Differential Geometry¶
Introduction¶
Base Class Reference¶

class
sympy.diffgeom.
Manifold
(name, dim, **kwargs)[source]¶ A mathematical manifold.
 Parameters
name : str
The name of the manifold.
dim : int
The dimension of the manifold.
Explanation
A manifold is a topological space that locally resembles Euclidean space near each point [1].
This class does not provide any means to study the topological characteristics of the manifold that it represents, though.
Examples
>>> from sympy.diffgeom import Manifold >>> m = Manifold('M', 2) >>> m M >>> m.dim 2
References

class
sympy.diffgeom.
Patch
(name, manifold, **kwargs)[source]¶ A patch on a manifold.
 Parameters
name : str
The name of the patch.
manifold : Manifold
The manifold on which the patch is defined.
Explanation
Coordinate patch, or patch in short, is a simplyconnected open set around a point in the manifold [1]. On a manifold one can have many patches that do not always include the whole manifold. On these patches coordinate charts can be defined that permit the parameterization of any point on the patch in terms of a tuple of real numbers (the coordinates).
This class does not provide any means to study the topological characteristics of the patch that it represents.
Examples
>>> from sympy.diffgeom import Manifold, Patch >>> m = Manifold('M', 2) >>> p = Patch('P', m) >>> p P >>> p.dim 2
References
 R142
Sussman, J. Wisdom, W. Farr, Functional Differential Geometry (2013)

class
sympy.diffgeom.
CoordSystem
(name, patch, symbols=None, relations={}, **kwargs)[source]¶ A coordinate system defined on the patch.
 Parameters
name : str
The name of the coordinate system.
patch : Patch
The patch where the coordinate system is defined.
symbols : list of Symbols, optional
Defines the names and assumptions of coordinate symbols.
relations : dict, optional
 keytuple of two strings, who are the names of systems where
the coordinates transform from and transform to.
value : Lambda returning the transformed coordinates.
Explanation
Coordinate system is a system that uses one or more coordinates to uniquely determine the position of the points or other geometric elements on a manifold [1].
By passing Symbols to symbols parameter, user can define the name and assumptions of coordinate symbols of the coordinate system. If not passed, these symbols are generated automatically and are assumed to be real valued. By passing relations parameter, user can define the tranform relations of coordinate systems. Inverse transformation and indirect transformation can be found automatically. If this parameter is not passed, coordinate transformation cannot be done.
Examples
>>> from sympy import symbols, pi, Lambda, Matrix, sqrt, atan2, cos, sin >>> from sympy.diffgeom import Manifold, Patch, CoordSystem >>> m = Manifold('M', 2) >>> p = Patch('P', m)
>>> x, y = symbols('x y', real=True) >>> r, theta = symbols('r theta', nonnegative=True) >>> relation_dict = { ... ('Car2D', 'Pol'): Lambda((x, y), Matrix([sqrt(x**2 + y**2), atan2(y, x)])), ... ('Pol', 'Car2D'): Lambda((r, theta), Matrix([r*cos(theta), r*sin(theta)])) ... } >>> Car2D = CoordSystem('Car2D', p, [x, y], relation_dict) >>> Pol = CoordSystem('Pol', p, [r, theta], relation_dict)
>>> Car2D Car2D >>> Car2D.dim 2 >>> Car2D.symbols [x, y] >>> Car2D.transformation(Pol) Lambda((x, y), Matrix([ [sqrt(x**2 + y**2)], [ atan2(y, x)]]))
>>> Car2D.transform(Pol) Matrix([ [sqrt(x**2 + y**2)], [ atan2(y, x)]]) >>> Car2D.transform(Pol, [1, 2]) Matrix([ [sqrt(5)], [atan(2)]])
>>> Pol.jacobian(Car2D) Matrix([ [cos(theta), r*sin(theta)], [sin(theta), r*cos(theta)]]) >>> Pol.jacobian(Car2D, [1, pi/2]) Matrix([ [0, 1], [1, 0]])
References

base_oneform
(coord_index)[source]¶ Return a basis 1form field. The basis oneform field for this coordinate system. It is also an operator on vector fields.

