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518 lines
15 KiB
518 lines
15 KiB
from sympy.vector.coordsysrect import CoordSys3D
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from sympy.vector.deloperator import Del
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from sympy.vector.scalar import BaseScalar
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from sympy.vector.vector import Vector, BaseVector
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from sympy.vector.operators import gradient, curl, divergence
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from sympy.core.function import diff
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from sympy.core.singleton import S
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from sympy.integrals.integrals import integrate
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from sympy.simplify.simplify import simplify
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from sympy.core import sympify
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from sympy.vector.dyadic import Dyadic
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def express(expr, system, system2=None, variables=False):
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"""
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Global function for 'express' functionality.
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Re-expresses a Vector, Dyadic or scalar(sympyfiable) in the given
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coordinate system.
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If 'variables' is True, then the coordinate variables (base scalars)
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of other coordinate systems present in the vector/scalar field or
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dyadic are also substituted in terms of the base scalars of the
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given system.
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Parameters
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==========
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expr : Vector/Dyadic/scalar(sympyfiable)
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The expression to re-express in CoordSys3D 'system'
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system: CoordSys3D
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The coordinate system the expr is to be expressed in
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system2: CoordSys3D
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The other coordinate system required for re-expression
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(only for a Dyadic Expr)
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variables : boolean
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Specifies whether to substitute the coordinate variables present
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in expr, in terms of those of parameter system
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Examples
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========
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>>> from sympy.vector import CoordSys3D
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>>> from sympy import Symbol, cos, sin
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>>> N = CoordSys3D('N')
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>>> q = Symbol('q')
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>>> B = N.orient_new_axis('B', q, N.k)
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>>> from sympy.vector import express
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>>> express(B.i, N)
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(cos(q))*N.i + (sin(q))*N.j
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>>> express(N.x, B, variables=True)
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B.x*cos(q) - B.y*sin(q)
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>>> d = N.i.outer(N.i)
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>>> express(d, B, N) == (cos(q))*(B.i|N.i) + (-sin(q))*(B.j|N.i)
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True
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"""
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if expr in (0, Vector.zero):
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return expr
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if not isinstance(system, CoordSys3D):
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raise TypeError("system should be a CoordSys3D \
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instance")
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if isinstance(expr, Vector):
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if system2 is not None:
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raise ValueError("system2 should not be provided for \
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Vectors")
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# Given expr is a Vector
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if variables:
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# If variables attribute is True, substitute
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# the coordinate variables in the Vector
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system_list = {x.system for x in expr.atoms(BaseScalar, BaseVector)} - {system}
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subs_dict = {}
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for f in system_list:
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subs_dict.update(f.scalar_map(system))
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expr = expr.subs(subs_dict)
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# Re-express in this coordinate system
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outvec = Vector.zero
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parts = expr.separate()
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for x in parts:
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if x != system:
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temp = system.rotation_matrix(x) * parts[x].to_matrix(x)
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outvec += matrix_to_vector(temp, system)
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else:
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outvec += parts[x]
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return outvec
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elif isinstance(expr, Dyadic):
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if system2 is None:
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system2 = system
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if not isinstance(system2, CoordSys3D):
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raise TypeError("system2 should be a CoordSys3D \
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instance")
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outdyad = Dyadic.zero
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var = variables
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for k, v in expr.components.items():
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outdyad += (express(v, system, variables=var) *
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(express(k.args[0], system, variables=var) |
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express(k.args[1], system2, variables=var)))
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return outdyad
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else:
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if system2 is not None:
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raise ValueError("system2 should not be provided for \
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Vectors")
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if variables:
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# Given expr is a scalar field
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system_set = set()
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expr = sympify(expr)
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# Substitute all the coordinate variables
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for x in expr.atoms(BaseScalar):
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if x.system != system:
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system_set.add(x.system)
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subs_dict = {}
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for f in system_set:
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subs_dict.update(f.scalar_map(system))
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return expr.subs(subs_dict)
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return expr
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def directional_derivative(field, direction_vector):
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"""
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Returns the directional derivative of a scalar or vector field computed
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along a given vector in coordinate system which parameters are expressed.
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Parameters
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==========
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field : Vector or Scalar
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The scalar or vector field to compute the directional derivative of
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direction_vector : Vector
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The vector to calculated directional derivative along them.
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Examples
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========
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>>> from sympy.vector import CoordSys3D, directional_derivative
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>>> R = CoordSys3D('R')
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>>> f1 = R.x*R.y*R.z
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>>> v1 = 3*R.i + 4*R.j + R.k
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>>> directional_derivative(f1, v1)
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R.x*R.y + 4*R.x*R.z + 3*R.y*R.z
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>>> f2 = 5*R.x**2*R.z
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>>> directional_derivative(f2, v1)
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5*R.x**2 + 30*R.x*R.z
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"""
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from sympy.vector.operators import _get_coord_systems
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coord_sys = _get_coord_systems(field)
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if len(coord_sys) > 0:
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# TODO: This gets a random coordinate system in case of multiple ones:
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coord_sys = next(iter(coord_sys))
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field = express(field, coord_sys, variables=True)
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i, j, k = coord_sys.base_vectors()
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x, y, z = coord_sys.base_scalars()
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out = Vector.dot(direction_vector, i) * diff(field, x)
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out += Vector.dot(direction_vector, j) * diff(field, y)
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out += Vector.dot(direction_vector, k) * diff(field, z)
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if out == 0 and isinstance(field, Vector):
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out = Vector.zero
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return out
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elif isinstance(field, Vector):
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return Vector.zero
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else:
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return S.Zero
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def laplacian(expr):
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"""
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Return the laplacian of the given field computed in terms of
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the base scalars of the given coordinate system.
