def test_vector():
# Tests the effects of orientation of coordinate systems on
# basic vector operations.
N = CoordSysCartesian('N')
A = N.orient_new_axis('A', q1, N.k)
B = A.orient_new_axis('B', q2, A.i)
C = B.orient_new_axis('C', q3, B.j)
# Test to_matrix
v1 = a * N.i + b * N.j + c * N.k
assert v1.to_matrix(A) == Matrix([[a * cos(q1) + b * sin(q1)],
[-a * sin(q1) + b * cos(q1)], [c]])
# Test dot
assert N.i.dot(A.i) == cos(q1)
assert N.i.dot(A.j) == -sin(q1)
assert N.i.dot(A.k) == 0
assert N.j.dot(A.i) == sin(q1)
assert N.j.dot(A.j) == cos(q1)
assert N.j.dot(A.k) == 0
assert N.k.dot(A.i) == 0
assert N.k.dot(A.j) == 0
assert N.k.dot(A.k) == 1
assert N.i.dot(A.i + A.j) == -sin(q1) + cos(q1) == \
(A.i + A.j).dot(N.i)
assert A.i.dot(C.i) == cos(q3)
assert A.i.dot(C.j) == 0
assert A.i.dot(C.k) == sin(q3)
assert A.j.dot(C.i) == sin(q2) * sin(q3)
assert A.j.dot(C.j) == cos(q2)
assert A.j.dot(C.k) == -sin(q2) * cos(q3)
assert A.k.dot(C.i) == -cos(q2) * sin(q3)
assert A.k.dot(C.j) == sin(q2)
assert A.k.dot(C.k) == cos(q2) * cos(q3)
pytest.raises(TypeError, lambda: N.i.dot(A.x))
# Test cross
assert N.i.cross(A.i) == sin(q1) * A.k
assert N.i.cross(A.j) == cos(q1) * A.k
assert N.i.cross(A.k) == -sin(q1) * A.i - cos(q1) * A.j
assert N.j.cross(A.i) == -cos(q1) * A.k
assert N.j.cross(A.j) == sin(q1) * A.k
assert N.j.cross(A.k) == cos(q1) * A.i - sin(q1) * A.j
assert N.k.cross(A.i) == A.j
assert N.k.cross(A.j) == -A.i
assert N.k.cross(A.k) == Vector.zero
assert N.i.cross(A.i) == sin(q1) * A.k
assert N.i.cross(A.j) == cos(q1) * A.k
assert N.i.cross(A.i + A.j) == sin(q1) * A.k + cos(q1) * A.k
assert (A.i + A.j).cross(N.i) == (-sin(q1) - cos(q1)) * N.k
assert A.i.cross(C.i) == sin(q3) * C.j
assert A.i.cross(C.j) == -sin(q3) * C.i + cos(q3) * C.k
assert A.i.cross(C.k) == -cos(q3) * C.j
assert C.i.cross(A.i) == (-sin(q3)*cos(q2))*A.j + \
(-sin(q2)*sin(q3))*A.k
assert C.j.cross(A.i) == (sin(q2)) * A.j + (-cos(q2)) * A.k
assert express(C.k.cross(A.i), C).trigsimp() == cos(q3) * C.j
pytest.raises(TypeError, lambda: N.i.cross(A.x))
def test_coordinate_vars():
