def test_parametric_region_list():
point = Point(-5, 12)
assert parametric_region_list(point) == [ParametricRegion((-5, 12))]
e = Ellipse(Point(2, 8), 2, 6)
assert parametric_region_list(e, t) == [ParametricRegion((2*cos(t) + 2, 6*sin(t) + 8), (t, 0, 2*pi))]
c = Curve((t, t**3), (t, 5, 3))
assert parametric_region_list(c) == [ParametricRegion((t, t**3), (t, 5, 3))]
s = Segment(Point(2, 11, -6), Point(0, 2, 5))
assert parametric_region_list(s, t) == [ParametricRegion((2 - 2*t, 11 - 9*t, 11*t - 6), (t, 0, 1))]
s1 = Segment(Point(0, 0), (1, 0))
assert parametric_region_list(s1, t) == [ParametricRegion((t, 0), (t, 0, 1))]
s2 = Segment(Point(1, 2, 3), Point(1, 2, 5))
assert parametric_region_list(s2, t) == [ParametricRegion((1, 2, 2*t + 3), (t, 0, 1))]
s3 = Segment(Point(12, 56), Point(12, 56))
assert parametric_region_list(s3) == [ParametricRegion((12, 56))]
poly = Polygon((1,3), (-3, 8), (2, 4))
assert parametric_region_list(poly, t) == [ParametricRegion((1 - 4*t, 5*t + 3), (t, 0, 1)), ParametricRegion((5*t - 3, 8 - 4*t), (t, 0, 1)), ParametricRegion((2 - t, 4 - t), (t, 0, 1))]
p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7,8)))
raises(ValueError, lambda: parametric_region_list(p1))
def test_parametric_surfaceintegrals():
semisphere = ParametricRegion((2*sin(phi)*cos(theta), 2*sin(phi)*sin(theta), 2*cos(phi)),\
(theta, 0, 2*pi), (phi, 0, pi/2))
assert ParametricIntegral(C.z, semisphere) == 8*pi
cylinder = ParametricRegion((sqrt(3)*cos(theta), sqrt(3)*sin(theta), z), (z, 0, 6), (theta, 0, 2*pi))
assert ParametricIntegral(C.y, cylinder) == 0
cone = ParametricRegion((v*cos(u), v*sin(u), v), (u, 0, 2*pi), (v, 0, 1))
assert ParametricIntegral(C.x*C.i + C.y*C.j + C.z**4*C.k, cone) == pi/3
triangle1 = ParametricRegion((x, y), (x, 0, 2), (y, 0, 10 - 5*x))
triangle2 = ParametricRegion((x, y), (y, 0, 10 - 5*x), (x, 0, 2))
assert ParametricIntegral(-15.6*C.y*C.k, triangle1) == ParametricIntegral(-15.6*C.y*C.k, triangle2)
assert ParametricIntegral(C.z, triangle1) == 10*C.z
def test_volumeintegrals():
cube = ParametricRegion((x, y, z), (x, 0, 1), (y, 0, 1), (z, 0, 1))
assert ParametricIntegral(1, cube) == 1
solidsphere = ParametricRegion((r*sin(phi)*cos(theta), r*sin(phi)*sin(theta), r*cos(phi)),\
(r, 0, 2), (theta, 0, 2*pi), (phi, 0, pi))
assert ParametricIntegral(C.x**2 + C.y**2, solidsphere) == -256 * pi / 15
region_under_plane1 = ParametricRegion((x, y, z), (x, 0, 3), (y, 0, -2*x/3 + 2),\
(z, 0, 6 - 2*x - 3*y))
region_under_plane2 = ParametricRegion((x, y, z), (x, 0, 3), (z, 0, 6 - 2*x - 3*y),\
(y, 0, -2*x/3 + 2))
assert ParametricIntegral(C.x*C.i + C.j - 100*C.k, region_under_plane1) == \
ParametricIntegral(C.x*C.i + C.j - 100*C.k, region_under_plane2)
assert ParametricIntegral(2 * C.x, region_under_plane2) == -9
def test_vector_integrate():
halfdisc = ParametricRegion((r * cos(theta), r * sin(theta)), (r, -2, 2),
(theta, 0, pi))
assert vector_integrate(C.x**2, halfdisc) == 4 * pi
vector_integrate(C.x, ParametricRegion(
(t, t**2), (t, 2, 3))) == -17 * sqrt(17) / 12 + 37 * sqrt(37) / 12