base_oneforms
()[source]¶ Returns a list of all base oneforms. For more details see the
base_oneform
method of this class.

base_scalar
(coord_index)[source]¶ Return
BaseScalarField
that takes a point and returns one of the coordinates.

base_scalars
()[source]¶ Returns a list of all coordinate functions. For more details see the
base_scalar
method of this class.

base_vector
(coord_index)[source]¶ Return a basis vector field. The basis vector field for this coordinate system. It is also an operator on scalar fields.

base_vectors
()[source]¶ Returns a list of all base vectors. For more details see the
base_vector
method of this class.

coord_function
(coord_index)[source]¶ Return
BaseScalarField
that takes a point and returns one of the coordinates.

coord_functions
()[source]¶ Returns a list of all coordinate functions. For more details see the
base_scalar
method of this class.

jacobian_determinant
(sys, coordinates=None)[source]¶ Return the jacobian determinant of a transformation.

class
sympy.diffgeom.
CoordinateSymbol
(coord_sys, index, **assumptions)[source]¶ A symbol which denotes an abstract value of ith coordinate of the coordinate system with given context.
 Parameters
coord_sys : CoordSystem
index : integer
Explanation
Each coordinates in coordinate system are represented by unique symbol, such as x, y, z in Cartesian coordinate system.
You may not construct this class directly. Instead, use \(symbols\) method of CoordSystem.
Examples
>>> from sympy import symbols >>> from sympy.diffgeom import Manifold, Patch, CoordSystem >>> m = Manifold('M', 2) >>> p = Patch('P', m) >>> _x, _y = symbols('x y', nonnegative=True)
>>> C = CoordSystem('C', p, [_x, _y]) >>> x, y = C.symbols
>>> x.name 'x' >>> x.coord_sys == C True >>> x.index 0 >>> x.is_nonnegative True

class
sympy.diffgeom.
Point
(coord_sys, coords, **kwargs)[source]¶ Point defined in a coordinate system.
 Parameters
coord_sys : CoordSystem
coords : list
The coordinates of the point.
Explanation
Mathematically, point is defined in the manifold and does not have any coordinates by itself. Coordinate system is what imbues the coordinates to the point by coordinate chart. However, due to the difficulty of realizing such logic, you must supply a coordinate system and coordinates to define a Point here.
The usage of this object after its definition is independent of the coordinate system that was used in order to define it, however due to limitations in the simplification routines you can arrive at complicated expressions if you use inappropriate coordinate systems.
Examples
>>> from sympy import pi >>> from sympy.diffgeom import Point >>> from sympy.diffgeom.rn import R2, R2_r, R2_p >>> rho, theta = R2_p.symbols
>>> p = Point(R2_p, [rho, 3*pi/4])
>>> p.manifold == R2 True
>>> p.coords() Matrix([ [ rho], [3*pi/4]]) >>> p.coords(R2_r) Matrix([ [sqrt(2)*rho/2], [ sqrt(2)*rho/2]])

class
sympy.diffgeom.
BaseScalarField
(coord_sys, index, **kwargs)[source]¶ Base scalar field over a manifold for a given coordinate system.
 Parameters
coord_sys : CoordSystem
index : integer
Explanation
A scalar field takes a point as an argument and returns a scalar. A base scalar field of a coordinate system takes a point and returns one of the coordinates of that point in the coordinate system in question.
To define a scalar field you need to choose the coordinate system and the index of the coordinate.
The use of the scalar field after its definition is independent of the coordinate system in which it was defined, however due to limitations in the simplification routines you may arrive at more complicated expression if you use unappropriate coordinate systems. You can build complicated scalar fields by just building up SymPy expressions containing
BaseScalarField
instances.Examples
>>> from sympy import Function, pi >>> from sympy.diffgeom import BaseScalarField >>> from sympy.diffgeom.rn import R2_r, R2_p >>> rho, _ = R2_p.symbols >>> point = R2_p.point([rho, 0]) >>> fx, fy = R2_r.base_scalars() >>> ftheta = BaseScalarField(R2_r, 1)
>>> fx(point) rho >>> fy(point) 0
>>> (fx**2+fy**2).rcall(point) rho**2
>>> g = Function('g') >>> fg = g(fthetapi) >>> fg.rcall(point) g(pi)