# Tests the coordinate variables functionality with respect to
# reorientation of coordinate systems.
A = CoordSysCartesian('A')
# Note that the name given on the lhs is different from A.x._name
assert BaseScalar('A.x', 0, A, 'A_x', r'\mathbf{{x}_{A}}') == A.x
assert BaseScalar('A.y', 1, A, 'A_y', r'\mathbf{{y}_{A}}') == A.y
assert BaseScalar('A.z', 2, A, 'A_z', r'\mathbf{{z}_{A}}') == A.z
assert hash(BaseScalar('A.x', 0, A, 'A_x',
r'\mathbf{{x}_{A}}')) == hash(A.x)
assert all(isinstance(_, BaseScalar) for _ in (A.x, A.y, A.z))
assert A.x * A.y == A.y * A.x
pytest.raises(TypeError, lambda: BaseScalar('Ax', 0, 1, ' ', ' '))
pytest.raises(ValueError, lambda: BaseScalar('Ax', 5, A, ' ', ' '))
assert A.scalar_map(A) == {A.x: A.x, A.y: A.y, A.z: A.z}
assert A.x.system == A
assert A.x.diff(A.x) == 1
B = A.orient_new_axis('B', q, A.k)
assert B.scalar_map(A) == {
B.z: A.z,
B.y: -A.x * sin(q) + A.y * cos(q),
B.x: A.x * cos(q) + A.y * sin(q)
}
assert A.scalar_map(B) == {
A.x: B.x * cos(q) - B.y * sin(q),
A.y: B.x * sin(q) + B.y * cos(q),
A.z: B.z
}
assert express(B.x, A, variables=True) == A.x * cos(q) + A.y * sin(q)
assert express(B.y, A, variables=True) == -A.x * sin(q) + A.y * cos(q)
assert express(B.z, A, variables=True) == A.z
assert express(B.x*B.y*B.z, A, variables=True) == \
A.z*(-A.x*sin(q) + A.y*cos(q))*(A.x*cos(q) + A.y*sin(q))
assert express(B.x*B.i + B.y*B.j + B.z*B.k, A) == \
(B.x*cos(q) - B.y*sin(q))*A.i + (B.x*sin(q) +
B.y*cos(q))*A.j + B.z*A.k
assert simplify(express(B.x*B.i + B.y*B.j + B.z*B.k, A,
variables=True)) == \
A.x*A.i + A.y*A.j + A.z*A.k
assert express(A.x*A.i + A.y*A.j + A.z*A.k, B) == \
(A.x*cos(q) + A.y*sin(q))*B.i + \
(-A.x*sin(q) + A.y*cos(q))*B.j + A.z*B.k
assert simplify(express(A.x*A.i + A.y*A.j + A.z*A.k, B,
variables=True)) == \
B.x*B.i + B.y*B.j + B.z*B.k
pytest.raises(TypeError, lambda: express(A.x, 1))
pytest.raises(ValueError, lambda: express(A.i, B, A))
pytest.raises(TypeError, lambda: express(A.i | A.j, B, 1))
pytest.raises(ValueError, lambda: express(1, B, 1))
N = B.orient_new_axis('N', -q, B.k)
assert N.scalar_map(A) == \
{N.x: A.x, N.z: A.z, N.y: A.y}
C = A.orient_new_axis('C', q, A.i + A.j + A.k)
mapping = A.scalar_map(C)
assert mapping[A.x] == (C.x * (2 * cos(q) + 1) / 3 + C.y *
(-2 * sin(q + pi / 6) + 1) / 3 + C.z *
(-2 * cos(q + pi / 3) + 1) / 3)
assert mapping[A.y] == (C.x * (-2 * cos(q + pi / 3) + 1) / 3 + C.y *
(2 * cos(q) + 1) / 3 + C.z *
(-2 * sin(q + pi / 6) + 1) / 3)
assert mapping[A.z] == (C.x * (-2 * sin(q + pi / 6) + 1) / 3 + C.y *
(-2 * cos(q + pi / 3) + 1) / 3 + C.z *
(2 * cos(q) + 1) / 3)
D = A.locate_new('D', a * A.i + b * A.j + c * A.k)
assert D.scalar_map(A) == {D.z: A.z - c, D.x: A.x - a, D.y: A.y - b}
E = A.orient_new_axis('E', a, A.k, a * A.i + b * A.j + c * A.k)
assert A.scalar_map(E) == {
A.z: E.z + c,
A.x: E.x * cos(a) - E.y * sin(a) + a,
A.y: E.x * sin(a) + E.y * cos(a) + b
}
assert E.scalar_map(A) == {
E.x: (A.x - a) * cos(a) + (A.y - b) * sin(a),
E.y: (-A.x + a) * sin(a) + (A.y - b) * cos(a),
E.z: A.z - c
}
F = A.locate_new('F', Vector.zero)
