def test_expression_6(self):
function = Function('Tensor')(Symbol('g'))
tensor = Tensor(function, dimension=3, weight=2)
self.assertEqual(
Parser._generate_liedrv(tensor, function, 'beta'),
r'\mathcal{L}_\text{beta} g = \text{beta}^a \partial_a g + (2)(\partial_a \text{beta}^a) g'
)
function = Function('Tensor')(Symbol('gUU'), Symbol('i'), Symbol('j'))
tensor = Tensor(function, 3)
self.assertEqual(
Parser._generate_liedrv(tensor, function, 'beta'),
r'\mathcal{L}_\text{beta} g^{i j} = \text{beta}^a \partial_a g^{i j} - (\partial_a \text{beta}^i) g^{a j} - (\partial_a \text{beta}^j) g^{i a}'
)
function = Function('Tensor')(Symbol('gUD'), Symbol('i'), Symbol('j'))
tensor = Tensor(function, 3)
self.assertEqual(
Parser._generate_liedrv(tensor, function, 'beta'),
r'\mathcal{L}_\text{beta} g^i_j = \text{beta}^a \partial_a g^i_j - (\partial_a \text{beta}^i) g^a_j + (\partial_j \text{beta}^a) g^i_a'
)
function = Function('Tensor')(Symbol('gDD'), Symbol('i'), Symbol('j'))
tensor = Tensor(function, 3)
self.assertEqual(
Parser._generate_liedrv(tensor, function, 'beta'),
r'\mathcal{L}_\text{beta} g_{i j} = \text{beta}^a \partial_a g_{i j} + (\partial_i \text{beta}^a) g_{a j} + (\partial_j \text{beta}^a) g_{i a}'
)
def test_assignment_8(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% vardef -metric 'gDD' (4D)
\gamma_{ij} = g_{ij}
""")), {'gUU', 'gdet', 'epsilonUUUU', 'gDD', 'gammaDD'})
self.assertEqual(
str(gammaDD),
'[[gDD11, gDD12, gDD13], [gDD12, gDD22, gDD23], [gDD13, gDD23, gDD33]]'
)
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% vardef -nosym 'TUU' (3D)
% vardef 'vD' (2D)
% keydef index i (2D)
w^\mu = T^{\mu i} v_i
""")), {'TUU', 'vD', 'wU'})
self.assertEqual(
str(wU),
'[TUU01*vD0 + TUU02*vD1, TUU11*vD0 + TUU12*vD1, TUU21*vD0 + TUU22*vD1]'
)
def test_example_6(self):
Parser.continue_parsing(True)
parse(r"""
% define basis [t, r, \theta, \phi];
\begin{align}
R^\alpha{}_{\beta\mu\nu} &= \partial_\mu \Gamma^\alpha_{\beta\nu} - \partial_\nu \Gamma^\alpha_{\beta\mu} + \Gamma^\alpha_{\mu\gamma}\Gamma^\gamma_{\beta\nu} - \Gamma^\alpha_{\nu\sigma}\Gamma^\sigma_{\beta\mu} \\
R^{\alpha\beta\mu\nu} &= g^{\beta a} g^{\mu b} g^{\nu c} R^\alpha_{a b c} \\
R_{\alpha\beta\mu\nu} &= g_{\alpha a} R^a_{\beta\mu\nu} \\
K &= R^{\alpha\beta\mu\nu} R_{\alpha\beta\mu\nu} \\
R_{\beta\nu} &= R^\alpha_{\beta\alpha\nu} \\
R &= g^{\beta\nu} R_{\beta\nu} \\
G_{\beta\nu} &= R_{\beta\nu} - \frac{1}{2}g_{\beta\nu}R
\end{align}
""")
self.assertEqual(0, simplify(GammaUDD[0][0][1] - GammaUDD[0][1][0]))
self.assertEqual(str(GammaUDD[0][0][1]), '-G*M/(r**2*(2*G*M/r - 1))')
self.assertEqual(str(GammaUDD[1][0][0]), 'G*M*(-2*G*M/r + 1)/r**2')
