def test_christoffel_1_gtnsr():
x1, x2 = symbols('x1, x2')
var_list = [x1, x2]
g = Arraypy([2, 2, 1])
g = g.to_tensor((-1, -1))
g[1, 1] = cos(x2)**2
g[1, 2] = 0
g[2, 1] = 0
g[2, 2] = 1
res_arr = Arraypy([3, 2, 1])
res_arr[1, 1, 1] = 0
res_arr[1, 1, 2] = sin(x2) * cos(x2)
res_arr[1, 2, 1] = -sin(x2) * cos(x2)
res_arr[2, 1, 1] = -sin(x2) * cos(x2)
res_arr[1, 2, 2] = 0
res_arr[2, 2, 1] = 0
res_arr[2, 1, 2] = 0
res_arr[2, 2, 2] = 0
res_ten = res_arr.to_tensor((-1, -1, -1))
print('test_christoffel_1_gtnsr_t <=== actual test code')
assert christoffel_1(g, var_list) == res_ten
assert isinstance(christoffel_1(g, var_list), TensorArray)
assert christoffel_1(g, var_list).type_pq == (0, 3)
assert christoffel_1(g, var_list, 't') == res_ten
assert isinstance(christoffel_1(g, var_list, 't'), TensorArray)
assert christoffel_1(g, var_list).type_pq == (0, 3)
print('test_christoffel_1_gtnsr_a <=== actual test code')
assert christoffel_1(g, var_list, 'a') == res_arr
assert isinstance(christoffel_1(g, var_list, 'a'), Arraypy)
def test_grad_gtnsr():
x1, x2, x3 = symbols('x1 x2 x3')
f = x1**2 * x2 + sin(x2 * x3 - x2)
var1 = [x1, x2, x3]
k1 = Arraypy([1, 3, 0]).to_tensor(1)
k1[0] = x1
k1[1] = x2
k1[2] = x3
# g задано tensor, индекс с 1 и var-list
a = Arraypy([2, 3, 1])
b = a.to_tensor((-1, -1))
b[1, 1] = 2
b[1, 2] = 1
b[1, 3] = 0
b[2, 1] = 1
b[2, 2] = 3
b[2, 3] = 0
b[3, 1] = 0
b[3, 2] = 0
b[3, 3] = 1
res_ar = Arraypy([1, 3, 1])
res_ar[1] = -x1**2 / 5 + 6 * x1 * x2 / 5 - (x3 - 1) * cos(x2 * x3 - x2) / 5
res_ar[2] = 2 * x1**2 / 5 - 2 * x1 * x2 / \
5 + cos(x2 * x3 - x2) * 2 * (x3 - 1) / 5
res_ar[3] = x2 * cos(x2 * x3 - x2)
res_ten = res_ar.to_tensor(1)
res_ar1 = Arraypy([1, 3, 0])
res_ar1[0] = 2 * x1 * x2
res_ar1[1] = x1**2 + (x3 - 1) * cos(x2 * x3 - x2)
res_ar1[2] = x2 * cos(x2 * x3 - x2)
assert str(
grad(
f,
var1,
b,
'l')) == '[-x1**2/5 + 6*x1*x2/5 - (x3 - 1)*cos(x2*x3 - x2)/5, 2*x1**2/5 - 2*x1*x2/5 + 2*(x3 - 1)*cos(x2*x3 - x2)/5, x2*cos(x2*x3 - x2)]'
assert isinstance(grad(f, var1, b, 'l'), list)
assert grad(f, var1, b, 'a') == res_ar
assert isinstance(grad(f, var1, b, 'a'), Arraypy)
assert grad(f, k1, output_type='a') == res_ar1
assert isinstance(grad(f, k1, output_type='a'), Arraypy)
assert grad(f, var1, b, 't') == res_ten
assert isinstance(grad(f, var1, b, 't'), TensorArray)
assert grad(f, var1, b, 't').type_pq == (1, 0)
assert grad(f, var1, b) == res_ten
assert isinstance(grad(f, var1, b, 't'), TensorArray)
assert grad(f, var1, b, 't').type_pq == (1, 0)
def test_grad_varlist():
x1, x2, x3 = symbols('x1 x2 x3')
f = x1**2 * x2 + sin(x2 * x3 - x2)
var1 = [x1, x2, x3]
res_ar1 = Arraypy([1, 3, 0])
res_ar1[0] = 2 * x1 * x2
res_ar1[1] = x1**2 + (x3 - 1) * cos(x2 * x3 - x2)
res_ar1[2] = x2 * cos(x2 * x3 - x2)
res_ten1 = res_ar1.to_tensor(1)
res_ar = Arraypy([1, 3, 0])
res_ar[0] = -x1**2 / 5 + 6 * x1 * x2 / 5 - (x3 - 1) * cos(x2 * x3 - x2) / 5
res_ar[1] = 2 * x1**2 / 5 - 2 * x1 * x2 / \
5 + cos(x2 * x3 - x2) * 2 * (x3 - 1) / 5
res_ar[2] = x2 * cos(x2 * x3 - x2)
res_ten = res_ar.to_tensor(1)
g = Matrix([[2, 1, 0], [1, 3, 0], [0, 0, 1]])
assert grad(f, var1, output_type='l') == [
2 * x1 * x2, x1**2 + (x3 - 1) * cos(x2 * x3 - x2), x2 *
cos(x2 * x3 - x2)]
assert isinstance(grad(f, var1, output_type='l'), list)
assert grad(f, var1) == [
2 * x1 * x2, x1**2 + (x3 - 1) * cos(x2 * x3 - x2), x2 *
cos(x2 * x3 - x2)]
assert isinstance(grad(f, var1), list)
assert grad(f, var1, output_type='a') == res_ar1
assert isinstance(grad(f, var1, output_type='t'), Arraypy)
assert grad(f, var1, output_type='t') == res_ten1
assert isinstance(grad(f, var1, output_type='t'), TensorArray)
assert grad(f, var1, output_type='t').type_pq == (1, 0)
assert str(
grad(
f,
var1,
g,
output_type='l')) == '[-x1**2/5 + 6*x1*x2/5 - (x3 - 1)*cos(x2*x3 - x2)/5, 2*x1**2/5 - 2*x1*x2/5 + 2*(x3 - 1)*cos(x2*x3 - x2)/5, x2*cos(x2*x3 - x2)]'
assert isinstance(grad(f, var1, g, output_type='l'), list)
assert grad(f, var1, g, output_type='a') == res_ar
assert isinstance(grad(f, var1, g, output_type='a'), Arraypy)
assert grad(f, var1, g, output_type='t') == res_ten
assert isinstance(grad(f, var1, g, output_type='t'), TensorArray)
assert grad(f, var1, g, output_type='t').type_pq == (1, 0)
def test_covar_der():
x1, x2 = symbols('x1, x2')
var_list = [x1, x2]
g = Matrix([[cos(x2)**2, 0], [0, 1]])
X = [x1 * x2**3, x1 - cos(x2)]
res_arr = Arraypy([2, 2, 0])
res_arr[0, 0] = x2**3 - (x1 - cos(x2)) * sin(x2) / cos(x2)
res_arr[0, 1] = x1 * x2**3 * sin(x2) * cos(x2) + 1
res_arr[1, 0] = -x1 * x2**3 * sin(x2) / cos(x2) + 3 * x1 * x2**2
res_arr[1, 1] = sin(x2)
res_ten = res_arr.to_tensor((1, -1))
print('test_covar_der_t <=== actual test code')
assert covar_der(X, g, var_list) == res_ten
assert isinstance(covar_der(X, g, var_list), TensorArray)
assert covar_der(X, g, var_list).type_pq == (1, 1)
assert covar_der(X, g, var_list, 't') == res_ten
assert isinstance(covar_der(X, g, var_list, 't'), TensorArray)
assert covar_der(X, g, var_list, 't').type_pq == (1, 1)
print('test_covar_der_a <=== actual test code')
assert covar_der(X, g, var_list, 'a') == res_arr
assert isinstance(covar_der(X, g, var_list, 'a'), Arraypy)
def test_grad_gm_vl():
x1, x2, x3 = symbols('x1 x2 x3')
f = x1**2 * x2 + sin(x2 * x3 - x2)
var1 = [x1, x2, x3]
g = Matrix([[2, 1, 0], [1, 3, 0], [0, 0, 1]])
k0 = Arraypy([1, 3, 1]).to_tensor(1)
k0[1] = x1
k0[2] = x2
k0[3] = x3
res_ar = Arraypy([1, 3, 0])
res_ar[0] = -x1**2 / 5 + 6 * x1 * x2 / 5 - (x3 - 1) * cos(x2 * x3 - x2) / 5
res_ar[1] = 2 * x1**2 / 5 - 2 * x1 * x2 / \
5 + cos(x2 * x3 - x2) * 2 * (x3 - 1) / 5
res_ar[2] = x2 * cos(x2 * x3 - x2)
res_ten = res_ar.to_tensor(1)
assert str(
grad(
f,
k0,
g,
'l')) == '[-x1**2/5 + 6*x1*x2/5 - (x3 - 1)*cos(x2*x3 - x2)/5, 2*x1**2/5 - 2*x1*x2/5 + 2*(x3 - 1)*cos(x2*x3 - x2)/5, x2*cos(x2*x3 - x2)]'
assert isinstance(grad(f, k0, g, 'l'), list)