class
sympy.diffgeom.
BaseVectorField
(coord_sys, index, **kwargs)[source]¶ Base vector field over a manifold for a given coordinate system.
 Parameters
coord_sys : CoordSystem
index : integer
Explanation
A vector field is an operator taking a scalar field and returning a directional derivative (which is also a scalar field). A base vector field is the same type of operator, however the derivation is specifically done with respect to a chosen coordinate.
To define a base vector field you need to choose the coordinate system and the index of the coordinate.
The use of the vector field after its definition is independent of the coordinate system in which it was defined, however due to limitations in the simplification routines you may arrive at more complicated expression if you use unappropriate coordinate systems.
Examples
>>> from sympy import Function >>> from sympy.diffgeom.rn import R2_p, R2_r >>> from sympy.diffgeom import BaseVectorField >>> from sympy import pprint
>>> x, y = R2_r.symbols >>> rho, theta = R2_p.symbols >>> fx, fy = R2_r.base_scalars() >>> point_p = R2_p.point([rho, theta]) >>> point_r = R2_r.point([x, y])
>>> g = Function('g') >>> s_field = g(fx, fy)
>>> v = BaseVectorField(R2_r, 1) >>> pprint(v(s_field)) / d \ (g(x, xi)) \dxi /xi=y >>> pprint(v(s_field).rcall(point_r).doit()) d (g(x, y)) dy >>> pprint(v(s_field).rcall(point_p)) / d \ (g(rho*cos(theta), xi)) \dxi /xi=rho*sin(theta)

class
sympy.diffgeom.
Commutator
(v1, v2)[source]¶ Commutator of two vector fields.
Explanation
The commutator of two vector fields \(v_1\) and \(v_2\) is defined as the vector field \([v_1, v_2]\) that evaluated on each scalar field \(f\) is equal to \(v_1(v_2(f))  v_2(v_1(f))\).
Examples
>>> from sympy.diffgeom.rn import R2_p, R2_r >>> from sympy.diffgeom import Commutator >>> from sympy.simplify import simplify
>>> fx, fy = R2_r.base_scalars() >>> e_x, e_y = R2_r.base_vectors() >>> e_r = R2_p.base_vector(0)
>>> c_xy = Commutator(e_x, e_y) >>> c_xr = Commutator(e_x, e_r) >>> c_xy 0
Unfortunately, the current code is not able to compute everything:
>>> c_xr Commutator(e_x, e_rho) >>> simplify(c_xr(fy**2)) 2*cos(theta)*y**2/(x**2 + y**2)

class
sympy.diffgeom.
Differential
(form_field)[source]¶ Return the differential (exterior derivative) of a form field.
Explanation
The differential of a form (i.e. the exterior derivative) has a complicated definition in the general case. The differential \(df\) of the 0form \(f\) is defined for any vector field \(v\) as \(df(v) = v(f)\).
Examples
>>> from sympy import Function >>> from sympy.diffgeom.rn import R2_r >>> from sympy.diffgeom import Differential >>> from sympy import pprint
>>> fx, fy = R2_r.base_scalars() >>> e_x, e_y = R2_r.base_vectors() >>> g = Function('g') >>> s_field = g(fx, fy) >>> dg = Differential(s_field)
>>> dg d(g(x, y)) >>> pprint(dg(e_x)) / d \ (g(xi, y)) \dxi /xi=x >>> pprint(dg(e_y)) / d \ (g(x, xi)) \dxi /xi=y
Applying the exterior derivative operator twice always results in:
>>> Differential(dg) 0