self.assertEqual(str(GammaUDD[1][1][1]), '-G*M/(r**2*(-2*G*M/r + 1))')
self.assertEqual(str(GammaUDD[1][3][3]),
'-r*(-2*G*M/r + 1)*sin(theta)**2')
self.assertEqual(0, simplify(GammaUDD[2][1][2] - GammaUDD[2][2][1]))
self.assertEqual(str(GammaUDD[2][1][2]), '1/r')
self.assertEqual(str(GammaUDD[2][3][3]), '-sin(theta)*cos(theta)')
self.assertEqual(0, simplify(GammaUDD[2][1][3] - GammaUDD[2][3][1]))
self.assertEqual(str(GammaUDD[3][1][3]), '1/r')
self.assertEqual(0, simplify(GammaUDD[3][2][3] - GammaUDD[3][3][2]))
self.assertEqual(str(GammaUDD[3][2][3]), 'cos(theta)/sin(theta)')
self.assertEqual(str(simplify(K)), '48*G**2*M**2/r**6')
self.assertEqual(simplify(R), 0)
self.assertEqual(
True,
all(component == 0
for component in (row for row in simplify(Matrix(GDD)))))
def test_metric_inverse(self):
# TODO: simplify -> assert_equal
for DIM in range(2, 5):
Parser.clear_namespace()
parse(r"""
% vardef -metric 'gDD' ({DIM}D)
\delta^a_c = g^{{ab}} g_{{bc}}
""".format(DIM=DIM))
# for i in range(DIM):
# for j in range(DIM):
# if i == j:
# assert_equal(deltaUD[i][j], 1)
# else:
# assert_equal(deltaUD[i][j], 0)
kronecker = simplify(Matrix(deltaUD))
self.assertEqual(
True,
all((simplify(kronecker[i, j]) == 1 if i ==
j else simplify(kronecker[i, j]) == 0 for j in range(DIM))
for i in range(DIM)))
for DIM in range(2, 5):
Parser.clear_namespace()
parse(r"""
% vardef -metric 'gUU' ({DIM}D)
\delta^a_c = g^{{ab}} g_{{bc}}
""".format(DIM=DIM))
kronecker = simplify(Matrix(deltaUD))
self.assertEqual(
True,
all((simplify(kronecker[i, j]) == 1 if i ==
j else simplify(kronecker[i, j]) == 0 for j in range(DIM))
for i in range(DIM)))
def test_expression_4(self):
function = Function('Tensor')(Symbol('T'))
tensor = Tensor(function, 4)
self.assertEqual(
Parser._generate_covdrv(tensor, function, [('beta', 'D')]),
r'\nabla_\beta T = \partial_\beta (T)')
function = Function('Tensor')(Symbol('TUU'), Symbol('mu'),
Symbol('nu'))
tensor = Tensor(function, 4)
self.assertEqual(
Parser._generate_covdrv(tensor, function, [('beta', 'D')]),
r'\nabla_\beta T^{\mu \nu} = \partial_\beta (T^{\mu \nu}) + \Gamma^\mu_{a \beta} (T^{a \nu}) + \Gamma^\nu_{a \beta} (T^{\mu a})'
)
function = Function('Tensor')(Symbol('TUD'), Symbol('mu'),
Symbol('nu'))
tensor = Tensor(function, 4)
self.assertEqual(
Parser._generate_covdrv(tensor, function, [('beta', 'D')]),
r'\nabla_\beta T^\mu_\nu = \partial_\beta (T^\mu_\nu) + \Gamma^\mu_{a \beta} (T^a_\nu) - \Gamma^a_{\nu \beta} (T^\mu_a)'
)
function = Function('Tensor')(Symbol('TDD'), Symbol('mu'),
Symbol('nu'))
tensor = Tensor(function, 4)
self.assertEqual(
Parser._generate_covdrv(tensor, function, [('beta', 'D')]),