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Parameters
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==========
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expr : SymPy Expr or Vector
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expr denotes a scalar or vector field.
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Examples
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========
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>>> from sympy.vector import CoordSys3D, laplacian
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>>> R = CoordSys3D('R')
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>>> f = R.x**2*R.y**5*R.z
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>>> laplacian(f)
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20*R.x**2*R.y**3*R.z + 2*R.y**5*R.z
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>>> f = R.x**2*R.i + R.y**3*R.j + R.z**4*R.k
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>>> laplacian(f)
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2*R.i + 6*R.y*R.j + 12*R.z**2*R.k
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"""
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delop = Del()
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if expr.is_Vector:
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return (gradient(divergence(expr)) - curl(curl(expr))).doit()
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return delop.dot(delop(expr)).doit()
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def is_conservative(field):
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"""
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Checks if a field is conservative.
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Parameters
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==========
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field : Vector
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The field to check for conservative property
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Examples
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========
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>>> from sympy.vector import CoordSys3D
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>>> from sympy.vector import is_conservative
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>>> R = CoordSys3D('R')
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>>> is_conservative(R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k)
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True
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>>> is_conservative(R.z*R.j)
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False
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"""
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# Field is conservative irrespective of system
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# Take the first coordinate system in the result of the
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# separate method of Vector
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if not isinstance(field, Vector):
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raise TypeError("field should be a Vector")
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if field == Vector.zero:
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return True
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return curl(field).simplify() == Vector.zero
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def is_solenoidal(field):
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"""
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Checks if a field is solenoidal.
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Parameters
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==========
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field : Vector
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The field to check for solenoidal property
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Examples
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========
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>>> from sympy.vector import CoordSys3D
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>>> from sympy.vector import is_solenoidal
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>>> R = CoordSys3D('R')
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>>> is_solenoidal(R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k)
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True
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>>> is_solenoidal(R.y * R.j)
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False
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"""
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# Field is solenoidal irrespective of system
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# Take the first coordinate system in the result of the
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# separate method in Vector
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if not isinstance(field, Vector):
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raise TypeError("field should be a Vector")
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if field == Vector.zero:
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return True
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return divergence(field).simplify() is S.Zero
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def scalar_potential(field, coord_sys):
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"""
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Returns the scalar potential function of a field in a given
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coordinate system (without the added integration constant).
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Parameters
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==========
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field : Vector
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The vector field whose scalar potential function is to be
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calculated
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coord_sys : CoordSys3D
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The coordinate system to do the calculation in
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Examples
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========
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>>> from sympy.vector import CoordSys3D
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>>> from sympy.vector import scalar_potential, gradient
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>>> R = CoordSys3D('R')
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>>> scalar_potential(R.k, R) == R.z
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True
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>>> scalar_field = 2*R.x**2*R.y*R.z
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>>> grad_field = gradient(scalar_field)
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>>> scalar_potential(grad_field, R)
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2*R.x**2*R.y*R.z
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"""
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# Check whether field is conservative
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if not is_conservative(field):
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raise ValueError("Field is not conservative")
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if field == Vector.zero:
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return S.Zero
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# Express the field exntirely in coord_sys
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# Substitute coordinate variables also
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if not isinstance(coord_sys, CoordSys3D):
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raise TypeError("coord_sys must be a CoordSys3D")
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field = express(field, coord_sys, variables=True)
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dimensions = coord_sys.base_vectors()
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scalars = coord_sys.base_scalars()
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# Calculate scalar potential function
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temp_function = integrate(field.dot(dimensions[0]), scalars[0])
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for i, dim in enumerate(dimensions[1:]):
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partial_diff = diff(temp_function, scalars[i + 1])
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partial_diff = field.dot(dim) - partial_diff
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temp_function += integrate(partial_diff, scalars[i + 1])
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return temp_function
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def scalar_potential_difference(field, coord_sys, point1, point2):
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"""
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Returns the scalar potential difference between two points in a
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certain coordinate system, wrt a given field.
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If a scalar field is provided, its values at the two points are
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considered. If a conservative vector field is provided, the values
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of its scalar potential function at the two points are used.
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Returns (potential at point2) - (potential at point1)
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The position vectors of the two Points are calculated wrt the
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origin of the coordinate system provided.