class
sympy.diffgeom.
TensorProduct
(*args)[source]¶ Tensor product of forms.
Explanation
The tensor product permits the creation of multilinear functionals (i.e. higher order tensors) out of lower order fields (e.g. 1forms and vector fields). However, the higher tensors thus created lack the interesting features provided by the other type of product, the wedge product, namely they are not antisymmetric and hence are not form fields.
Examples
>>> from sympy.diffgeom.rn import R2_r >>> from sympy.diffgeom import TensorProduct
>>> fx, fy = R2_r.base_scalars() >>> e_x, e_y = R2_r.base_vectors() >>> dx, dy = R2_r.base_oneforms()
>>> TensorProduct(dx, dy)(e_x, e_y) 1 >>> TensorProduct(dx, dy)(e_y, e_x) 0 >>> TensorProduct(dx, fx*dy)(fx*e_x, e_y) x**2 >>> TensorProduct(e_x, e_y)(fx**2, fy**2) 4*x*y >>> TensorProduct(e_y, dx)(fy) dx
You can nest tensor products.
>>> tp1 = TensorProduct(dx, dy) >>> TensorProduct(tp1, dx)(e_x, e_y, e_x) 1
You can make partial contraction for instance when ‘raising an index’. Putting
None
in the second argument ofrcall
means that the respective position in the tensor product is left as it is.>>> TP = TensorProduct >>> metric = TP(dx, dx) + 3*TP(dy, dy) >>> metric.rcall(e_y, None) 3*dy
Or automatically pad the args with
None
without specifying them.>>> metric.rcall(e_y) 3*dy

class
sympy.diffgeom.
WedgeProduct
(*args)[source]¶ Wedge product of forms.
Explanation
In the context of integration only completely antisymmetric forms make sense. The wedge product permits the creation of such forms.
Examples
>>> from sympy.diffgeom.rn import R2_r >>> from sympy.diffgeom import WedgeProduct
>>> fx, fy = R2_r.base_scalars() >>> e_x, e_y = R2_r.base_vectors() >>> dx, dy = R2_r.base_oneforms()
>>> WedgeProduct(dx, dy)(e_x, e_y) 1 >>> WedgeProduct(dx, dy)(e_y, e_x) 1 >>> WedgeProduct(dx, fx*dy)(fx*e_x, e_y) x**2 >>> WedgeProduct(e_x, e_y)(fy, None) e_x
You can nest wedge products.
>>> wp1 = WedgeProduct(dx, dy) >>> WedgeProduct(wp1, dx)(e_x, e_y, e_x) 0

class
sympy.diffgeom.
LieDerivative
(v_field, expr)[source]¶ Lie derivative with respect to a vector field.
Explanation
The transport operator that defines the Lie derivative is the pushforward of the field to be derived along the integral curve of the field with respect to which one derives.
Examples
>>> from sympy.diffgeom.rn import R2_r, R2_p >>> from sympy.diffgeom import (LieDerivative, TensorProduct)
>>> fx, fy = R2_r.base_scalars() >>> e_x, e_y = R2_r.base_vectors() >>> e_rho, e_theta = R2_p.base_vectors() >>> dx, dy = R2_r.base_oneforms()
>>> LieDerivative(e_x, fy) 0 >>> LieDerivative(e_x, fx) 1 >>> LieDerivative(e_x, e_x) 0
The Lie derivative of a tensor field by another tensor field is equal to their commutator:
>>> LieDerivative(e_x, e_rho) Commutator(e_x, e_rho) >>> LieDerivative(e_x + e_y, fx) 1
>>> tp = TensorProduct(dx, dy) >>> LieDerivative(e_x, tp) LieDerivative(e_x, TensorProduct(dx, dy)) >>> LieDerivative(e_x, tp) LieDerivative(e_x, TensorProduct(dx, dy))