r'\nabla_\beta T_{\mu \nu} = \partial_\beta (T_{\mu \nu}) - \Gamma^a_{\mu \beta} (T_{a \nu}) - \Gamma^a_{\nu \beta} (T_{\mu a})'
)
def test_example_BSSN():
import NRPy_param_funcs as par, reference_metric as rfm, BSSN.BSSN_RHSs as Brhs
Parser.clear_namespace()
parse(r"""
% define deltaDD (3D), sym01 hDD (3D), sym01 aDD (3D), vetU (3D)
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign symbolic <H> gammahatDD
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign numeric hDD, aDD, vetU, gammabarDD
% assign metric gammahatDD, gammabarDD
% srepl "\bar{A}" -> "a", "\beta" -> "\text{vet}"
% parse \bar{A}^i_j = \bar{\gamma}^{ik} \bar{A}_{kj}
\begin{align}
% define basis [x, y, z]
%% parse \partial_k \bar{\gamma}_{ij} = \vphantom{upwind} \partial_k h_{ij} + \partial_k \hat{\gamma}_{ij}
% srepl "\text{vet}^<1> \partial_<1>" -> "\text{vet}^<1> \vphantom{upwind} \partial_<1>" %% (inside Lie derivative expansion)
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\text{vet}^k \vphantom{symbolic} \partial_k \text{gammabardet}}{2 \text{gammabardet}})"
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{ij} &= \mathcal{L}_\beta \bar{\gamma}_{ij}
+ \frac{2}{3} \bar{\gamma}_{ij} \left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{ij}
% srepl "K" -> "\text{trK}", "\phi" -> "\text{cf}", "\partial_t \text{cf}" -> "\text{cf_rhs}"
\partial_t \phi &= \mathcal{L}_\beta \phi - \frac{1}{3} \phi \left(\bar{D}_k \beta^k - \alpha K \right)
\end{align}
""")
par.set_parval_from_str('reference_metric::CoordSystem', 'Cartesian')
rfm.reference_metric()
Brhs.BSSN_RHSs()
assert_equal({
'h_rhsDD': h_rhsDD,
'cf_rhs': cf_rhs
}, {
'h_rhsDD': Brhs.h_rhsDD,
'cf_rhs': Brhs.cf_rhs
})
def test_assignment_7(self):
Parser.clear_namespace()
parse(r"""
% define basis [\theta, \phi]
% define const r, deltaDD (2D)
% define index [a-z] (2D)
% parse g_{\mu\nu} = \delta_{\mu\nu}
\begin{align*}
g_{0 0} &= r^{{2}} \\
g_{1 1} &= r^{{2}} \sin^2(\theta)
\end{align*}
% assign metric gDD
\begin{align*}
R^\alpha_{\beta\mu\nu} &= \partial_\mu \Gamma^\alpha_{\beta\nu} - \partial_\nu \Gamma^\alpha_{\beta\mu} + \Gamma^\alpha_{\mu\gamma}\Gamma^\gamma_{\beta\nu} - \Gamma^\alpha_{\nu\sigma}\Gamma^\sigma_{\beta\mu} \\
R_{\alpha\beta\mu\nu} &= g_{\alpha a} R^a_{\beta\mu\nu} \\
R_{\beta\nu} &= R^\alpha_{\beta\alpha\nu} \\
R &= g^{\beta\nu} R_{\beta\nu}
\end{align*}
""")
self.assertEqual(str(GammaUDD[0][1][1]), '-sin(theta)*cos(theta)')
self.assertEqual(0, simplify(GammaUDD[1][0][1] - GammaUDD[1][1][0]))
self.assertEqual(str(GammaUDD[1][0][1]), 'cos(theta)/sin(theta)')
self.assertEqual(
0,
simplify(RDDDD[0][1][0][1] - (-RDDDD[0][1][1][0]) +