assert vector_integrate(C.y**3 * C.z, (C.x, 0, 3),
(C.y, -1, 4)) == 765 * C.z / 4
s1 = Segment(Point(0, 0), Point(0, 1))
assert vector_integrate(-15 * C.y, s1) == S(-15) / 2
s2 = Segment(Point(4, 3, 9), Point(1, 1, 7))
assert vector_integrate(C.y * C.i, s2) == -6
curve = Curve((sin(t), cos(t)), (t, 0, 2))
assert vector_integrate(5 * C.z, curve) == 10 * C.z
c1 = Circle(Point(2, 3), 6)
assert vector_integrate(C.x * C.y, c1) == 72 * pi
c2 = Circle(Point(0, 0), Point(1, 1), Point(1, 0))
assert vector_integrate(1, c2) == c2.circumference
triangle = Polygon((0, 0), (1, 0), (1, 1))
assert vector_integrate(C.x * C.i - 14 * C.y * C.j, triangle) == 0
p1, p2, p3, p4 = [(0, 0), (1, 0), (5, 1), (0, 1)]
poly = Polygon(p1, p2, p3, p4)
assert vector_integrate(-23 * C.z,
poly) == -161 * C.z - 23 * sqrt(17) * C.z
point = Point(2, 3)
assert vector_integrate(C.i * C.y - C.z, point) == ParametricIntegral(
C.y * C.i, ParametricRegion((2, 3)))
c3 = ImplicitRegion((x, y), x**2 + y**2 - 4)
assert vector_integrate(45, c3) == 360 * pi
c4 = ImplicitRegion((x, y), (x - 3)**2 + (y - 4)**2 - 9)
assert vector_integrate(1, c4) == 12 * pi
pl = Plane(Point(1, 1, 1), Point(2, 3, 4), Point(2, 2, 2))
raises(ValueError, lambda: vector_integrate(C.x * C.z * C.i + C.k, pl))
def test_parametric_lineintegrals():
halfcircle = ParametricRegion((4*cos(theta), 4*sin(theta)), (theta, -pi/2, pi/2))
assert ParametricIntegral(C.x*C.y**4, halfcircle) == S(8192)/5
curve = ParametricRegion((t, t**2, t**3), (t, 0, 1))
field1 = 8*C.x**2*C.y*C.z*C.i + 5*C.z*C.j - 4*C.x*C.y*C.k
assert ParametricIntegral(field1, curve) == 1
line = ParametricRegion((4*t - 1, 2 - 2*t, t), (t, 0, 1))
assert ParametricIntegral(C.x*C.z*C.i - C.y*C.z*C.k, line) == 3
assert ParametricIntegral(4*C.x**3, ParametricRegion((1, t), (t, 0, 2))) == 8
helix = ParametricRegion((cos(t), sin(t), 3*t), (t, 0, 4*pi))
assert ParametricIntegral(C.x*C.y*C.z, helix) == -3*sqrt(10)*pi
field2 = C.y*C.i + C.z*C.j + C.z*C.k
assert ParametricIntegral(field2, ParametricRegion((cos(t), sin(t), t**2), (t, 0, pi))) == -5*pi/2 + pi**4/2
def test_ParametricRegion():
point = ParametricRegion((3, 4))
assert point.definition == (3, 4)
assert point.parameters == ()
assert point.limits == {}
assert point.dimensions == 0
# line x = y
line_xy = ParametricRegion((y, y), (y, 1, 5))
assert line_xy .definition == (y, y)
assert line_xy.parameters == (y,)
assert line_xy.dimensions == 1
# line y = z
line_yz = ParametricRegion((x,t,t), x, (t, 1, 2))
assert line_yz.definition == (x,t,t)
assert line_yz.parameters == (x, t)
assert line_yz.limits == {t: (1, 2)}
assert line_yz.dimensions == 1
p1 = ParametricRegion((9*a, -16*b), (a, 0, 2), (b, -1, 5))
assert p1.definition == (9*a, -16*b)
assert p1.parameters == (a, b)
assert p1.limits == {a: (0, 2), b: (-1, 5)}
assert p1.dimensions == 2
p2 = ParametricRegion((t, t**3), t)
assert p2.parameters == (t,)
assert p2.limits == {}
assert p2.dimensions == 0
circle = ParametricRegion((r*cos(theta), r*sin(theta)), r, (theta, 0, 2*pi))
assert circle.definition == (r*cos(theta), r*sin(theta))