assert grad(f, k0, g, 'a') == res_ar
assert isinstance(grad(f, k0, g, 'a'), Arraypy)
assert grad(f, k0, g, 't') == res_ten
assert isinstance(grad(f, k0, g, 't'), TensorArray)
assert grad(f, k0, g, 't').type_pq == (1, 0)
def test_covar_der_xy():
x1, x2 = symbols('x1, x2')
var_list = [x1, x2]
g = Matrix([[cos(x2)**2, 0], [0, 1]])
X = [x1 * x2**3, x1 - cos(x2)]
Y = [1, 2]
res_arr = Arraypy([1, 2, 0])
res_arr[0] = -2 * x1 * x2**3 * \
sin(x2) / cos(x2) + 6 * x1 * x2**2 + \
x2**3 - (x1 - cos(x2)) * sin(x2) / cos(x2)
res_arr[1] = x1 * x2**3 * sin(x2) * cos(x2) + 2 * sin(x2) + 1
res_ten = res_arr.to_tensor(1)
print('test_covar_der_xy_t <=== actual test code')
assert covar_der_xy(X, Y, g, var_list) == res_ten
assert isinstance(covar_der_xy(X, Y, g, var_list), TensorArray)
assert covar_der_xy(X, Y, g, var_list).type_pq == (1, 0)
assert covar_der_xy(X, Y, g, var_list, 't') == res_ten
assert isinstance(covar_der_xy(X, Y, g, var_list, 't'), TensorArray)
assert covar_der_xy(X, Y, g, var_list, 't').type_pq == (1, 0)
print('test_covar_der_xy_a <=== actual test code')
assert covar_der_xy(X, Y, g, var_list, 'a') == res_arr
assert isinstance(covar_der_xy(X, Y, g, var_list, 'a'), Arraypy)
def test_df_var_tnsr1():
x1, x2, x3 = symbols('x1 x2 x3')
f = x1**2 * x2 + sin(x2 * x3 - x2)
var_tnsr1 = Arraypy([1, 3, 1]).to_tensor(1)
var_tnsr1[1] = x1
var_tnsr1[2] = x2
var_tnsr1[3] = x3
res_ar1 = Arraypy([1, 3, 1])
res_ar1[1] = 2 * x1 * x2
res_ar1[2] = x1**2 + (x3 - 1) * cos(x2 * x3 - x2)
res_ar1[3] = x2 * cos(x2 * x3 - x2)
res_ten1 = res_ar1.to_tensor(-1)
assert df(f, var_tnsr1) == [
2 * x1 * x2, x1**2 + (x3 - 1) * cos(x2 * x3 - x2), x2 *
cos(x2 * x3 - x2)]
assert isinstance(df(f, var_tnsr1), list)
assert df(f, var_tnsr1, 'l') == [
2 * x1 * x2, x1**2 + (x3 - 1) * cos(x2 * x3 - x2), x2 *
cos(x2 * x3 - x2)]
assert isinstance(df(f, var_tnsr1, 'l'), list)
assert df(f, var_tnsr1, 'a') == res_ar1
assert isinstance(df(f, var_tnsr1, 'a'), Arraypy)
assert df(f, var_tnsr1, 't') == res_ten1
assert isinstance(df(f, var_tnsr1, 't'), TensorArray)
assert df(f, var_tnsr1, 't').type_pq == (0, 1)
def test_christoffel_1_gm():
x1, x2 = symbols('x1, x2')
var_list = [x1, x2]
g = Matrix([[cos(x2)**2, 0], [0, 1]])
res_arr = Arraypy([3, 2, 0])
res_arr[0, 0, 0] = 0
res_arr[0, 0, 1] = sin(x2) * cos(x2)
res_arr[0, 1, 0] = -sin(x2) * cos(x2)
res_arr[1, 0, 0] = -sin(x2) * cos(x2)
res_arr[0, 1, 1] = 0
res_arr[1, 1, 0] = 0
res_arr[1, 0, 1] = 0
res_arr[1, 1, 1] = 0
res_ten = res_arr.to_tensor((-1, -1, -1))
print('test_christoffel_1_gm_t <=== actual test code')
assert christoffel_1(g, var_list) == res_ten
assert isinstance(christoffel_1(g, var_list), TensorArray)
assert christoffel_1(g, var_list).type_pq == (0, 3)
assert christoffel_1(g, var_list, 't') == res_ten
assert isinstance(christoffel_1(g, var_list, 't'), TensorArray)
assert christoffel_1(g, var_list).type_pq == (0, 3)
print('test_christoffel_1_gm_a <=== actual test code')
assert christoffel_1(g, var_list, 'a') == res_arr
assert isinstance(christoffel_1(g, var_list, 'a'), Arraypy)
def test_ricci_riemtnsr0():
x1, x2 = symbols('x1, x2')
var_list = [x1, x2]
riemann_arr = Arraypy([4, 2, 0])
riemann_arr[0, 0, 0, 0] = 0
riemann_arr[0, 0, 0, 1] = 0
riemann_arr[0, 0, 1, 1] = 0
riemann_arr[0, 1, 1, 1] = 0
riemann_arr[1, 1, 1, 1] = 0
riemann_arr[1, 1, 1, 0] = 0
riemann_arr[1, 1, 0, 0] = 0
riemann_arr[1, 0, 0, 0] = 0
riemann_arr[1, 0, 1, 0] = -1
riemann_arr[0, 1, 0, 1] = -cos(x2)**2
riemann_arr[1, 1, 0, 1] = 0
riemann_arr[0, 0, 1, 0] = 0
riemann_arr[1, 0, 1, 1] = 0
riemann_arr[0, 1, 0, 0] = 0
riemann_arr[0, 1, 1, 0] = 1
riemann_arr[1, 0, 0, 1] = cos(x2)**2
riemann_ten = riemann_arr.to_tensor((1, -1, -1, -1))
res_arr = Arraypy([2, 2, 0])
res_arr[0, 0] = cos(x2)**2
res_arr[0, 1] = 0
res_arr[1, 1] = 0
res_arr[1, 1] = 1
res_ten = res_arr.to_tensor((-1, -1))
print('test_ricci_riemtnsr0_t <=== actual test code')
assert ricci(riemann_ten, var_list) == res_ten
assert isinstance(ricci(riemann_ten, var_list), TensorArray)
assert ricci(riemann_ten, var_list).type_pq == (0, 2)
assert ricci(riemann_ten, var_list, 't') == res_ten
assert isinstance(ricci(riemann_ten, var_list, 't'), TensorArray)
assert ricci(riemann_ten, var_list, 't').type_pq == (0, 2)
print('test_ricci_riemtnsr0_a <=== actual test code')
assert ricci(riemann_ten, var_list, 'a') == res_arr
assert isinstance(ricci(riemann_ten, var_list, 'a'), Arraypy)
def wedgearray2tensor(B):
"""Function takes the Wedge_array(or dictionary) and creates a full skew
array.
Examples:
=========
>>> from sympy import symbols
>>> from tensor_analysis.arraypy import Arraypy, TensorArray
>>> from tensor_analysis.tensor_fields import tensor2wedgearray, \
wedgearray2tensor
>>> x1, x2, x3 = symbols('x1 x2 x3')
>>> y2 = TensorArray(Arraypy((3, 3)), (-1, -1))
>>> y2[0, 1] = x3
>>> y2[0, 2] = -x2
>>> y2[1, 0] = -x3
>>> y2[1, 2] = x1
>>> y2[2, 0] = x2
>>> y2[2, 1] = -x1
>>> A=tensor2wedgearray(y2)
>>> wdg=wedgearray2tensor(A)
>>> print(wdg)
0 x3 -x2
-x3 0 x1
x2 -x1 0
"""
if isinstance(B, Wedge_array):
B = B._output
else:
raise TypeError('Input tensor must be of Wedge_array or dict type')
list_indices = sorted(B.keys())
# (min_index, max_index) - the range of indexes
max_index = max(max(list_indices))
# for the start index of arraypy
min_index = min(min(list_indices))
# shape output array
rank = len(list_indices[0])
# Creating the output array
arr = Arraypy([rank, max_index - min_index + 1, min_index])
valence_list = [(-1) for k in range(max_index - min_index)]
tensor = arr.to_tensor(valence_list)
for key, value in B.items():
tensor[key] = value
# multiply by the sign of the permutation
for p in permutations(list(key)):
tensor[p] = sign_permutations(list(p)) * value
return tensor
def wedgearray2tensor(B):
"""Function takes the Wedge_array(or dictionary) and creates a full skew
array.
