def test_functional_diffgeom_ch3():
x0, y0 = symbols('x0, y0', extended_real=True)
x, y, t = symbols('x, y, t', extended_real=True)
f = Function('f')
b1 = Function('b1')
b2 = Function('b2')
p_r = R2_r.point([x0, y0])
s_field = f(R2.x, R2.y)
v_field = b1(R2.x) * R2.e_x + b2(R2.y) * R2.e_y
assert v_field.rcall(s_field).rcall(p_r).doit() == b1(x0) * Derivative(
f(x0, y0), x0) + b2(y0) * Derivative(f(x0, y0), y0)
assert R2.e_x(R2.r**2).rcall(p_r) == 2 * x0
v = R2.e_x + 2 * R2.e_y
s = R2.r**2 + 3 * R2.x
assert v.rcall(s).rcall(p_r).doit() == 2 * x0 + 4 * y0 + 3
circ = -R2.y * R2.e_x + R2.x * R2.e_y
series = intcurve_series(circ, t, R2_r.point([1, 0]), coeffs=True)
series_x, series_y = zip(*series)
assert all(term == cos(t).taylor_term(i, t)
for i, term in enumerate(series_x))
assert all(term == sin(t).taylor_term(i, t)
for i, term in enumerate(series_y))
def test_sympyissue_11799():
n = 2
M = Manifold('M', n)
P = Patch('P', M)
coord = CoordSystem('coord', P, ['x%s' % i for i in range(n)])
x = coord.coord_functions()
dx = coord.base_oneforms()
f = Function('f')
g = [[f(x[0], x[1])**2, 0], [0, f(x[0], x[1])**2]]
metric = sum(g[i][j]*TP(dx[i], dx[j]) for i in range(n) for j in range(n))
R = metric_to_Riemann_components(metric)
d = Symbol('d')
assert (R[0, 1, 0, 1] ==
-Subs(Derivative(f(d, x[1]), d, d), (d, x[0]))/f(x[0], x[1]) -
Subs(Derivative(f(x[0], d), d, d), (d, x[1]))/f(x[0], x[1]) +
Subs(Derivative(f(d, x[1]), d), (d, x[0]))**2/f(x[0], x[1])**2 +
Subs(Derivative(f(x[0], d), d), (d, x[1]))**2/f(x[0], x[1])**2)
def test_Derivative():
assert mcode(Derivative(f(x), x, x)) == 'Hold[D[f[x], x, x]]'
assert mcode(Derivative(sin(x), x)) == "Hold[D[Sin[x], x]]"
assert mcode(Derivative(x, x)) == "Hold[D[x, x]]"
assert mcode(Derivative(sin(x)*y**4, x, 2)) == "Hold[D[y^4*Sin[x], x, x]]"
assert mcode(Derivative(sin(x)*y**4, x, y, x)) == "Hold[D[y^4*Sin[x], x, y, x]]"
assert mcode(Derivative(sin(x)*y**4, x, y, 3, x)) == "Hold[D[y^4*Sin[x], x, y, y, y, x]]"
def make_expression(terms):
product = []
for term, rat, sym, deriv in terms:
if deriv is not None:
var, order = deriv
while order > 0:
term, order = Derivative(term, var), order - 1
if sym is None:
if rat is S.One:
product.append(term)
else:
product.append(Pow(term, rat))
else:
product.append(Pow(term, rat * sym))
return Mul(*product)
def test_functional_diffgeom_ch4():
x0, y0, theta0 = symbols('x0, y0, theta0', extended_real=True)
x, y, r, theta = symbols('x, y, r, theta', extended_real=True)
r0 = symbols('r0', positive=True)
f = Function('f')
b1 = Function('b1')
b2 = Function('b2')
p_r = R2_r.point([x0, y0])
p_p = R2_p.point([r0, theta0])
f_field = b1(R2.x, R2.y) * R2.dx + b2(R2.x, R2.y) * R2.dy
assert f_field.rcall(R2.e_x).rcall(p_r) == b1(x0, y0)
assert f_field.rcall(R2.e_y).rcall(p_r) == b2(x0, y0)
s_field_r = f(R2.x, R2.y)
df = Differential(s_field_r)
assert df(R2.e_x).rcall(p_r).doit() == Derivative(f(x0, y0), x0)
assert df(R2.e_y).rcall(p_r).doit() == Derivative(f(x0, y0), y0)
s_field_p = f(R2.r, R2.theta)
df = Differential(s_field_p)
assert trigsimp(df(R2.e_x).rcall(p_p).doit()) == (
cos(theta0) * Derivative(f(r0, theta0), r0) -
sin(theta0) * Derivative(f(r0, theta0), theta0) / r0)
assert trigsimp(df(R2.e_y).rcall(p_p).doit()) == (
sin(theta0) * Derivative(f(r0, theta0), r0) +
cos(theta0) * Derivative(f(r0, theta0), theta0) / r0)
assert R2.dx(R2.e_x).rcall(p_r) == 1
assert R2.dx(R2.e_x) == 1
assert R2.dx(R2.e_y).rcall(p_r) == 0
assert R2.dx(R2.e_y) == 0
circ = -R2.y * R2.e_x + R2.x * R2.e_y
assert R2.dx(circ).rcall(p_r).doit() == -y0
assert R2.dy(circ).rcall(p_r) == x0
assert R2.dr(circ).rcall(p_r) == 0
assert simplify(R2.dtheta(circ).rcall(p_r)) == 1
assert (circ - R2.e_theta).rcall(s_field_r).rcall(p_r) == 0
def test_Derivative():
assert mcode(Derivative(f(x), x, x)) == 'D[f[x], x, x]'