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Parameters
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==========
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field : Vector/Expr
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The field to calculate wrt
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coord_sys : CoordSys3D
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The coordinate system to do the calculations in
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point1 : Point
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The initial Point in given coordinate system
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position2 : Point
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The second Point in the given coordinate system
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Examples
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========
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>>> from sympy.vector import CoordSys3D
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>>> from sympy.vector import scalar_potential_difference
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>>> R = CoordSys3D('R')
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>>> P = R.origin.locate_new('P', R.x*R.i + R.y*R.j + R.z*R.k)
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>>> vectfield = 4*R.x*R.y*R.i + 2*R.x**2*R.j
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>>> scalar_potential_difference(vectfield, R, R.origin, P)
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2*R.x**2*R.y
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>>> Q = R.origin.locate_new('O', 3*R.i + R.j + 2*R.k)
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>>> scalar_potential_difference(vectfield, R, P, Q)
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-2*R.x**2*R.y + 18
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"""
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if not isinstance(coord_sys, CoordSys3D):
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raise TypeError("coord_sys must be a CoordSys3D")
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if isinstance(field, Vector):
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# Get the scalar potential function
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scalar_fn = scalar_potential(field, coord_sys)
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else:
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# Field is a scalar
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scalar_fn = field
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# Express positions in required coordinate system
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origin = coord_sys.origin
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position1 = express(point1.position_wrt(origin), coord_sys,
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variables=True)
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position2 = express(point2.position_wrt(origin), coord_sys,
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variables=True)
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# Get the two positions as substitution dicts for coordinate variables
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subs_dict1 = {}
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subs_dict2 = {}
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scalars = coord_sys.base_scalars()
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for i, x in enumerate(coord_sys.base_vectors()):
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subs_dict1[scalars[i]] = x.dot(position1)
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subs_dict2[scalars[i]] = x.dot(position2)
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return scalar_fn.subs(subs_dict2) - scalar_fn.subs(subs_dict1)
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def matrix_to_vector(matrix, system):
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"""
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Converts a vector in matrix form to a Vector instance.
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It is assumed that the elements of the Matrix represent the
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measure numbers of the components of the vector along basis
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vectors of 'system'.
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Parameters
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==========
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matrix : SymPy Matrix, Dimensions: (3, 1)
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The matrix to be converted to a vector
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system : CoordSys3D
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The coordinate system the vector is to be defined in
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Examples
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========
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>>> from sympy import ImmutableMatrix as Matrix
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>>> m = Matrix([1, 2, 3])
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>>> from sympy.vector import CoordSys3D, matrix_to_vector
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>>> C = CoordSys3D('C')
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>>> v = matrix_to_vector(m, C)
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>>> v
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C.i + 2*C.j + 3*C.k
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>>> v.to_matrix(C) == m
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True
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"""
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outvec = Vector.zero
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vects = system.base_vectors()
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for i, x in enumerate(matrix):
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outvec += x * vects[i]
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return outvec
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def _path(from_object, to_object):
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"""
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Calculates the 'path' of objects starting from 'from_object'
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to 'to_object', along with the index of the first common
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ancestor in the tree.
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Returns (index, list) tuple.
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"""
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if from_object._root != to_object._root:
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raise ValueError("No connecting path found between " +
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str(from_object) + " and " + str(to_object))
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other_path = []
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obj = to_object
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while obj._parent is not None:
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other_path.append(obj)
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obj = obj._parent
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other_path.append(obj)
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object_set = set(other_path)
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from_path = []
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obj = from_object
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while obj not in object_set:
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from_path.append(obj)
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obj = obj._parent
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index = len(from_path)
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i = other_path.index(obj)
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while i >= 0:
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from_path.append(other_path[i])
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i -= 1
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return index, from_path
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def orthogonalize(*vlist, orthonormal=False):
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"""
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Takes a sequence of independent vectors and orthogonalizes them
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using the Gram - Schmidt process. Returns a list of
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orthogonal or orthonormal vectors.
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Parameters
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==========
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vlist : sequence of independent vectors to be made orthogonal.
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orthonormal : Optional parameter
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Set to True if the vectors returned should be
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orthonormal.
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Default: False
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Examples
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========
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>>> from sympy.vector.coordsysrect import CoordSys3D
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>>> from sympy.vector.functions import orthogonalize
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>>> C = CoordSys3D('C')
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>>> i, j, k = C.base_vectors()
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>>> v1 = i + 2*j
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>>> v2 = 2*i + 3*j
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>>> orthogonalize(v1, v2)
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[C.i + 2*C.j, 2/5*C.i + (-1/5)*C.j]
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Gram-Schmidt_process
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"""
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if not all(isinstance(vec, Vector) for vec in vlist):
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raise TypeError('Each element must be of Type Vector')
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ortho_vlist = []
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for i, term in enumerate(vlist):
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for j in range(i):
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term -= ortho_vlist[j].projection(vlist[i])
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# TODO : The following line introduces a performance issue
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# and needs to be changed once a good solution for issue #10279 is
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# found.
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if simplify(term).equals(Vector.zero):
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raise ValueError("Vector set not linearly independent")
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ortho_vlist.append(term)
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if orthonormal:
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ortho_vlist = [vec.normalize() for vec in ortho_vlist]
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return ortho_vlist
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