Examples:
=========
>>> from sympy import symbols
>>> from tensor_analysis.arraypy import Arraypy, TensorArray
>>> from tensor_analysis.tensor_fields import tensor2wedgearray, \
wedgearray2tensor
>>> x1, x2, x3 = symbols('x1 x2 x3')
>>> y2 = TensorArray(Arraypy((3, 3)), (-1, -1))
>>> y2[0, 1] = x3
>>> y2[0, 2] = -x2
>>> y2[1, 0] = -x3
>>> y2[1, 2] = x1
>>> y2[2, 0] = x2
>>> y2[2, 1] = -x1
>>> A=tensor2wedgearray(y2)
>>> wdg=wedgearray2tensor(A)
>>> print(wdg)
0 x3 -x2
-x3 0 x1
x2 -x1 0
"""
if isinstance(B, Wedge_array):
B = B._output
else:
raise TypeError('Input tensor must be of Wedge_array or dict type')
list_indices = sorted(B.keys())
# (min_index, max_index) - the range of indexes
max_index = max(max(list_indices))
# for the start index of arraypy
min_index = min(min(list_indices))
# shape output array
rank = len(list_indices[0])
# Creating the output array
arr = Arraypy([rank, max_index - min_index + 1, min_index])
valence_list = [(-1) for k in range(max_index - min_index)]
tensor = arr.to_tensor(valence_list)
for key, value in B.items():
tensor[key] = value
# multiply by the sign of the permutation
for p in permutations(list(key)):
tensor[p] = sign_permutations(list(p)) * value
return tensor
def test_riemann_gtnsr():
x1, x2 = symbols('x1, x2')
var_list = [x1, x2]
g = Arraypy([2, 2, 1])
g = g.to_tensor((-1, -1))
g[1, 1] = cos(x2)**2
g[1, 2] = 0
g[2, 1] = 0
g[2, 2] = 1
res_arr = Arraypy([4, 2, 1])
res_arr[1, 1, 1, 1] = 0
res_arr[1, 1, 1, 2] = 0
res_arr[1, 1, 2, 1] = 0
res_arr[1, 1, 2, 2] = 0
res_arr[1, 2, 1, 1] = 0
res_arr[1, 2, 2, 1] = 1
res_arr[1, 2, 2, 2] = 0
res_arr[1, 2, 1, 2] = -cos(x2)**2
res_arr[2, 1, 1, 1] = 0
res_arr[2, 1, 1, 2] = cos(x2)**2
res_arr[2, 2, 1, 1] = 0
res_arr[2, 2, 2, 1] = 0
res_arr[2, 1, 2, 2] = 0
res_arr[2, 1, 2, 1] = -1
res_arr[2, 2, 1, 2] = 0
res_arr[2, 2, 2, 2] = 0
res_ten = res_arr.to_tensor((1, -1, -1, -1))
print('test_riemann_gtnsr_t <=== actual test code')
assert riemann(g, var_list) == res_ten
assert isinstance(riemann(g, var_list), TensorArray)
assert riemann(g, var_list).type_pq == (1, 3)
assert riemann(g, var_list, 't') == res_ten
assert isinstance(riemann(g, var_list, 't'), TensorArray)
assert riemann(g, var_list, 't').type_pq == (1, 3)
print('test_riemann_gtnsr_a <=== actual test code')
assert riemann(g, var_list, 'a') == res_arr
assert isinstance(riemann(g, var_list, 'a'), Arraypy)
def test_riemann_Li():
x1, x2 = symbols('x1, x2')
var_list = [x1, x2]
C = Arraypy([3, 2, 0]).to_tensor((1, -1, -1))
C[0, 0, 0] = 0
C[0, 0, 1] = sin(x2) * cos(x2)
C[0, 1, 1] = 0
C[1, 1, 1] = 0
C[1, 0, 1] = 0
C[1, 1, 0] = 0
C[1, 0, 0] = -sin(x2) * cos(x2)
C[0, 1, 0] = -sin(x2) * cos(x2)
g = Arraypy((2, 2)).to_tensor((-1, -1))
g[0, 0] = cos(x2)**2
g[0, 1] = 0
g[1, 0] = 0
g[1, 1] = 1
res_arr = Arraypy([4, 2, 0])
res_arr[0, 0, 0, 0] = -0.25 * sin(x2)**2 * cos(x2)**2
res_arr[0, 0, 0, 1] = 0
res_arr[0, 0, 1, 1] = 0
res_arr[0, 1, 1, 1] = 0
res_arr[1, 1, 1, 1] = 0
res_arr[1, 1, 1, 0] = 0
res_arr[1, 1, 0, 0] = 0
res_arr[1, 0, 0, 0] = 0
res_arr[1, 0, 1, 0] = 0
res_arr[0, 1, 0, 1] = 0
res_arr[1, 1, 0, 1] = 0
res_arr[0, 0, 1, 0] = 0
res_arr[1, 0, 1, 1] = 0
res_arr[0, 1, 0, 0] = 0
res_arr[0, 1, 1, 0] = 0
res_arr[1, 0, 0, 1] = 0
res_ten = res_arr.to_tensor((1, -1, -1, -1))
print('test_riemann_li_t <=== actual test code')
assert riemann_li(C, g, var_list) == res_ten
assert isinstance(riemann_li(C, g, var_list), TensorArray)
assert riemann_li(C, g, var_list).type_pq == (1, 3)
assert riemann_li(C, g, var_list, 't') == res_ten
assert isinstance(riemann_li(C, g, var_list, 't'), TensorArray)
assert riemann_li(C, g, var_list, 't').type_pq == (1, 3)
print('test_riemann_li_a <=== actual test code')
assert riemann_li(C, g, var_list, 'a') == res_arr
assert isinstance(riemann_li(C, g, var_list, 'a'), Arraypy)
def test_kulkarni_nomizu():
x1, x2 = symbols('x1, x2')
var_list = [x1, x2]
h = Arraypy((2, 2)).to_tensor((-1, -1))
h[0, 0] = x1
h[0, 1] = 0
h[1, 0] = 0
h[1, 1] = x2
k = Arraypy((2, 2)).to_tensor((-1, -1))
k[0, 0] = x2
k[0, 1] = 0
k[1, 0] = 0
k[1, 1] = x1
res_arr = Arraypy([4, 2, 0])
res_arr[0, 0, 0, 0] = 0
res_arr[0, 0, 0, 1] = 0
res_arr[0, 0, 1, 1] = 0
res_arr[0, 1, 1, 1] = 0
res_arr[1, 1, 1, 1] = 0
res_arr[1, 1, 1, 0] = 0
res_arr[1, 1, 0, 0] = 0
res_arr[1, 0, 0, 0] = 0
res_arr[1, 0, 1, 0] = x1**2 + x2**2
res_arr[0, 1, 0, 1] = x1**2 + x2**2
res_arr[1, 1, 0, 1] = 0
res_arr[0, 0, 1, 0] = 0
res_arr[1, 0, 1, 1] = 0
res_arr[0, 1, 0, 0] = 0
res_arr[0, 1, 1, 0] = -x1**2 - x2**2
res_arr[1, 0, 0, 1] = -x1**2 - x2**2
res_ten = res_arr.to_tensor((-1, -1, -1, -1))
print('test_kulkarni_nomizu_t <=== actual test code')
assert kulkarni_nomizu(h, k, var_list) == res_ten
assert isinstance(kulkarni_nomizu(h, k, var_list), TensorArray)
assert kulkarni_nomizu(h, k, var_list).type_pq == (0, 4)
assert kulkarni_nomizu(h, k, var_list, 't') == res_ten
assert isinstance(kulkarni_nomizu(h, k, var_list, 't'), TensorArray)
assert kulkarni_nomizu(h, k, var_list, 't').type_pq == (0, 4)
print('test_kulkarni_nomizu_a <=== actual test code')
assert kulkarni_nomizu(h, k, var_list, 'a') == res_arr
assert isinstance(kulkarni_nomizu(h, k, var_list, 'a'), Arraypy)
def test_k_sigma():
x1, x2 = symbols('x1, x2')
var_list = [x1, x2]
g = Matrix([[cos(x2)**2, 0], [0, 1]])
X = [1, 2]
Y = [3, 4]
g_ten = Arraypy([2, 2, 1]).to_tensor((-1, -1))