assert A.scalar_map(F) == {A.z: F.z, A.x: F.x, A.y: F.y}
def test_dyadic():
a, b = symbols('a, b')
assert Dyadic.zero != 0
assert isinstance(Dyadic.zero, DyadicZero)
assert BaseDyadic(A.i, A.j) != BaseDyadic(A.j, A.i)
assert (BaseDyadic(Vector.zero, A.i) == BaseDyadic(A.i, Vector.zero) ==
Dyadic.zero)
d1 = A.i | A.i
d2 = A.j | A.j
d3 = A.i | A.j
assert isinstance(d1, BaseDyadic)
d_mul = a * d1
assert isinstance(d_mul, DyadicMul)
assert d_mul.base_dyadic == d1
assert d_mul.measure_number == a
assert isinstance(a * d1 + b * d3, DyadicAdd)
assert d1 == A.i.outer(A.i)
assert d3 == A.i.outer(A.j)
v1 = a * A.i - A.k
v2 = A.i + b * A.j
assert v1 | v2 == v1.outer(v2) == (a * (A.i | A.i) + (a * b) *
(A.i | A.j) - (A.k | A.i) - b *
(A.k | A.j))
assert d1 * 0 == Dyadic.zero
assert d1 != Dyadic.zero
assert d1 * 2 == 2 * (A.i | A.i)
assert d1 / 2. == 0.5 * d1
assert d1.dot(0 * d1) == Vector.zero
assert d1 & d2 == Dyadic.zero
assert d1.dot(A.i) == A.i == d1 & A.i
assert d1.cross(Vector.zero) == Dyadic.zero
assert d1.cross(A.i) == Dyadic.zero
assert d1 ^ A.j == d1.cross(A.j)
assert d1.cross(A.k) == -A.i | A.j
assert d2.cross(A.i) == -A.j | A.k == d2 ^ A.i
assert A.i ^ d1 == Dyadic.zero
assert A.j.cross(d1) == -A.k | A.i == A.j ^ d1
assert Vector.zero.cross(d1) == Dyadic.zero
assert A.k ^ d1 == A.j | A.i
assert A.i.dot(d1) == A.i & d1 == A.i
assert A.j.dot(d1) == Vector.zero
assert Vector.zero.dot(d1) == Vector.zero
assert A.j & d2 == A.j
assert d1.dot(d3) == d1 & d3 == A.i | A.j == d3
assert d3 & d1 == Dyadic.zero
q = symbols('q')
B = A.orient_new_axis('B', q, A.k)
assert express(d1, B) == express(d1, B, B)
assert express(d1, B) == ((cos(q)**2) * (B.i | B.i) + (-sin(q) * cos(q)) *
(B.i | B.j) + (-sin(q) * cos(q)) * (B.j | B.i) +
(sin(q)**2) * (B.j | B.j))
assert express(d1, B,
A) == (cos(q)) * (B.i | A.i) + (-sin(q)) * (B.j | A.i)
assert express(d1, A,
B) == (cos(q)) * (A.i | B.i) + (-sin(q)) * (A.i | B.j)
assert d1.to_matrix(A) == Matrix([[1, 0, 0], [0, 0, 0], [0, 0, 0]])
assert d1.to_matrix(A, B) == Matrix([[cos(q), -sin(q), 0], [0, 0, 0],
[0, 0, 0]])
assert d3.to_matrix(A) == Matrix([[0, 1, 0], [0, 0, 0], [0, 0, 0]])
a, b, c, d, e, f = symbols('a, b, c, d, e, f')
v1 = a * A.i + b * A.j + c * A.k
v2 = d * A.i + e * A.j + f * A.k
d4 = v1.outer(v2)
assert d4.to_matrix(A) == Matrix([[a * d, a * e, a * f],
[b * d, b * e, b * f],
[c * d, c * e, c * f]])
d5 = v1.outer(v1)
C = A.orient_new_axis('C', q, A.i)
for expected, actual in zip(
C.rotation_matrix(A) * d5.to_matrix(A) * C.rotation_matrix(A).T,
d5.to_matrix(C)):
assert (expected - actual).simplify() == 0
"""\
N_j + ⎛ 2 ⌠ ⎞ N_k\n\
⎜C_x - ⎮ f(b) db⎟ \n\
⎝ ⌡ ⎠ \
"""
pretty_v_8 = \
"""\
N_j + / / \\\n\
| 2 | |\n\
|C_x - | f(b) db|\n\
| | |\n\
\\ / / \
"""
v.append(N.i + C.k)
v.append(express(N.i, C))
v.append((a**2 + b) * N.i + (Integral(f(b))) * N.k)
upretty_v_11 = \
"""\
⎛ 2 ⎞ N_i + ⎛⌠ ⎞ N_k\n\
⎝a + b⎠ ⎜⎮ f(b) db⎟ \n\
⎝⌡ ⎠ \
"""
pretty_v_11 = \
"""\
/ 2 \\ + / / \\\n\
\\a + b/ N_i| | |\n\
| | f(b) db|\n\
| | |\n\
\\/ / \
"""
"""\
N_j + ⎛ 2 ⌠ ⎞ N_k\n\
⎜C_x - ⎮ f(b) db⎟ \n\