(-RDDDD[1][0][0][1]) - RDDDD[1][0][1][0]))
self.assertEqual(str(RDDDD[0][1][0][1]), 'r**2*sin(theta)**2')
self.assertEqual(RDD[0][0], 1)
self.assertEqual(str(RDD[1][1]), 'sin(theta)**2')
self.assertEqual(0, simplify(RDD[0][1] - RDD[1][0]))
self.assertEqual(RDD[0][1], 0)
self.assertEqual(str(R), '2/r**2')
def test_example_1(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% define nosym hUD (4D)
h = h^\mu{}_\mu
""")), {'hUD', 'h'})
self.assertEqual(str(h), 'hUD00 + hUD11 + hUD22 + hUD33')
def test_assignment_1(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% define vU (2D), wU (2D);
T^{ab}_c = \vphantom{_d} \partial_c (v^a w^b)
""")), {'vU', 'wU', 'vU_dD', 'wU_dD', 'TUUD'})
self.assertEqual(str(TUUD[0][0][0]), 'vU0*wU_dD00 + vU_dD00*wU0')
def test_assignment_2(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% define vU (2D), const w;
T^a_c = \vphantom{_d} \partial_c (v^a w)
""")), {'vU', 'w', 'vU_dD', 'TUD'})
self.assertEqual(str(TUD),
'[[vU_dD00*w, vU_dD01*w], [vU_dD10*w, vU_dD11*w]]')
def test_assignment_1(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% vardef -numeric 'vU' (2D), 'wU' (2D)
% keydef index [a-z] (2D)
T^{ab}_c = \partial_c (v^a w^b)
""")), {'vU', 'wU', 'vU_dD', 'wU_dD', 'TUUD'})
self.assertEqual(str(TUUD[0][0][0]), 'vU0*wU_dD00 + vU_dD00*wU0')
def test_example_4_1(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% define anti01 FUU (4D), metric gDD (4D), const k;
J^\mu = (4\pi k)^{-1} \vphantom{_d} \nabla_\nu F^{\mu\nu}
""")), {
'FUU', 'gUU', 'gdet', 'gDD', 'k', 'FUU_dD', 'gDD_dD',
'GammaUDD', 'FUU_cdD', 'JU'
})
def test_assignment_3(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% define metric gDD (4D), vU (4D);
T^{\mu\nu} = \vphantom{_d} \nabla^\nu v^\mu
""")), {
'gUU', 'gdet', 'gDD', 'vU', 'vU_dD', 'gDD_dD', 'GammaUDD',
'vU_cdD', 'vU_cdU', 'TUU'
})
def test_example_4_2(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% define anti01 FUU (4D), metric ghatDD (4D), const k
J^\mu = (4\pi k)^{-1} \vphantom{numeric} \hat{\nabla}_\nu F^{\mu\nu}
""")), {
'FUU', 'ghatUU', 'ghatDD', 'FUU_dD', 'ghatDD_dD',
'GammahatUDD', 'FUU_cdhatD', 'JU'
})
def test_assignment_2(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% define vU (2D), const w
% assign numeric vU
% define index [a-z] (2D)
T^a_c = \partial_c (v^a w)
""")), {'vU', 'vU_dD', 'TUD'})
self.assertEqual(str(TUD),
'[[vU_dD00*w, vU_dD01*w], [vU_dD10*w, vU_dD11*w]]')
def test_example_3(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% define epsilonDDD (3D)
% define vU (3D), wU (3D)
u_i = \epsilon_{ijk} v^j w^k
""")), {'epsilonDDD', 'vU', 'wU', 'uD'})
self.assertEqual(
str(uD),
'[vU1*wU2 - vU2*wU1, -vU0*wU2 + vU2*wU0, vU0*wU1 - vU1*wU0]')