g_ten[1, 1] = cos(x2)**2
g_ten[1, 2] = 0
g_ten[2, 1] = 0
g_ten[2, 2] = 1
g_ten0 = Arraypy([2, 2, 0]).to_tensor((-1, -1))
g_ten0[0, 0] = cos(x2)**2
g_ten0[0, 1] = 0
g_ten0[1, 0] = 0
g_ten0[1, 1] = 1
riemann_arr = Arraypy([4, 2, 0])
riemann_arr[0, 0, 0, 0] = 0
riemann_arr[0, 0, 0, 1] = 0
riemann_arr[0, 0, 1, 1] = 0
riemann_arr[0, 1, 1, 1] = 0
riemann_arr[1, 1, 1, 1] = 0
riemann_arr[1, 1, 1, 0] = 0
riemann_arr[1, 1, 0, 0] = 0
riemann_arr[1, 0, 0, 0] = 0
riemann_arr[1, 0, 1, 0] = -1
riemann_arr[0, 1, 0, 1] = -cos(x2)**2
riemann_arr[1, 1, 0, 1] = 0
riemann_arr[0, 0, 1, 0] = 0
riemann_arr[1, 0, 1, 1] = 0
riemann_arr[0, 1, 0, 0] = 0
riemann_arr[0, 1, 1, 0] = 1
riemann_arr[1, 0, 0, 1] = cos(x2)**2
riemann_ten = riemann_arr.to_tensor((-1, -1, -1, 1))
assert k_sigma(X, Y, riemann_ten, g, var_list) == 1
assert k_sigma(X, Y, riemann_ten, g_ten, var_list) == 1
assert k_sigma(X, Y, riemann_ten, g_ten0, var_list) == 1
def test_inner_product():
x1, x2, x3, l, m, n = symbols('x1 x2 x3 l m n')
omega2=Arraypy([2,3,1]).to_tensor((-1,-1))
omega2[1,2]=x3
omega2[1,3]=-x2
omega2[2,1]=-x3
omega2[2,3]=x1
omega2[3,1]=x2
omega2[3,2]=-x1
X_t=Arraypy([1,3,1]).to_tensor((1))
X_t[1]=l
X_t[2]=m
X_t[3]=n
res_ar1 = Arraypy([1, 3, 1])
res_ar1[1] = -m*x3 + n*x2
res_ar1[2] = l*x3 - n*x1
res_ar1[3] = -l*x2 + m*x1
res_ten1 = res_ar1.to_tensor(-1)
assert inner_product(omega2, X_t)==res_ten1
def test_riemann_gm():
x1, x2 = symbols('x1, x2')
var_list = [x1, x2]
g = Matrix([[cos(x2)**2, 0], [0, 1]])
res_arr = Arraypy([4, 2, 0])
res_arr[0, 0, 0, 0] = 0
res_arr[0, 0, 0, 1] = 0
res_arr[0, 0, 1, 1] = 0
res_arr[0, 1, 1, 1] = 0
res_arr[1, 1, 1, 1] = 0
res_arr[1, 1, 1, 0] = 0
res_arr[1, 1, 0, 0] = 0
res_arr[1, 0, 0, 0] = 0
res_arr[1, 0, 1, 0] = -1
res_arr[0, 1, 0, 1] = -cos(x2)**2
res_arr[1, 1, 0, 1] = 0
res_arr[0, 0, 1, 0] = 0
res_arr[1, 0, 1, 1] = 0
res_arr[0, 1, 0, 0] = 0
res_arr[0, 1, 1, 0] = 1
res_arr[1, 0, 0, 1] = cos(x2)**2
res_ten = res_arr.to_tensor((1, -1, -1, -1))
print('test_riemann_gm_t <=== actual test code')
assert riemann(g, var_list) == res_ten
assert isinstance(riemann(g, var_list), TensorArray)
assert riemann(g, var_list).type_pq == (1, 3)
assert riemann(g, var_list, 't') == res_ten
assert isinstance(riemann(g, var_list, 't'), TensorArray)
assert riemann(g, var_list, 't').type_pq == (1, 3)
print('test_riemann_gm_a <=== actual test code')
assert riemann(g, var_list, 'a') == res_arr
assert isinstance(riemann(g, var_list, 'a'), Arraypy)
def curl(X, args, output_type=None):
"""Return the rotor vector field curl(X) of a vector field X in R^3
(curl, rotation, rotor, vorticity).
A rotor can be calculated for only in three-dimensional Euclidean space.
Examples:
=========
>>> from tensor_analysis.tensor_fields import curl
>>> from sympy import symbols, cos
>>> from tensor_analysis.arraypy import Arraypy, TensorArray
>>> x1, x2, x3 = symbols('x1 x2 x3')
X is a vector field, args it's a list of symbol arguments of a vector field
X. They can be a list, array arraypy or contravariant tensor:
>>> X=Arraypy(3)
>>> X_t=TensorArray(X,(1))
>>> X_t[0]=x1*x2**3
>>> X_t[1]=x2-cos(x3)
>>> X_t[2]=x3**3-x1
>>> arg=[x1,x2,x3]
>>> r=curl(X_t,arg,'t')
>>> print(r)
-sin(x3) 1 -3*x1*x2**2
>>> r.type_pq
(1, 0)
"""
# Handling of a vector of arguments
check_vector_of_arguments(args)
if len(args) != 3:
raise ValueError("Three variables are required")
# The definition of the start index
if isinstance(args, list):
idx_args = 0
else:
idx_args = args.start_index[0]
# Handling of a vector field
check_the_vector_field(X)
if len(X) != 3:
raise ValueError("A three-dimensional vector field is necessary")
# The definition of the start index and type of output
if isinstance(X, (TensorArray, Arraypy)):
if isinstance(X, TensorArray):
out_t = 't'
idx_X = X.start_index[0]
else:
idx_X = 0
out_t = 'l'
# The definition type of output of an output array
if output_type is None:
if out_t is not None:
output_type = out_t
else:
output_type = 'a'
# The definition of the start index
if isinstance(X, type(args)) and (idx_X != idx_args):
raise ValueError(
"The start index of vector field and vector of arguments must be \
equal")
idx_st = idx_X
# Creating the output array in accordance with the start index
array = Arraypy([1, 3, idx_st])
# Calculation
if isinstance(X, (TensorArray, Arraypy)):
X = X.to_list()
if isinstance(args, (TensorArray, Arraypy)):
args = args.to_list()
array[idx_st] = (diff(X[2], args[1]) - diff(X[1], args[2]))
array[idx_st + 1] = diff(X[0], args[2]) - diff(X[2], args[0])
array[idx_st + 2] = diff(X[1], args[0]) - diff(X[0], args[1])
# Handling of an output array
if output_type == 't' or output_type == Symbol('t'):
rotor = Arraypy.to_tensor(array, 1)
elif output_type == 'a' or output_type == Symbol('a'):
rotor = array
elif output_type == 'l' or output_type == Symbol('l'):
rotor = Arraypy.to_list(array)
else:
raise TypeError(
"The third argument must be 't'-tensor,'a'-Arraypy, \
'l'-list")
# Output
return rotor
def lie_xy(X, Y, args, output_type=None):
"""Return the vector field [X,Y], Lie bracket (commutator) of a vector
fields X and Y.
Examples:
=========
>>> from tensor_analysis.tensor_fields import lie_xy
>>> from sympy import symbols, cos, sin
>>> x1, x2, x3 = symbols('x1 x2 x3')
X, Y is a vector field, args it's a list of symbol arguments.