assert circle.dimensions == 1
halfdisc = ParametricRegion((r*cos(theta), r*sin(theta)), (r, -2, 2), (theta, 0, pi))
assert halfdisc.definition == (r*cos(theta), r*sin(theta))
assert halfdisc.parameters == (r, theta)
assert halfdisc.limits == {r: (-2, 2), theta: (0, pi)}
assert halfdisc.dimensions == 2
ellipse = ParametricRegion((a*cos(t), b*sin(t)), (t, 0, 8))
assert ellipse.parameters == (t,)
assert ellipse.limits == {t: (0, 8)}
assert ellipse.dimensions == 1
cylinder = ParametricRegion((r*cos(theta), r*sin(theta), z), (r, 0, 1), (theta, 0, 2*pi), (z, 0, 4))
assert cylinder.parameters == (r, theta, z)
assert cylinder.dimensions == 3
sphere = ParametricRegion((r*sin(phi)*cos(theta),r*sin(phi)*sin(theta), r*cos(phi)),
r, (theta, 0, 2*pi), (phi, 0, pi))
assert sphere.definition == (r*sin(phi)*cos(theta),r*sin(phi)*sin(theta), r*cos(phi))
assert sphere.parameters == (r, theta, phi)
assert sphere.dimensions == 2
raises(ValueError, lambda: ParametricRegion((a*t**2, 2*a*t), (a, -2)))
raises(ValueError, lambda: ParametricRegion((a, b), (a**2, sin(b)), (a, 2, 4, 6)))
def test_parametricregion():
point = ParametricRegion((), (3, 4), {})
assert point.definition == (3, 4)
assert point.parameters == ()
assert point.limits == {}
# line x = y
line_xy = ParametricRegion(C, (C.y), limits={C.y: (-3, 3)})
assert line_xy.definition == (C.y, )
assert line_xy.parameters == (C.x, C.y, C.z)
# line y = z
line_yz = ParametricRegion((t), (C.x, t, t), limits={t: (1, 2)})
assert line_yz.definition == (C.x, t, t)
assert line_yz.parameters == (t, )
p1 = ParametricRegion((a, b), (9 * a, -16 * b),
limits={
a: (0, 2),
b: (-1, 5)
})
assert p1.definition == (9 * a, -16 * b)
assert p1.parameters == (a, b)
assert p1.limits == {a: (0, 2), b: (-1, 5)}
p2 = ParametricRegion(t, (t, t**3))
assert p2.parameters == (t, )
assert p2.limits == {}
circle = ParametricRegion((r, theta), (r * cos(theta), r * sin(theta)),
{theta: (0, 2 * pi)})
assert circle.definition == (r * cos(theta), r * sin(theta))
halfdisc = ParametricRegion((r, theta), (r * cos(theta), r * sin(theta)), {
r: (-2, 2),
theta: (0, pi)
})
assert halfdisc.definition == (r * cos(theta), r * sin(theta))
assert halfdisc.parameters == (r, theta)
assert halfdisc.limits == {r: (-2, 2), theta: (0, pi)}
ellipse = ParametricRegion(t, (a * cos(t), b * sin(t)), {t: (0, 8)})
assert ellipse.parameters == (t, )
assert ellipse.limits == {t: (0, 8)}
cylinder = ParametricRegion((r, theta, z),
(cos(theta), r * sin(theta), C.z), {
theta: (0, 2 * pi),
z: (0, 4)
})
assert cylinder.parameters == (r, theta, z)
sphere = ParametricRegion(
(r, theta, phi),
(r * sin(phi) * cos(theta), r * sin(phi) * sin(theta), r * cos(phi)), {
theta: (0, 2 * pi),
phi: (0, pi)
})
assert sphere.definition == (r * sin(phi) * cos(theta),
r * sin(phi) * sin(theta), r * cos(phi))
assert sphere.parameters == (r, theta, phi)
raises(ValueError, lambda: ParametricRegion(
(t), (a * t**2, 2 * a * t), {a: (-2, 2)}))
raises(ValueError, lambda: ParametricRegion(
(a, b), (a**2, sin(b)), {a: (2, 4, 6)}))