⎝ ⌡ ⎠ \
"""
pretty_v_8 = \
"""\
N_j + / / \\\n\
| 2 | |\n\
|C_x - | f(b) db|\n\
| | |\n\
\\ / / \
"""
v.append(N.i + C.k)
v.append(express(N.i, C))
v.append((a**2 + b)*N.i + (Integral(f(b)))*N.k)
upretty_v_11 = \
"""\
⎛ 2 ⎞ N_i + ⎛⌠ ⎞ N_k\n\
⎝a + b⎠ ⎜⎮ f(b) db⎟ \n\
⎝⌡ ⎠ \
"""
pretty_v_11 = \
"""\
/ 2 \\ + / / \\\n\
\\a + b/ N_i| | |\n\
| | f(b) db|\n\
| | |\n\
\\/ / \
"""
def test_express():
assert express(Vector.zero, N) == Vector.zero
assert express(Integer(0), N) == Integer(0)
assert express(A.i, C) == cos(q3)*C.i + sin(q3)*C.k
assert express(A.j, C) == sin(q2)*sin(q3)*C.i + cos(q2)*C.j - \
sin(q2)*cos(q3)*C.k
assert express(A.k, C) == -sin(q3)*cos(q2)*C.i + sin(q2)*C.j + \
cos(q2)*cos(q3)*C.k
assert express(A.i, N) == cos(q1)*N.i + sin(q1)*N.j
assert express(A.j, N) == -sin(q1)*N.i + cos(q1)*N.j
assert express(A.k, N) == N.k
assert express(A.i, A) == A.i
assert express(A.j, A) == A.j
assert express(A.k, A) == A.k
assert express(A.i, B) == B.i
assert express(A.j, B) == cos(q2)*B.j - sin(q2)*B.k
assert express(A.k, B) == sin(q2)*B.j + cos(q2)*B.k
assert express(A.i, C) == cos(q3)*C.i + sin(q3)*C.k
assert express(A.j, C) == sin(q2)*sin(q3)*C.i + cos(q2)*C.j - \
sin(q2)*cos(q3)*C.k
assert express(A.k, C) == -sin(q3)*cos(q2)*C.i + sin(q2)*C.j + \
cos(q2)*cos(q3)*C.k
# Check to make sure UnitVectors get converted properly
assert express(N.i, N) == N.i
assert express(N.j, N) == N.j
assert express(N.k, N) == N.k
assert express(N.i, A) == (cos(q1)*A.i - sin(q1)*A.j)
assert express(N.j, A) == (sin(q1)*A.i + cos(q1)*A.j)
assert express(N.k, A) == A.k
assert express(N.i, B) == (cos(q1)*B.i - sin(q1)*cos(q2)*B.j +
sin(q1)*sin(q2)*B.k)
assert express(N.j, B) == (sin(q1)*B.i + cos(q1)*cos(q2)*B.j -
sin(q2)*cos(q1)*B.k)
assert express(N.k, B) == (sin(q2)*B.j + cos(q2)*B.k)
assert express(N.i, C) == (
(cos(q1)*cos(q3) - sin(q1)*sin(q2)*sin(q3))*C.i -
sin(q1)*cos(q2)*C.j +
(sin(q3)*cos(q1) + sin(q1)*sin(q2)*cos(q3))*C.k)
assert express(N.j, C) == (
(sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*C.i +
cos(q1)*cos(q2)*C.j +
(sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*C.k)
assert express(N.k, C) == (-sin(q3)*cos(q2)*C.i + sin(q2)*C.j +
cos(q2)*cos(q3)*C.k)
assert express(A.i, N) == (cos(q1)*N.i + sin(q1)*N.j)
assert express(A.j, N) == (-sin(q1)*N.i + cos(q1)*N.j)
assert express(A.k, N) == N.k
assert express(A.i, A) == A.i
assert express(A.j, A) == A.j
assert express(A.k, A) == A.k
assert express(A.i, B) == B.i
assert express(A.j, B) == (cos(q2)*B.j - sin(q2)*B.k)
assert express(A.k, B) == (sin(q2)*B.j + cos(q2)*B.k)
assert express(A.i, C) == (cos(q3)*C.i + sin(q3)*C.k)
assert express(A.j, C) == (sin(q2)*sin(q3)*C.i + cos(q2)*C.j -
sin(q2)*cos(q3)*C.k)
assert express(A.k, C) == (-sin(q3)*cos(q2)*C.i + sin(q2)*C.j +
cos(q2)*cos(q3)*C.k)
assert express(B.i, N) == (cos(q1)*N.i + sin(q1)*N.j)
assert express(B.j, N) == (-sin(q1)*cos(q2)*N.i +
cos(q1)*cos(q2)*N.j + sin(q2)*N.k)
assert express(B.k, N) == (sin(q1)*sin(q2)*N.i -
sin(q2)*cos(q1)*N.j + cos(q2)*N.k)
assert express(B.i, A) == A.i
