if __name__ == '__main__':
#f = Function('f')(x, y)
#g1 = Function('g')(x, y)
a1, a2, a3, ao1, ao2, theta1, theta2, theta3, theta4 = symbols(
'a1,a2,a3,ao1,ao2,theta1,theta2,theta3,theta4')
m = Manifold('M', 4)
patch = Patch('P', m)
rect = CoordSystem('rect', patch)
polar = CoordSystem('polar', patch)
rect in patch.coord_systems
polar.connect_to(rect, [theta1, theta2, theta3, theta4], [
a1 * theta1 + a2 * theta2 + a3 * theta3 +
(theta1 * ao1 + theta2 * ao2) * theta4
])
print(polar.jacobian(rect, [theta1, theta2, theta3, theta4]))
g = polar.jacobian(rect, [theta1, theta2, theta3, theta4])
patch2 = Patch('P', m)
rect2 = CoordSystem('rect2', patch2)
polar2 = CoordSystem('polar2', patch2)
rect2 in patch2.coord_systems
polar2.connect_to(rect2, [theta1, theta2, theta3, theta4], g)
print(polar2.jacobian(rect2, [theta1, theta2, theta3, theta4]))
H = polar2.jacobian(rect2, [theta1, theta2, theta3, theta4])
'''
n = Manifold('M', 1)
patch3 = Patch('P', n)
rect3 = CoordSystem('rect3', patch3)
polar3 = CoordSystem('polar3', patch3)
rect3 in patch3.coord_systems
m = Manifold('M', 2)
patch = Patch('P', m)
rect = CoordSystem('rect', patch)
polar = CoordSystem('polar', patch)
print(rect in patch.coord_systems)
polar.connect_to(rect, [r, theta], [r*cos(theta), r*sin(theta)])
print(polar.coord_tuple_transform_to(rect, [0, 2]))
print(polar.coord_tuple_transform_to(rect, [2, pi/2]))
print(rect.coord_tuple_transform_to(polar, [1, 1]).applyfunc(simplify))
print(polar.jacobian(rect, [r, theta]))
p = polar.point([1, 3*pi/4])
print(rect.point_to_coords(p))
print(rect.coord_function(0)(p))
print(rect.coord_function(1)(p))
v_x = rect.base_vector(0)
x = rect.coord_function(0)
print(v_x(x))
print(v_x(v_x(x)))
v_r = polar.base_vector(0)
print(v_r(x))
# ^ ????
# 建立两个坐标系之间的转换关系，逆变换是自动的。 # 这个地方，如果笛卡尔是3维，另一个是2维曲面（但是也用u,v，不用极坐标，那会怎么样呢？） polar.connect_to(rect, [r, theta], [r * cos(theta), r * sin(theta)]) # t = polar.coord_tuple_transform_to(rect, [1, pi]) # 极坐标存在奇点，r = 0时，其他怎么取都是0 t = polar.coord_tuple_transform_to(rect, [0, pi]) print(t) t = polar.coord_tuple_transform_to(rect, [2, pi / 2]) print(t) t = rect.coord_tuple_transform_to(polar, [0, 2]) print(t) # 雅克比矩阵。极坐标到笛卡尔坐标系的变换矩阵函数，在每个点，是可以求出一个具体的矩阵的 # 但整体来说是一个函数。 m = polar.jacobian(rect, [r, theta]) print(m) plot_latex(latex(m)) # 笛卡尔坐标系到笛卡尔坐标系的雅克比矩阵是单位矩阵 m = rect.jacobian(rect, [x, y]) # plot_latex(latex(m)) p = rect.point([3, 2]) # base_vector返回一个向量场。给定一个点，会返回一个向量？ # v_x = rect.base_vector(0) # v_y = rect.base_vector(1) # pprint(v_x(p), v_y(p)) # 返回标量场。标量场的意思是取一个点，返回一个标量值。x y轴可以视为两个天然的标量场，但应该开可以返回其他标量场 x_scalar_field = rect.coord_function(0)