class
sympy.diffgeom.
BaseCovarDerivativeOp
(coord_sys, index, christoffel)[source]¶ Covariant derivative operator with respect to a base vector.
Examples
>>> from sympy.diffgeom.rn import R2_r >>> from sympy.diffgeom import BaseCovarDerivativeOp >>> from sympy.diffgeom import metric_to_Christoffel_2nd, TensorProduct
>>> TP = TensorProduct >>> fx, fy = R2_r.base_scalars() >>> e_x, e_y = R2_r.base_vectors() >>> dx, dy = R2_r.base_oneforms()
>>> ch = metric_to_Christoffel_2nd(TP(dx, dx) + TP(dy, dy)) >>> ch [[[0, 0], [0, 0]], [[0, 0], [0, 0]]] >>> cvd = BaseCovarDerivativeOp(R2_r, 0, ch) >>> cvd(fx) 1 >>> cvd(fx*e_x) e_x

class
sympy.diffgeom.
CovarDerivativeOp
(wrt, christoffel)[source]¶ Covariant derivative operator.
Examples
>>> from sympy.diffgeom.rn import R2_r >>> from sympy.diffgeom import CovarDerivativeOp >>> from sympy.diffgeom import metric_to_Christoffel_2nd, TensorProduct >>> TP = TensorProduct >>> fx, fy = R2_r.base_scalars() >>> e_x, e_y = R2_r.base_vectors() >>> dx, dy = R2_r.base_oneforms() >>> ch = metric_to_Christoffel_2nd(TP(dx, dx) + TP(dy, dy))
>>> ch [[[0, 0], [0, 0]], [[0, 0], [0, 0]]] >>> cvd = CovarDerivativeOp(fx*e_x, ch) >>> cvd(fx) x >>> cvd(fx*e_x) x*e_x

sympy.diffgeom.
intcurve_series
(vector_field, param, start_point, n=6, coord_sys=None, coeffs=False)[source]¶ Return the series expansion for an integral curve of the field.
 Parameters
vector_field
the vector field for which an integral curve will be given
param
the argument of the function \(\gamma\) from R to the curve
start_point
the point which corresponds to \(\gamma(0)\)
n
the order to which to expand
coord_sys
the coordinate system in which to expand coeffs (default False)  if True return a list of elements of the expansion
Explanation
Integral curve is a function \(\gamma\) taking a parameter in \(R\) to a point in the manifold. It verifies the equation:
\(V(f)\big(\gamma(t)\big) = \frac{d}{dt}f\big(\gamma(t)\big)\)
where the given
vector_field
is denoted as \(V\). This holds for any value \(t\) for the parameter and any scalar field \(f\).This equation can also be decomposed of a basis of coordinate functions \(V(f_i)\big(\gamma(t)\big) = \frac{d}{dt}f_i\big(\gamma(t)\big) \quad \forall i\)
This function returns a series expansion of \(\gamma(t)\) in terms of the coordinate system
coord_sys
. The equations and expansions are necessarily done in coordinatesystemdependent way as there is no other way to represent movement between points on the manifold (i.e. there is no such thing as a difference of points for a general manifold).Examples
Use the predefined R2 manifold:
>>> from sympy.abc import t, x, y >>> from sympy.diffgeom.rn import R2_p, R2_r >>> from sympy.diffgeom import intcurve_series
Specify a starting point and a vector field:
>>> start_point = R2_r.point([x, y]) >>> vector_field = R2_r.e_x
Calculate the series:
>>> intcurve_series(vector_field, t, start_point, n=3) Matrix([ [t + x], [ y]])
Or get the elements of the expansion in a list:
>>> series = intcurve_series(vector_field, t, start_point, n=3, coeffs=True) >>> series[0] Matrix([ [x], [y]]) >>> series[1] Matrix([ [t], [0]]) >>> series[2] Matrix([ [0], [0]])
The series in the polar coordinate system:
>>> series = intcurve_series(vector_field, t, start_point, ... n=3, coord_sys=R2_p, coeffs=True) >>> series[0] Matrix([ [sqrt(x**2 + y**2)], [ atan2(y, x)]]) >>> series[1] Matrix([ [t*x/sqrt(x**2 + y**2)], [ t*y/(x**2 + y**2)]]) >>> series[2] Matrix([ [t**2*(x**2/(x**2 + y**2)**(3/2) + 1/sqrt(x**2 + y**2))/2], [ t**2*x*y/(x**2 + y**2)**2]])
See also