It's can be a list, array arraypy or contravariant tensor:
>>> X=[x1*x2**3,x2-cos(x3),x3**3-x1]
>>> Y = [x1**3*x2**3, x2*x3 - sin(x1*x3), x3**3 - x1**2]
>>> arg = [x1, x2, x3]
The Lie brackets of two vector fields:
>>> lie = lie_xy(X, Y, arg,'a')
>>> print(lie)
2*x1**3*x2**6 + 3*x1**3*x2**2*(x2 - cos(x3)) - 3*x1*x2**2*(x2*x3 - \
sin(x1*x3))
-x1*x2**3*x3*cos(x1*x3) - x2*x3 + x3*(x2 - cos(x3)) + \
(-x1 + x3**3)*(-x1*cos(x1*x3) + x2) - (-x1**2 + x3**3)*sin(x3) + sin(x1*x3)
x1**3*x2**3 - 2*x1**2*x2**3 + 3*x3**2*(-x1 + x3**3) - \
3*x3**2*(-x1**2 + x3**3)
"""
# Handling of a vector of arguments
check_vector_of_arguments(args)
# The definition of the start index args
if isinstance(args, list):
idx_args = 0
else:
idx_args = args.start_index[0]
# Handling of the first vector field
check_the_vector_field(X)
# The definition of the start index X and type of output
if isinstance(X, (TensorArray, Arraypy)):
if isinstance(X, TensorArray):
out_t = 't'
idx_X = X.start_index[0]
else:
idx_X = 0
out_t = 'l'
# Handling of the second vector field
check_the_vector_field(Y)
# The definition of the start index Y
if isinstance(Y, (TensorArray, Arraypy)):
idx_Y = Y.start_index[0]
else:
idx_Y = 0
if len(Y) != len(X):
raise ValueError(
"The different number of arguments in the vector fields")
elif len(args) != len(X) or len(args) != len(Y):
raise ValueError(
"The different number of components in the vector field and \
vector of arguments")
# Define the start index in the output tensor
if type(Y) == type(X) == type(args):
if idx_Y != idx_X or idx_Y != idx_args or idx_X != idx_args:
raise ValueError(
"The start index of vector fields and vector of argements \
must be equal")
if idx_Y != idx_X:
raise ValueError("The start index of vector fields must be equal")
idx_st = idx_Y
if output_type is None:
if out_t is not None:
output_type = out_t
else:
output_type = 'a'
# Creating the output array in accordance with start indexes
Li = Arraypy([1, len(X), idx_st])
# Calculating
if isinstance(Y, (TensorArray, Arraypy)):
Y = Y.to_list()
if isinstance(X, (TensorArray, Arraypy)):
X = X.to_list()
if isinstance(args, (TensorArray, Arraypy)):
args = args.to_list()
if X == Y:
return 0
else:
indices = range(len(args))
for i in indices:
for k in indices:
Li[i + idx_st] += Add(
diff(Y[i], args[k]) * X[k] - diff(X[i], args[k]) * Y[k])
# Handling of an output array
if output_type == 't' or output_type == Symbol('t'):
Lie = Arraypy.to_tensor(Li, 1)
elif output_type == 'a' or output_type == Symbol('a'):
Lie = Li
elif output_type == 'l' or output_type == Symbol('l'):
Lie = Arraypy.to_list(Li)
else:
raise TypeError(
"The third argument must be 't'-TensorArray,'a'-Arraypy, \
'l'-list")
# Output
return Lie
def df(f, args, output_type='l'):
"""Return an the 1-form df, differential of function f.
Examples:
=========
>>> from tensor_analysis.tensor_fields import df
>>> from sympy import symbols, sin
>>> from tensor_analysis.arraypy import Arraypy
>>> x1, x2, x3= symbols('x1 x2 x3')
f it's a function the differential of that is calculated:
>>> f=x1**2*x2 + sin(x2*x3 - x2)
args it's a list of symbol arguments of the function f. It can be in list,
array of arraypy or contravariant tensor:
>>> args_t=Arraypy([1,3,1]).to_tensor(1)
>>> args_t[1]=x1
>>> args_t[2]=x2
>>> args_t[3]=x3
output_type it is an optional parameter accepting symbol value of
'l', 'a' or 't' and indicative on the type of result of calculations:
- 'l' it is a result as a list(list);
- 'a' it is a result as an unidimensional array of arraypy;
- 't' it is a result as an unidimensional covariant tensor.
Differential:
>>> d = df(f, args_t, 't')
>>> print(d)
2*x1*x2 x1**2 + (x3 - 1)*cos(x2*x3 - x2) x2*cos(x2*x3 - x2)
The valence of the returned tensor:
>>> d.type_pq
(0, 1)
"""
# Handling of a vector of arguments
check_vector_of_arguments(args)
# The definition of the start index
if isinstance(args, list):
idx_start = 0
else:
idx_start = args.start_index[0]
# Creating the output array in accordance with the start index
n = len(args)
array = Arraypy([1, n, idx_start])
# Calculation
for k in range(idx_start, idx_start + n):
array[k] = diff(f, args[k])
# Handling of an output array
if output_type == 't' or output_type == Symbol('t'):
differential = Arraypy.to_tensor(array, -1)
elif output_type == 'a' or output_type == Symbol('a'):
differential = array
elif output_type == 'l' or output_type == Symbol('l'):
differential = Arraypy.to_list(array)
else:
raise TypeError(
"The third argument must be 't' - TensorArray,'a' - Arraypy, \
'l' - list")
# Output
return differential
def lie_w(omega, X, args):
"""Return a skew-symmetric tensor of type (0,p).
Function lie_w calculates all the components of the Lie derivative
differential forms in a symbolic form.
Indexes the output tensor will start as well as the input tensor (array)
"omega". If all the input parameters of the same type, they must be equal
to the initial indexes.
Examples:
=========
>>> from tensor_analysis.tensor_fields import lie_w
>>> from sympy import symbols, cos
>>> from tensor_analysis.arraypy import Arraypy
>>> x1, x2, x3 = symbols('x1 x2 x3')
omega - skew-symmetric tensor. Can be a tensor of type (0,p) or an array
arraypy:
>>> omega=Arraypy([2,3,1]).to_tensor((-1,-1))
>>> omega[1,2]=x3
>>> omega[1,3]=-x2
>>> omega[2,1]=-x3
>>> omega[2,3]=x1
>>> omega[3,1]=x2
>>> omega[3,2]=-x1
X - the vector field along which the derivative is calculated:
>>> X = [x1*x2**3,x2-cos(x3),x3**3-x1]
args it's a list of symbol arguments.
It's can be in list, array of arraypy or contravariant tensor:
>>> arg = [x1, x2, x3]
Lie derivative of a differential form:
>>> li = lie_w(omega,X,arg)
>>> print(li)
0 x2**3*x3 + x3**3 + x3 -x2**4 - 3*x2*x3**2 - x2 + x3*sin(x3) + cos(x3)
-x2**3*x3 - x3**3 - x3 0 -2*x1*x2**3 + 3*x1*x3**2 + x1
x2**4 + 3*x2*x3**2 + x2 - x3*sin(x3) - cos(x3) 2*x1*x2**3 - 3*x1*x3**2 - x1 0
>>> li.type_pq
(0, 2)
"""
# Handling of a vector of arguments
check_vector_of_arguments(args)
if isinstance(args, list):
idx_args = 0
else:
idx_args = args.start_index[0]
# Handling of a vector field
check_the_vector_field(X)
if isinstance(X, (TensorArray, Arraypy)):
idx_X = X.start_index[0]
else:
idx_X = 0
# Handling of a differential form
if not isinstance(omega, (TensorArray, Arraypy)):
raise ValueError(
"The type of differential form must be TensorArray or Arraypy")
if not is_asymmetric(omega):
raise ValueError("The differential form must be a skew-symmetric")
idx_omega = omega.start_index[0]
# Define the start index in the output tensor
if type(omega) == type(X) == type(args):
if idx_omega != idx_X or idx_omega != idx_args or idx_X != idx_args:
raise ValueError(
"The start index of differential form, vector field and \
vector of argements must be equal")
if isinstance(omega, type(X)) and idx_omega != idx_X:
raise ValueError(
"The start index of differential form and vector field must be \
equal")
idx_st = idx_omega
# Creating the output array in accordance with start indexes
n = omega.shape[0] # the dimensionality of the input array
r = len(omega.shape) # the rank of the input array
a = Arraypy([r, n, idx_st])
valence_list = [(-1) for k in range(r)]
diff_Lie = a.to_tensor(valence_list)
# Calculation
idx = diff_Lie.start_index
if isinstance(args, (TensorArray, Arraypy)):
args = args.to_list()
if isinstance(X, (TensorArray, Arraypy)):
X = X.to_list()
l_idx = len(idx)
for p in range(len(diff_Lie)):
for k in range(l_idx + 1):
tuple_list_indx = [
replace_index_to_k(idx, f, k + idx_st) for f in range(l_idx)
]
# the intermediate value
diff_omega = diff(omega[idx], args[k]) * X[k]
for j in range(l_idx):
diff_Lie[idx] += diff(X[k], args[idx[j] - idx_st]) *\
omega[tuple_list_indx[j]]
diff_Lie[idx] = diff_Lie[idx] + diff_omega
idx = diff_Lie.next_index(idx)
# Output
return diff_Lie
def inner_product(w, X):
"""Calculation of the inner product of the form and the vector field.
