def vector_integrate(field, *region):
"""
Compute the integral of a vector/scalar field
over a a region or a set of parameters.
Examples
========
>>> from sympy.vector import CoordSys3D, ParametricRegion, vector_integrate
>>> from sympy.abc import t
>>> C = CoordSys3D('C')
>>> region = ParametricRegion((t, t**2), (t, 1, 5))
>>> vector_integrate(C.x*C.i, region)
12
Integrals over special regions can also be calculated using geometry module.
>>> from sympy.geometry import Point, Circle, Triangle
>>> c = Circle(Point(0, 2), 5)
>>> vector_integrate(C.x**2 + C.y**2, c)
290*pi
>>> triangle = Triangle(Point(-2, 3), Point(2, 3), Point(0, 5))
>>> vector_integrate(3*C.x**2*C.y*C.i + C.j, triangle)
-8
Integrals over some simple implicit regions can be computed. But in most cases,
it takes too long to compute over them.
>>> from sympy.abc import x, y
>>> from sympy.vector import ImplicitRegion
>>> c2 = ImplicitRegion((x, y), (x - 2)**2 + (y - 1)**2 - 9)
>>> vector_integrate(1, c2)
12*pi
>>> vector_integrate(12*C.y**3, (C.y, 1, 3))
240
>>> vector_integrate(C.x**2*C.z, C.x)
C.x**3*C.z/3
"""
if len(region) == 1:
if isinstance(region[0], ParametricRegion):
return ParametricIntegral(field, region[0])
if isinstance(region[0], ImplicitRegion):
region = parametric_region_list(region[0])[0]
return vector_integrate(field, region)
if isinstance(region[0], GeometryEntity):
regions_list = parametric_region_list(region[0])
result = 0
for reg in regions_list:
result += vector_integrate(field, reg)
return result
return integrate(field, *region)
def vector_integrate(field, *region):
"""
Compute the integral of a vector/scalar field
over a a region or a set of parameters.
Examples:
=========
>>> from sympy.vector import CoordSys3D, ParametricRegion, vector_integrate
>>> from sympy.abc import t
>>> C = CoordSys3D('C')
>>> region = ParametricRegion((t, t**2), (t, 1, 5))
>>> vector_integrate(C.x*C.i, region)
12
Integrals over special regions can also be calculated using geometry module.
>>> from sympy.geometry import Point, Circle, Triangle
>>> c = Circle(Point(0, 2), 5)
>>> vector_integrate(C.x**2 + C.y**2, c)
290*pi
>>> triangle = Triangle(Point(-2, 3), Point(2, 3), Point(0, 5))
>>> vector_integrate(3*C.x**2*C.y*C.i + C.j, triangle)
-8
>>> vector_integrate(12*C.y**3, (C.y, 1, 3))
240
>>> vector_integrate(C.x**2*C.z, C.x)
C.x**3*C.z/3
"""
if len(region) == 1:
if isinstance(region[0], ParametricRegion):
return ParametricIntegral(field, region[0])
if isinstance(region[0], GeometryEntity):
regions_list = parametric_region_list(region[0])
result = 0
for reg in regions_list:
result += vector_integrate(field, reg)
return result
return integrate(field, *region)
def vector_integrate(field, *region):
"""
Compute the integral of a vector/scalar field
over a a region or a set of parameters.
Examples
========
>>> from sympy.vector import CoordSys3D, ParametricRegion, vector_integrate
>>> from sympy.abc import x, y, t
>>> C = CoordSys3D('C')
>>> region = ParametricRegion((t, t**2), (t, 1, 5))
>>> vector_integrate(C.x*C.i, region)
12
Integrals over some objects of geometry module can also be calculated.
>>> from sympy.geometry import Point, Circle, Triangle
>>> c = Circle(Point(0, 2), 5)
>>> vector_integrate(C.x**2 + C.y**2, c)
290*pi
>>> triangle = Triangle(Point(-2, 3), Point(2, 3), Point(0, 5))
>>> vector_integrate(3*C.x**2*C.y*C.i + C.j, triangle)
-8
Integrals over some simple implicit regions can be computed. But in most cases,
it takes too long to compute over them. This is due to the expressions of parametric
representation becoming large.
>>> from sympy.vector import ImplicitRegion
>>> c2 = ImplicitRegion((x, y), (x - 2)**2 + (y - 1)**2 - 9)
>>> vector_integrate(1, c2)
6*pi
Integral of fields with respect to base scalars:
>>> vector_integrate(12*C.y**3, (C.y, 1, 3))
240
>>> vector_integrate(C.x**2*C.z, C.x)
C.x**3*C.z/3
>>> vector_integrate(C.x*C.i - C.y*C.k, C.x)
(Integral(C.x, C.x))*C.i + (Integral(-C.y, C.x))*C.k
>>> _.doit()
C.x**2/2*C.i + (-C.x*C.y)*C.k
"""
if len(region) == 1:
if isinstance(region[0], ParametricRegion):
return ParametricIntegral(field, region[0])
if isinstance(region[0], ImplicitRegion):
region = parametric_region_list(region[0])[0]
return vector_integrate(field, region)
if isinstance(region[0], GeometryEntity):
regions_list = parametric_region_list(region[0])
result = 0
for reg in regions_list:
result += vector_integrate(field, reg)
return result
return integrate(field, *region)