sympy.diffgeom.
intcurve_diffequ
(vector_field, param, start_point, coord_sys=None)[source]¶ Return the differential equation for an integral curve of the field.
 Parameters
vector_field
the vector field for which an integral curve will be given
param
the argument of the function \(\gamma\) from R to the curve
start_point
the point which corresponds to \(\gamma(0)\)
coord_sys
the coordinate system in which to give the equations
 Returns
a tuple of (equations, initial conditions)
Explanation
Integral curve is a function \(\gamma\) taking a parameter in \(R\) to a point in the manifold. It verifies the equation:
\(V(f)\big(\gamma(t)\big) = \frac{d}{dt}f\big(\gamma(t)\big)\)
where the given
vector_field
is denoted as \(V\). This holds for any value \(t\) for the parameter and any scalar field \(f\).This function returns the differential equation of \(\gamma(t)\) in terms of the coordinate system
coord_sys
. The equations and expansions are necessarily done in coordinatesystemdependent way as there is no other way to represent movement between points on the manifold (i.e. there is no such thing as a difference of points for a general manifold).Examples
Use the predefined R2 manifold:
>>> from sympy.abc import t >>> from sympy.diffgeom.rn import R2, R2_p, R2_r >>> from sympy.diffgeom import intcurve_diffequ
Specify a starting point and a vector field:
>>> start_point = R2_r.point([0, 1]) >>> vector_field = R2.y*R2.e_x + R2.x*R2.e_y
Get the equation:
>>> equations, init_cond = intcurve_diffequ(vector_field, t, start_point) >>> equations [f_1(t) + Derivative(f_0(t), t), f_0(t) + Derivative(f_1(t), t)] >>> init_cond [f_0(0), f_1(0)  1]
The series in the polar coordinate system:
>>> equations, init_cond = intcurve_diffequ(vector_field, t, start_point, R2_p) >>> equations [Derivative(f_0(t), t), Derivative(f_1(t), t)  1] >>> init_cond [f_0(0)  1, f_1(0)  pi/2]
See also

sympy.diffgeom.
vectors_in_basis
(expr, to_sys)[source]¶ Transform all base vectors in base vectors of a specified coord basis. While the new base vectors are in the new coordinate system basis, any coefficients are kept in the old system.
Examples
>>> from sympy.diffgeom import vectors_in_basis >>> from sympy.diffgeom.rn import R2_r, R2_p
>>> vectors_in_basis(R2_r.e_x, R2_p) y*e_theta/(x**2 + y**2) + x*e_rho/sqrt(x**2 + y**2) >>> vectors_in_basis(R2_p.e_r, R2_r) sin(theta)*e_y + cos(theta)*e_x

sympy.diffgeom.
twoform_to_matrix
(expr)[source]¶ Return the matrix representing the twoform.
For the twoform \(w\) return the matrix \(M\) such that \(M[i,j]=w(e_i, e_j)\), where \(e_i\) is the ith base vector field for the coordinate system in which the expression of \(w\) is given.
Examples
>>> from sympy.diffgeom.rn import R2 >>> from sympy.diffgeom import twoform_to_matrix, TensorProduct >>> TP = TensorProduct
>>> twoform_to_matrix(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy)) Matrix([ [1, 0], [0, 1]]) >>> twoform_to_matrix(R2.x*TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy)) Matrix([ [x, 0], [0, 1]]) >>> twoform_to_matrix(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy)  TP(R2.dx, R2.dy)/2) Matrix([ [ 1, 0], [1/2, 1]])