Examples:
=========
>>> from sympy import symbols
>>> from tensor_analysis.arraypy import Arraypy
>>> from tensor_analysis.tensor_fields import inner_product
>>> x1, x2, x3, l, m, n = symbols('x1 x2 x3 l m n')
>>> omega2=Arraypy([2,3,1]).to_tensor((-1,-1))
>>> omega2[1,2]=x3
>>> omega2[1,3]=-x2
>>> omega2[2,1]=-x3
>>> omega2[2,3]=x1
>>> omega2[3,1]=x2
>>> omega2[3,2]=-x1
>>> X_t=Arraypy([1,3,1]).to_tensor((1))
>>> X_t[1]=l
>>> X_t[2]=m
>>> X_t[3]=n
>>> print(inner_product(omega2, X_t))
-m*x3 + n*x2 l*x3 - n*x1 -l*x2 + m*x1
"""
# Handling of a differential form
if not isinstance(w, (TensorArray, Arraypy)):
raise ValueError(
"The type of differential form must be TensorArray or Arraypy")
# The valency (0,q)
if (len(w.type_pq) == 1 and w.type_pq[0] != 1) or (len(w.type_pq) > 1
and w.type_pq[0] != 0):
raise ValueError("The valency of tensor must be (0,q)")
if not is_asymmetric(w):
raise ValueError("The form must be a skew-symmetric")
# Handling of a vector field
check_the_vector_field(X)
if isinstance(X, (TensorArray, Arraypy)):
idx_start_X = X.start_index[0]
else:
idx_start_X = 0
# Define the start index in the output form
idx_start_w = w.start_index[0]
if idx_start_w != idx_start_X:
raise ValueError(
"The start index of tensor and vector field must be equal")
# Creating the output array in accordance with start indexes
n = w.shape[0] # the dimension of the input array
q = len(w.shape) # the rank of the input array
if (q != 1):
result_tensor = Arraypy([q - 1, n, idx_start_w])
# all indices are -1
valence_list = [(-1) for k in range(q - 1)]
# result tensor
int_prod = result_tensor.to_tensor(valence_list)
for index in int_prod.index_list:
# the sum on k.k is [0,n-1], n is len(X)
for k in range(len(X)):
idx = list(index)
idx.insert(0, k + idx_start_w)
int_prod[index] += w[tuple(idx)] * X[k + idx_start_w]
if q == 1:
# int_prod is number(symbolic expression)
int_prod = 0
for index in w.index_list:
for k in range(len(X)):
int_prod += w[index] * X[k + idx_start_w]
# Output
return int_prod
def test_lie_xy():
x1, x2, x3, t, l, a = symbols('x1 x2 x3 t l a')
X = [x1 * x2**3, x2 - cos(x3), x3**3 - x1]
Y = [x1**3 * x2**3, x2 * x3 - sin(x1 * x3), x3**3 - x1**2]
arg = [x1, x2, x3]
res_ar = Arraypy([1, 3, 0])
res_ar[0] = 2 * x1**3 * x2**6 + 3 * x1**3 * x2**2 * \
(x2 - cos(x3)) - 3 * x1 * x2**2 * (x2 * x3 - sin(x1 * x3))
res_ar[1] = -x1 * x2**3 * x3 * cos(x1 * x3) - x2 * x3 + x3 * (x2 - cos(
x3)) + (-x1 + x3**3) * (-x1 * cos(x1 * x3) + x2) - (-x1**2 + x3**3) * \
sin(x3) + sin(x1 * x3)
res_ar[2] = x1**3 * x2**3 - 2 * x1**2 * x2**3 + 3 * \
x3**2 * (-x1 + x3**3) - 3 * x3**2 * (-x1**2 + x3**3)
res_ten = res_ar.to_tensor(1)
assert lie_xy(X, Y, arg, 'l') == [2 *
x1**3 *
x2**6 +
3 *
x1**3 *
x2**2 *
(x2 -
cos(x3)) -
3 *
x1 *
x2**2 *
(x2 *
x3 -
sin(x1 *
x3)), -
x1 *
x2**3 *
x3 *
cos(x1 *
x3) -
x2 *
x3 +
x3 *
(x2 -
cos(x3)) +
(-
x1 +
x3**3) *
(-
x1 *
cos(x1 *
x3) +
x2) -
(-
x1**2 +
x3**3) *
sin(x3) +
sin(x1 *
x3), x1**3 *
x2**3 -
2 *
x1**2 *
x2**3 +
3 *
x3**2 *
(-
x1 +
x3**3) -
3 *
x3**2 *
(-
x1**2 +
x3**3)]
assert isinstance(lie_xy(X, Y, arg, 'l'), list)
assert lie_xy(X, Y, arg, 'a') == res_ar
assert isinstance(lie_xy(X, Y, arg, 'a'), Arraypy)
assert lie_xy(X, Y, arg, 't') == res_ten
assert isinstance(lie_xy(X, Y, arg, 't'), TensorArray)
assert lie_xy(X, Y, arg, 't').type_pq == (1, 0)
def inner_product(w, X):
"""Calculation of the inner product of the form and the vector field.
Examples:
=========
>>> from sympy import symbols
>>> from tensor_analysis.arraypy import Arraypy
>>> from tensor_analysis.tensor_fields import inner_product
>>> x1, x2, x3, l, m, n = symbols('x1 x2 x3 l m n')
>>> omega2=Arraypy([2,3,1]).to_tensor((-1,-1))
>>> omega2[1,2]=x3
>>> omega2[1,3]=-x2
>>> omega2[2,1]=-x3
>>> omega2[2,3]=x1
>>> omega2[3,1]=x2
>>> omega2[3,2]=-x1
>>> X_t=Arraypy([1,3,1]).to_tensor((1))
>>> X_t[1]=l
>>> X_t[2]=m
>>> X_t[3]=n
>>> print(inner_product(omega2, X_t))
-m*x3 + n*x2 l*x3 - n*x1 -l*x2 + m*x1
"""
# Handling of a differential form
if not isinstance(w, (TensorArray, Arraypy)):
raise ValueError(
"The type of differential form must be TensorArray or Arraypy")
# The valency (0,q)
if (len(w.type_pq) == 1 and w.type_pq[0] != 1) or (
len(w.type_pq) > 1 and w.type_pq[0] != 0):
raise ValueError("The valency of tensor must be (0,q)")
if not is_asymmetric(w):
raise ValueError("The form must be a skew-symmetric")
# Handling of a vector field
check_the_vector_field(X)
if isinstance(X, (TensorArray, Arraypy)):
idx_start_X = X.start_index[0]
else:
idx_start_X = 0
# Define the start index in the output form
idx_start_w = w.start_index[0]
if idx_start_w != idx_start_X:
raise ValueError(
"The start index of tensor and vector field must be equal")
# Creating the output array in accordance with start indexes
n = w.shape[0] # the dimension of the input array
q = len(w.shape) # the rank of the input array
if (q != 1):
result_tensor = Arraypy([q - 1, n, idx_start_w])
# all indices are -1
valence_list = [(-1) for k in range(q - 1)]
# result tensor
int_prod = result_tensor.to_tensor(valence_list)
for index in int_prod.index_list:
# the sum on k.k is [0,n-1], n is len(X)
for k in range(len(X)):
idx = list(index)
idx.insert(0, k + idx_start_w)
int_prod[index] += w[tuple(idx)] * X[k + idx_start_w]
if q == 1:
# int_prod is number(symbolic expression)
int_prod = 0
for index in w.index_list:
for k in range(len(X)):
int_prod += w[index] * X[k + idx_start_w]
# Output
return int_prod
def lie_w(omega, X, args):
"""Return a skew-symmetric tensor of type (0,p).
Function lie_w calculates all the components of the Lie derivative
differential forms in a symbolic form.
Indexes the output tensor will start as well as the input tensor (array)
"omega". If all the input parameters of the same type, they must be equal
to the initial indexes.
Examples:
=========
>>> from tensor_analysis.tensor_fields import lie_w
>>> from sympy import symbols, cos
>>> from tensor_analysis.arraypy import Arraypy
>>> x1, x2, x3 = symbols('x1 x2 x3')
omega - skew-symmetric tensor. Can be a tensor of type (0,p) or an array
arraypy:
>>> omega=Arraypy([2,3,1]).to_tensor((-1,-1))
>>> omega[1,2]=x3
>>> omega[1,3]=-x2
>>> omega[2,1]=-x3
>>> omega[2,3]=x1
>>> omega[3,1]=x2
>>> omega[3,2]=-x1
X - the vector field along which the derivative is calculated:
>>> X = [x1*x2**3,x2-cos(x3),x3**3-x1]
args it's a list of symbol arguments.