assert express(B.j, A) == (cos(q2)*A.j + sin(q2)*A.k)
assert express(B.k, A) == (-sin(q2)*A.j + cos(q2)*A.k)
assert express(B.i, B) == B.i
assert express(B.j, B) == B.j
assert express(B.k, B) == B.k
assert express(B.i, C) == (cos(q3)*C.i + sin(q3)*C.k)
assert express(B.j, C) == C.j
assert express(B.k, C) == (-sin(q3)*C.i + cos(q3)*C.k)
assert express(C.i, N) == (
(cos(q1)*cos(q3) - sin(q1)*sin(q2)*sin(q3))*N.i +
(sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*N.j -
sin(q3)*cos(q2)*N.k)
assert express(C.j, N) == (
-sin(q1)*cos(q2)*N.i + cos(q1)*cos(q2)*N.j + sin(q2)*N.k)
assert express(C.k, N) == (
(sin(q3)*cos(q1) + sin(q1)*sin(q2)*cos(q3))*N.i +
(sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*N.j +
cos(q2)*cos(q3)*N.k)
assert express(C.i, A) == (cos(q3)*A.i + sin(q2)*sin(q3)*A.j -
sin(q3)*cos(q2)*A.k)
assert express(C.j, A) == (cos(q2)*A.j + sin(q2)*A.k)
assert express(C.k, A) == (sin(q3)*A.i - sin(q2)*cos(q3)*A.j +
cos(q2)*cos(q3)*A.k)
assert express(C.i, B) == (cos(q3)*B.i - sin(q3)*B.k)
assert express(C.j, B) == B.j
assert express(C.k, B) == (sin(q3)*B.i + cos(q3)*B.k)
assert express(C.i, C) == C.i
assert express(C.j, C) == C.j
assert express(C.k, C) == C.k == (C.k)
# Check to make sure Vectors get converted back to UnitVectors
assert N.i == express((cos(q1)*A.i - sin(q1)*A.j), N).simplify()
assert N.j == express((sin(q1)*A.i + cos(q1)*A.j), N).simplify()
assert N.i == express((cos(q1)*B.i - sin(q1)*cos(q2)*B.j +
sin(q1)*sin(q2)*B.k), N).simplify()
assert N.j == express((sin(q1)*B.i + cos(q1)*cos(q2)*B.j -
sin(q2)*cos(q1)*B.k), N).simplify()
assert N.k == express((sin(q2)*B.j + cos(q2)*B.k), N).simplify()
assert A.i == express((cos(q1)*N.i + sin(q1)*N.j), A).simplify()
assert A.j == express((-sin(q1)*N.i + cos(q1)*N.j), A).simplify()
assert A.j == express((cos(q2)*B.j - sin(q2)*B.k), A).simplify()
assert A.k == express((sin(q2)*B.j + cos(q2)*B.k), A).simplify()
assert A.i == express((cos(q3)*C.i + sin(q3)*C.k), A).simplify()
assert A.j == express((sin(q2)*sin(q3)*C.i + cos(q2)*C.j -
sin(q2)*cos(q3)*C.k), A).simplify()
assert A.k == express((-sin(q3)*cos(q2)*C.i + sin(q2)*C.j +
cos(q2)*cos(q3)*C.k), A).simplify()
assert B.i == express((cos(q1)*N.i + sin(q1)*N.j), B).simplify()
assert B.j == express((-sin(q1)*cos(q2)*N.i +
cos(q1)*cos(q2)*N.j + sin(q2)*N.k), B).simplify()
assert B.k == express((sin(q1)*sin(q2)*N.i -
sin(q2)*cos(q1)*N.j + cos(q2)*N.k), B).simplify()
assert B.j == express((cos(q2)*A.j + sin(q2)*A.k), B).simplify()
assert B.k == express((-sin(q2)*A.j + cos(q2)*A.k), B).simplify()
assert B.i == express((cos(q3)*C.i + sin(q3)*C.k), B).simplify()
assert B.k == express((-sin(q3)*C.i + cos(q3)*C.k), B).simplify()
assert C.i == express((cos(q3)*A.i + sin(q2)*sin(q3)*A.j -
sin(q3)*cos(q2)*A.k), C).simplify()
assert C.j == express((cos(q2)*A.j + sin(q2)*A.k), C).simplify()
assert C.k == express((sin(q3)*A.i - sin(q2)*cos(q3)*A.j +
cos(q2)*cos(q3)*A.k), C).simplify()
assert C.i == express((cos(q3)*B.i - sin(q3)*B.k), C).simplify()
assert C.k == express((sin(q3)*B.i + cos(q3)*B.k), C).simplify()
