hat{r}}^i=x_i/\sqrt{x^2+y^2+z^2}$ on the grid. Doesn't depend on the shape of the surface. More...
struct	RhatCompute
struct	RicciScalar
	Ricci scalar is the two-dimensional intrinsic Ricci scalar curvature of a Strahlkorper. More...
struct	RicciScalarCompute
	Computes the two-dimensional intrinsic Ricci scalar of a Strahlkorper. More...
struct	Strahlkorper
	Tag referring to a ylm::Strahlkorper More...
struct	Tangents
	Tangents(i,j) is $\partial x_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^i/\partial q^j$, where $x_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^i$ are the Cartesian coordinates of the surface (i.e. CartesianCoords) and are considered functions of $(\theta ,\varphi )$. More...
struct	TangentsCompute
struct	ThetaPhi
	$(\theta ,\varphi )$ on the grid. Doesn't depend on the shape of the surface. More...
struct	ThetaPhiCompute
struct	TimeDerivStrahlkorper
	Tag to compute the time derivative of the coefficients of a Strahlkorper from a number of previous Strahlkorpers. More...
struct	TimeDerivStrahlkorperCompute
struct	UnitNormalOneForm
	The unit normal one-form $s_j$ to the horizon. More...
struct	UnitNormalOneFormCompute
	Computes the unit one-form perpendicular to the horizon. More...
struct	UnitNormalVector
	UnitNormalVector is defined as $S^i={\gamma }^{ij}{S}_j$, where $S_j$ is the unit normal one form and ${\gamma }^{ij}$ is the inverse spatial metric. More...
struct	UnitNormalVectorCompute
	Computes the UnitNormalVector perpendicular to the horizon. More...

Typedefs
template<typename Frame >
using	items_tags = implementation defined
template<typename Frame >
using	compute_items_tags = implementation defined

Detailed Description

Holds tags and ComputeItems associated with a ylm::Strahlkorper.