It's can be in list, array of arraypy or contravariant tensor:
>>> arg = [x1, x2, x3]
Lie derivative of a differential form:
>>> li = lie_w(omega,X,arg)
>>> print(li)
0 x2**3*x3 + x3**3 + x3 -x2**4 - 3*x2*x3**2 - x2 + x3*sin(x3) + cos(x3)
-x2**3*x3 - x3**3 - x3 0 -2*x1*x2**3 + 3*x1*x3**2 + x1
x2**4 + 3*x2*x3**2 + x2 - x3*sin(x3) - cos(x3) 2*x1*x2**3 - 3*x1*x3**2 - x1 0
>>> li.type_pq
(0, 2)
"""
# Handling of a vector of arguments
check_vector_of_arguments(args)
if isinstance(args, list):
idx_args = 0
else:
idx_args = args.start_index[0]
# Handling of a vector field
check_the_vector_field(X)
if isinstance(X, (TensorArray, Arraypy)):
idx_X = X.start_index[0]
else:
idx_X = 0
# Handling of a differential form
if not isinstance(omega, (TensorArray, Arraypy)):
raise ValueError(
"The type of differential form must be TensorArray or Arraypy")
if not is_asymmetric(omega):
raise ValueError("The differential form must be a skew-symmetric")
idx_omega = omega.start_index[0]
# Define the start index in the output tensor
if type(omega) == type(X) == type(args):
if idx_omega != idx_X or idx_omega != idx_args or idx_X != idx_args:
raise ValueError(
"The start index of differential form, vector field and \
vector of argements must be equal")
if isinstance(omega, type(X)) and idx_omega != idx_X:
raise ValueError(
"The start index of differential form and vector field must be \
equal")
idx_st = idx_omega
# Creating the output array in accordance with start indexes
n = omega.shape[0] # the dimensionality of the input array
r = len(omega.shape) # the rank of the input array
a = Arraypy([r, n, idx_st])
valence_list = [(-1) for k in range(r)]
diff_Lie = a.to_tensor(valence_list)
# Calculation
idx = diff_Lie.start_index
if isinstance(args, (TensorArray, Arraypy)):
args = args.to_list()
if isinstance(X, (TensorArray, Arraypy)):
X = X.to_list()
l_idx = len(idx)
for p in range(len(diff_Lie)):
for k in range(l_idx + 1):
tuple_list_indx = [
replace_index_to_k(idx, f, k + idx_st) for f in range(l_idx)]
# the intermediate value
diff_omega = diff(omega[idx], args[k]) * X[k]
for j in range(l_idx):
diff_Lie[idx] += diff(X[k], args[idx[j] - idx_st]) *\
omega[tuple_list_indx[j]]
diff_Lie[idx] = diff_Lie[idx] + diff_omega
idx = diff_Lie.next_index(idx)
# Output
return diff_Lie
def df(f, args, output_type='l'):
"""Return an the 1-form df, differential of function f.
Examples:
=========
>>> from tensor_analysis.tensor_fields import df
>>> from sympy import symbols, sin
>>> from tensor_analysis.arraypy import Arraypy
>>> x1, x2, x3= symbols('x1 x2 x3')
f it's a function the differential of that is calculated:
>>> f=x1**2*x2 + sin(x2*x3 - x2)
args it's a list of symbol arguments of the function f. It can be in list,
array of arraypy or contravariant tensor:
>>> args_t=Arraypy([1,3,1]).to_tensor(1)
>>> args_t[1]=x1
>>> args_t[2]=x2
>>> args_t[3]=x3
output_type it is an optional parameter accepting symbol value of
'l', 'a' or 't' and indicative on the type of result of calculations:
- 'l' it is a result as a list(list);
- 'a' it is a result as an unidimensional array of arraypy;
- 't' it is a result as an unidimensional covariant tensor.
Differential:
>>> d = df(f, args_t, 't')
>>> print(d)
2*x1*x2 x1**2 + (x3 - 1)*cos(x2*x3 - x2) x2*cos(x2*x3 - x2)
The valence of the returned tensor:
>>> d.type_pq
(0, 1)
"""
# Handling of a vector of arguments
check_vector_of_arguments(args)
# The definition of the start index
if isinstance(args, list):
idx_start = 0
else:
idx_start = args.start_index[0]
# Creating the output array in accordance with the start index
n = len(args)
array = Arraypy([1, n, idx_start])
# Calculation
for k in range(idx_start, idx_start + n):
array[k] = diff(f, args[k])
# Handling of an output array
if output_type == 't' or output_type == Symbol('t'):
differential = Arraypy.to_tensor(array, -1)
elif output_type == 'a' or output_type == Symbol('a'):
differential = array
elif output_type == 'l' or output_type == Symbol('l'):
differential = Arraypy.to_list(array)
else:
raise TypeError(
"The third argument must be 't' - TensorArray,'a' - Arraypy, \
'l' - list")
# Output
return differential
def curl(X, args, output_type=None):
"""Return the rotor vector field curl(X) of a vector field X in R^3
(curl, rotation, rotor, vorticity).
A rotor can be calculated for only in three-dimensional Euclidean space.
Examples:
=========
>>> from tensor_analysis.tensor_fields import curl
>>> from sympy import symbols, cos
>>> from tensor_analysis.arraypy import Arraypy, TensorArray
>>> x1, x2, x3 = symbols('x1 x2 x3')
X is a vector field, args it's a list of symbol arguments of a vector field
X. They can be a list, array arraypy or contravariant tensor:
>>> X=Arraypy(3)
>>> X_t=TensorArray(X,(1))
>>> X_t[0]=x1*x2**3
>>> X_t[1]=x2-cos(x3)
>>> X_t[2]=x3**3-x1
>>> arg=[x1,x2,x3]
>>> r=curl(X_t,arg,'t')
>>> print(r)
-sin(x3) 1 -3*x1*x2**2
>>> r.type_pq
(1, 0)
"""
# Handling of a vector of arguments
check_vector_of_arguments(args)
if len(args) != 3:
raise ValueError("Three variables are required")
# The definition of the start index
if isinstance(args, list):
idx_args = 0
else:
idx_args = args.start_index[0]
# Handling of a vector field
check_the_vector_field(X)
if len(X) != 3:
raise ValueError("A three-dimensional vector field is necessary")
# The definition of the start index and type of output
if isinstance(X, (TensorArray, Arraypy)):
if isinstance(X, TensorArray):
out_t = 't'
idx_X = X.start_index[0]
else:
idx_X = 0
out_t = 'l'
# The definition type of output of an output array
if output_type is None:
if out_t is not None:
output_type = out_t
else:
output_type = 'a'
# The definition of the start index
if isinstance(X, type(args)) and (idx_X != idx_args):
raise ValueError(
"The start index of vector field and vector of arguments must be \
equal")
idx_st = idx_X
# Creating the output array in accordance with the start index
array = Arraypy([1, 3, idx_st])
# Calculation
if isinstance(X, (TensorArray, Arraypy)):
X = X.to_list()
if isinstance(args, (TensorArray, Arraypy)):
args = args.to_list()
array[idx_st] = (diff(X[2], args[1]) - diff(X[1], args[2]))
array[idx_st + 1] = diff(X[0], args[2]) - diff(X[2], args[0])
array[idx_st + 2] = diff(X[1], args[0]) - diff(X[0], args[1])
# Handling of an output array
if output_type == 't' or output_type == Symbol('t'):
rotor = Arraypy.to_tensor(array, 1)
elif output_type == 'a' or output_type == Symbol('a'):
rotor = array
elif output_type == 'l' or output_type == Symbol('l'):
rotor = Arraypy.to_list(array)
else:
raise TypeError("The third argument must be 't'-tensor,'a'-Arraypy, \
'l'-list")
# Output
return rotor
def grad(f, args, g=None, output_type=None):
"""Return the vector field gradient of a function f(x).
Examples:
=========
>>> from tensor_analysis.tensor_fields import grad
>>> from sympy import symbols, sin
>>> from tensor_analysis.arraypy import Arraypy
>>> x1, x2, x3 = symbols('x1 x2 x3')
f it's a function the differential of that is calculated:
>>> f=x1**2*x2 + sin(x2*x3 - x2)
args it's a list of symbol arguments of function of f.
It can be in list, array of arraypy or contravariant tensor:
>>> args=[x1,x2,x3]
g - optional parameter, metric tensor, which can be a matrix "Matrix",
array of arraypy or covariant tensor:
>>> g=Arraypy([2,3,1])
>>> g_t=g.to_tensor((-1,-1))
>>> g_t[1,1]=2
>>> g_t[1,2]=1
>>> g_t[1,3]=0
>>> g_t[2,1]=1
>>> g_t[2,2]=3
>>> g_t[2,3]=0
>>> g_t[3,1]=0
>>> g_t[3,2]=0
>>> g_t[3,3]=1
output _ type it is an optional parameter accepting symbol value of
'l', 'a' or 't' and indicative on the type of result of calculations:
- 'l' it is a result as a list(list);
- 'a' it is a result as an unidimensional array of arraypy;
- 't' it is a result as an unidimensional covariant tensor.