def test_assignment_2(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% vardef -const 'w'
% vardef -numeric 'vU' (2D)
% keydef index [a-z] (2D)
T^a_c = \partial_c (v^a w)
""")), {'vU', 'vU_dD', 'TUD'})
self.assertEqual(str(TUD),
'[[vU_dD00*w, vU_dD01*w], [vU_dD10*w, vU_dD11*w]]')
def test_example_2(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% define metric gUU (3D), vD (3D)
v^\mu = g^{\mu\nu} v_\nu
""")), {'gDD', 'gUU', 'vD', 'vU'})
self.assertEqual(
str(vU),
'[gUU00*vD0 + gUU01*vD1 + gUU02*vD2, gUU01*vD0 + gUU11*vD1 + gUU12*vD2, gUU02*vD0 + gUU12*vD1 + gUU22*vD2]'
)
def test_example_4_2(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% vardef -anti01 'FUU' (4D)
% vardef -metric 'ghatDD' (4D)
% vardef -const 'k'
J^\mu = (4\pi k)^{-1} \vphantom{numeric} \hat{\nabla}_\nu F^{\mu\nu}
""")), {
'FUU', 'ghatUU', 'ghatdet', 'epsilonUUUU', 'ghatDD',
'FUU_dD', 'ghatDD_dD', 'GammahatUDD', 'FUU_cdhatD', 'JU'
})
def test_assignment_4(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% define basis [x, y], deriv _d;
% define vD (2D);
v_0 = x^{{2}} + 2x;
v_1 = y\sqrt{x};
T_{\mu\nu} = (\partial_\nu v_\mu)\,\partial^2_x v_0
""")), {'vD', 'x', 'y', 'vD_dD', 'TDD'})
self.assertEqual(str(TDD),
'[[2*vD_dD00, 2*vD_dD01], [2*vD_dD10, 2*vD_dD11]]')
def test_example_2(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% vardef -metric 'gUU' (3D)
% vardef 'vD' (3D)
v^\mu = g^{\mu\nu} v_\nu
""")), {'gDD', 'gdet', 'epsilonDDD', 'gUU', 'vD', 'vU'})
self.assertEqual(
str(vU),
'[gUU00*vD0 + gUU01*vD1 + gUU02*vD2, gUU01*vD0 + gUU11*vD1 + gUU12*vD2, gUU02*vD0 + gUU12*vD1 + gUU22*vD2]'
)
def test_assignment_3(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% vardef -numeric -metric 'gDD' (4D)
% vardef -numeric 'vU' (4D)
% keydef index [a-z] (4D)
T^{ab} = \nabla^b v^a
""")), {
'gUU', 'gdet', 'epsilonUUUU', 'gDD', 'vU', 'vU_dD',
'gDD_dD', 'GammaUDD', 'vU_cdD', 'vU_cdU', 'TUU'
})
def test_assignment_3(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% define metric gDD (4D), vU (4D)
% assign numeric gDD, vU
% define index [a-z] (4D)
T^{ab} = \nabla^b v^a
""")), {
'gUU', 'gDD', 'vU', 'vU_dD', 'gDD_dD', 'GammaUDD',
'vU_cdD', 'vU_cdU', 'TUU'
})
def test_expression_5(self):
Parser.clear_namespace()
parse(r"""
% vardef -numeric -metric 'gDD' (4D)
% vardef -numeric 'vU' (4D)
T^\mu_b = \nabla_b v^\mu
""")
tensor = Parser._namespace['vU_cdD']
function = Parser._namespace['vU_cdD'].equation[0]
self.assertEqual(
Parser._generate_covdrv(tensor, function, 'a'),
r'\nabla_a \nabla_b v^\mu = \partial_a (\nabla_b v^\mu) + \text{Gamma}^\mu_{c a} (\nabla_b v^c) - \text{Gamma}^c_{b a} (\nabla_c v^\mu)'