sympy.diffgeom.
metric_to_Christoffel_1st
(expr)[source]¶ Return the nested list of Christoffel symbols for the given metric. This returns the Christoffel symbol of first kind that represents the LeviCivita connection for the given metric.
Examples
>>> from sympy.diffgeom.rn import R2 >>> from sympy.diffgeom import metric_to_Christoffel_1st, TensorProduct >>> TP = TensorProduct
>>> metric_to_Christoffel_1st(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy)) [[[0, 0], [0, 0]], [[0, 0], [0, 0]]] >>> metric_to_Christoffel_1st(R2.x*TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy)) [[[1/2, 0], [0, 0]], [[0, 0], [0, 0]]]

sympy.diffgeom.
metric_to_Christoffel_2nd
(expr)[source]¶ Return the nested list of Christoffel symbols for the given metric. This returns the Christoffel symbol of second kind that represents the LeviCivita connection for the given metric.
Examples
>>> from sympy.diffgeom.rn import R2 >>> from sympy.diffgeom import metric_to_Christoffel_2nd, TensorProduct >>> TP = TensorProduct
>>> metric_to_Christoffel_2nd(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy)) [[[0, 0], [0, 0]], [[0, 0], [0, 0]]] >>> metric_to_Christoffel_2nd(R2.x*TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy)) [[[1/(2*x), 0], [0, 0]], [[0, 0], [0, 0]]]

sympy.diffgeom.
metric_to_Riemann_components
(expr)[source]¶ Return the components of the Riemann tensor expressed in a given basis.
Given a metric it calculates the components of the Riemann tensor in the canonical basis of the coordinate system in which the metric expression is given.
Examples
>>> from sympy import exp >>> from sympy.diffgeom.rn import R2 >>> from sympy.diffgeom import metric_to_Riemann_components, TensorProduct >>> TP = TensorProduct
>>> metric_to_Riemann_components(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy)) [[[[0, 0], [0, 0]], [[0, 0], [0, 0]]], [[[0, 0], [0, 0]], [[0, 0], [0, 0]]]] >>> non_trivial_metric = exp(2*R2.r)*TP(R2.dr, R2.dr) + R2.r**2*TP(R2.dtheta, R2.dtheta) >>> non_trivial_metric exp(2*rho)*TensorProduct(drho, drho) + rho**2*TensorProduct(dtheta, dtheta) >>> riemann = metric_to_Riemann_components(non_trivial_metric) >>> riemann[0, :, :, :] [[[0, 0], [0, 0]], [[0, exp(2*rho)*rho], [exp(2*rho)*rho, 0]]] >>> riemann[1, :, :, :] [[[0, 1/rho], [1/rho, 0]], [[0, 0], [0, 0]]]

sympy.diffgeom.
metric_to_Ricci_components
(expr)[source]¶ Return the components of the Ricci tensor expressed in a given basis.
Given a metric it calculates the components of the Ricci tensor in the canonical basis of the coordinate system in which the metric expression is given.
Examples
>>> from sympy import exp >>> from sympy.diffgeom.rn import R2 >>> from sympy.diffgeom import metric_to_Ricci_components, TensorProduct >>> TP = TensorProduct
>>> metric_to_Ricci_components(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy)) [[0, 0], [0, 0]] >>> non_trivial_metric = exp(2*R2.r)*TP(R2.dr, R2.dr) + R2.r**2*TP(R2.dtheta, R2.dtheta) >>> non_trivial_metric exp(2*rho)*TensorProduct(drho, drho) + rho**2*TensorProduct(dtheta, dtheta) >>> metric_to_Ricci_components(non_trivial_metric) [[1/rho, 0], [0, exp(2*rho)*rho]]