Gradient:
>>> gr=grad(f,args,g_t,'a')
>>> print(gr)
-x1**2/5 + 6*x1*x2/5 - (x3 - 1)*cos(x2*x3 - x2)/5 2*x1**2/5 - 2*x1*x2/5 + \
2*(x3 - 1)*cos(x2*x3 - x2)/5 x2*cos(x2*x3 - x2)
"""
# Handling of a vector of arguments
check_vector_of_arguments(args)
# The definition of the start index
if isinstance(args, list):
idx_args = 0
else:
idx_args = args.start_index[0]
# Handling of the metric tensor
# 1. if g is not NULL
if g is not None:
if output_type is None:
output_type = 't'
check_metric_tensor(g)
# The definition of the start index
if isinstance(g, Matrix):
idx_st = 0
else:
idx_st = g.start_index[0]
# The start index is the same
if isinstance(g, type(args)) and idx_st != idx_args:
raise ValueError(
"The start index of the metric tensor and vector of arguments \
must be equal")
if isinstance(g, (TensorArray, Arraypy)):
g = g.to_matrix()
# 2.if g is NULL
else:
# g - the identity matrix
g = eye(len(args))
idx_st = 0
# Creating the output array in accordance with start indexes
n = len(args)
array = Arraypy([1, n, idx_st])
indices = range(idx_st, idx_st + n)
# Calculating
g_inv = g.inv()
if isinstance(args, (TensorArray, Arraypy)):
args = args.to_list()
for i in indices:
for j in indices:
array[i] += (g_inv[i - idx_st, j - idx_st] * diff(f, args[j -
idx_st]))
# Handling of an output array
if output_type == 't' or output_type == Symbol('t'):
gradient = Arraypy.to_tensor(array, 1)
elif output_type == 'a' or output_type == Symbol('a'):
gradient = array
elif output_type == 'l' or output_type == Symbol('l') or output_type is \
None:
gradient = Arraypy.to_list(array)
else:
raise TypeError(
"The third argument must be 't' - tensor,'a' - Arraypy, \
'l' - list")
# Output
return gradient
def lie_xy(X, Y, args, output_type=None):
"""Return the vector field [X,Y], Lie bracket (commutator) of a vector
fields X and Y.
Examples:
=========
>>> from tensor_analysis.tensor_fields import lie_xy
>>> from sympy import symbols, cos, sin
>>> x1, x2, x3 = symbols('x1 x2 x3')
X, Y is a vector field, args it's a list of symbol arguments.
It's can be a list, array arraypy or contravariant tensor:
>>> X=[x1*x2**3,x2-cos(x3),x3**3-x1]
>>> Y = [x1**3*x2**3, x2*x3 - sin(x1*x3), x3**3 - x1**2]
>>> arg = [x1, x2, x3]
The Lie brackets of two vector fields:
>>> lie = lie_xy(X, Y, arg,'a')
>>> print(lie)
2*x1**3*x2**6 + 3*x1**3*x2**2*(x2 - cos(x3)) - 3*x1*x2**2*(x2*x3 - \
sin(x1*x3))
-x1*x2**3*x3*cos(x1*x3) - x2*x3 + x3*(x2 - cos(x3)) + \
(-x1 + x3**3)*(-x1*cos(x1*x3) + x2) - (-x1**2 + x3**3)*sin(x3) + sin(x1*x3)
x1**3*x2**3 - 2*x1**2*x2**3 + 3*x3**2*(-x1 + x3**3) - \
3*x3**2*(-x1**2 + x3**3)
"""
# Handling of a vector of arguments
check_vector_of_arguments(args)
# The definition of the start index args
if isinstance(args, list):
idx_args = 0
else:
idx_args = args.start_index[0]
# Handling of the first vector field
check_the_vector_field(X)
# The definition of the start index X and type of output
if isinstance(X, (TensorArray, Arraypy)):
if isinstance(X, TensorArray):
out_t = 't'
idx_X = X.start_index[0]
else:
idx_X = 0
out_t = 'l'
# Handling of the second vector field
check_the_vector_field(Y)
# The definition of the start index Y
if isinstance(Y, (TensorArray, Arraypy)):
idx_Y = Y.start_index[0]
else:
idx_Y = 0
if len(Y) != len(X):
raise ValueError(
"The different number of arguments in the vector fields")
elif len(args) != len(X) or len(args) != len(Y):
raise ValueError(
"The different number of components in the vector field and \
vector of arguments")
# Define the start index in the output tensor
if type(Y) == type(X) == type(args):
if idx_Y != idx_X or idx_Y != idx_args or idx_X != idx_args:
raise ValueError(
"The start index of vector fields and vector of argements \
must be equal")
if idx_Y != idx_X:
raise ValueError(
"The start index of vector fields must be equal")
idx_st = idx_Y
if output_type is None:
if out_t is not None:
output_type = out_t
else:
output_type = 'a'
# Creating the output array in accordance with start indexes
Li = Arraypy([1, len(X), idx_st])
# Calculating
if isinstance(Y, (TensorArray, Arraypy)):
Y = Y.to_list()
if isinstance(X, (TensorArray, Arraypy)):
X = X.to_list()
if isinstance(args, (TensorArray, Arraypy)):
args = args.to_list()
if X == Y:
return 0
else:
indices = range(len(args))
for i in indices:
for k in indices:
Li[i +
idx_st] += Add(diff(Y[i], args[k]) *
X[k] -
diff(X[i], args[k]) *
Y[k])
# Handling of an output array
if output_type == 't' or output_type == Symbol('t'):
Lie = Arraypy.to_tensor(Li, 1)
elif output_type == 'a' or output_type == Symbol('a'):
Lie = Li
elif output_type == 'l' or output_type == Symbol('l'):
Lie = Arraypy.to_list(Li)
else:
raise TypeError(
"The third argument must be 't'-TensorArray,'a'-Arraypy, \
'l'-list")
# Output
return Lie
def grad(f, args, g=None, output_type=None):
"""Return the vector field gradient of a function f(x).
Examples:
=========
>>> from tensor_analysis.tensor_fields import grad
>>> from sympy import symbols, sin
>>> from tensor_analysis.arraypy import Arraypy
>>> x1, x2, x3 = symbols('x1 x2 x3')
f it's a function the differential of that is calculated:
>>> f=x1**2*x2 + sin(x2*x3 - x2)
args it's a list of symbol arguments of function of f.
It can be in list, array of arraypy or contravariant tensor:
>>> args=[x1,x2,x3]
g - optional parameter, metric tensor, which can be a matrix "Matrix",
array of arraypy or covariant tensor:
>>> g=Arraypy([2,3,1])
>>> g_t=g.to_tensor((-1,-1))
>>> g_t[1,1]=2
>>> g_t[1,2]=1
>>> g_t[1,3]=0
>>> g_t[2,1]=1
>>> g_t[2,2]=3
>>> g_t[2,3]=0
>>> g_t[3,1]=0
>>> g_t[3,2]=0
>>> g_t[3,3]=1
output _ type it is an optional parameter accepting symbol value of
'l', 'a' or 't' and indicative on the type of result of calculations:
- 'l' it is a result as a list(list);
- 'a' it is a result as an unidimensional array of arraypy;
- 't' it is a result as an unidimensional covariant tensor.
Gradient:
>>> gr=grad(f,args,g_t,'a')
>>> print(gr)
-x1**2/5 + 6*x1*x2/5 - (x3 - 1)*cos(x2*x3 - x2)/5 2*x1**2/5 - 2*x1*x2/5 + \
2*(x3 - 1)*cos(x2*x3 - x2)/5 x2*cos(x2*x3 - x2)
"""
# Handling of a vector of arguments
check_vector_of_arguments(args)
# The definition of the start index
if isinstance(args, list):
idx_args = 0
else:
idx_args = args.start_index[0]
# Handling of the metric tensor
# 1. if g is not NULL
if g is not None:
if output_type is None:
output_type = 't'
check_metric_tensor(g)
# The definition of the start index
if isinstance(g, Matrix):
idx_st = 0
else:
idx_st = g.start_index[0]
# The start index is the same
if isinstance(g, type(args)) and idx_st != idx_args:
raise ValueError(
"The start index of the metric tensor and vector of arguments \
must be equal")
if isinstance(g, (TensorArray, Arraypy)):
g = g.to_matrix()
# 2.if g is NULL
else:
# g - the identity matrix
g = eye(len(args))
idx_st = 0
# Creating the output array in accordance with start indexes
n = len(args)
array = Arraypy([1, n, idx_st])
indices = range(idx_st, idx_st + n)
# Calculating
g_inv = g.inv()
if isinstance(args, (TensorArray, Arraypy)):
args = args.to_list()
for i in indices:
for j in indices:
array[i] += (g_inv[i - idx_st, j - idx_st] *
diff(f, args[j - idx_st]))
# Handling of an output array
if output_type == 't' or output_type == Symbol('t'):
gradient = Arraypy.to_tensor(array, 1)
elif output_type == 'a' or output_type == Symbol('a'):
gradient = array
elif output_type == 'l' or output_type == Symbol('l') or output_type is \
None:
gradient = Arraypy.to_list(array)
else:
raise TypeError(
"The third argument must be 't' - tensor,'a' - Arraypy, \
'l' - list")
# Output
return gradient