)
def test_assignment_6(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% define vD (2D), uD (2D), wD (2D)
% define index [a-z] (2D)
T_{abc} = \vphantom{numeric} ((v_a + u_a)_{,b} - w_{a,b})_{,c}
""")), {
'vD', 'uD', 'wD', 'TDDD', 'uD_dD', 'vD_dD', 'wD_dD',
'wD_dDD', 'uD_dDD', 'vD_dDD'
})
self.assertEqual(str(TDDD[0][0][0]),
'uD_dDD000 + vD_dDD000 - wD_dDD000')
def test_srepl_macro(self):
Parser.clear_namespace()
parse(r"""
% srepl "<1>'" -> "\text{<1>prime}"
% srepl "\text{<1..>}_<2>" -> "\text{(<1..>)<2>}"
% srepl "<1>_{<2>}" -> "<1>_<2>", "<1>_<2>" -> "\text{<1>_<2>}"
% srepl "\text{(<1..>)<2>}" -> "\text{<1..>_<2>}"
% srepl "<1>^{<2>}" -> "<1>^<2>", "<1>^<2>" -> "<1>^{{<2>}}"
""")
expr = r"x_n^4 + x'_n \exp(x_n y_n^2)"
self.assertEqual(str(parse_expr(expr)),
"x_n**4 + xprime_n*exp(x_n*y_n**2)")
Parser.clear_namespace()
parse(r""" % srepl "<1>'^{<2..>}" -> "\text{<1>prime}" """)
expr = r"v'^{label}"
self.assertEqual(str(parse_expr(expr)), "vprime")
def test_expression_5(self):
tensor = Tensor(Function('Tensor')(Symbol('vU'), Symbol('mu')), 4)
self.assertEqual(
Parser._generate_covdrv(tensor, [('a', 'U'), ('b', 'D')],
'symbolic'),
r'\nabla_a \nabla_b v^\mu = \partial_a (\partial_b (v^\mu) + \Gamma^\mu_{c b} (v^c)) + \Gamma^\mu_{c a} (\partial_b (v^c) + \Gamma^c_{d b} (v^d))'
)
def test_expression_5(self):
function = Function('Tensor')(Symbol('vU'), Symbol('mu'))
tensor = Tensor(function, 4)
self.assertEqual(
Parser._generate_covdrv(tensor, function, [('a', 'D'),
('b', 'D')]),
r'\nabla_a \nabla_b v^\mu = \partial_a (\partial_b (v^\mu) + \Gamma^\mu_{c b} (v^c)) + \Gamma^\mu_{c a} (\partial_b (v^c) + \Gamma^c_{d b} (v^d)) - \Gamma^c_{b a} (\partial_c (v^\mu) + \Gamma^\mu_{b c} (v^b))'
)
def test_assignment_4(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% keydef basis [x, y]
% vardef 'uD' (2D), 'wD' (2D)
% keydef index [a-z] (2D)
u_0 = x^{{2}} + 2x \\
u_1 = y\sqrt{x} \\
v_a = u_a + w_a \\
% assign -numeric 'wD', 'vD'
T_{ab} = \partial^2_x v_0 (\partial_b v_a)
""")), {'uD', 'wD', 'vD', 'vD_dD', 'wD_dD', 'TDD'})
self.assertEqual(
str(TDD),
'[[2*wD_dD00 + 4*x + 4, 2*wD_dD01], [2*wD_dD10 + y/sqrt(x), 2*wD_dD11 + 2*sqrt(x)]]'
)
def test_assignment_5(self):
Parser.clear_namespace()
self.assertEqual(
set(
parse(r"""
% define basis [x, y]
% define uD (2D), wD (2D)
% assign symbolic <H> uD
% define index [a-z] (2D)
u_0 = x^{{2}} + 2x \\
u_1 = y\sqrt{x} \\
v_a = u_a + w_a \\
T_{ab} = \partial^2_x v_0 (\vphantom{numeric} \partial_b v_a)
""")), {'uD', 'wD', 'vD', 'vD_dD', 'wD_dD', 'TDD'})
self.assertEqual(
str(TDD),
'[[2*wD_dD00 + 4*x + 4, 2*wD_dD01], [2*wD_dD10 + y/sqrt(x), 2*wD_dD11 + 2*sqrt(x)]]'
)