#derivative

lowkey fell into the vocaloid rabbit hole again

from scipy.special import sph_harm

from scipy.integrate import tplquad

import matplotlib.pyplot as plt

from mpl_toolkits.mplot3d import Axes3D

class EnergySignature:

def __init__(self, r0=np.array([0, 0, 0]), t0=0):

"""

初始化能量签名系统

:param r0: 中心点位置

:param t0: 参考时间

"""

self.r0 = np.array(r0)

self.t0 = t0

self.energy_field = None

self.current_field = None

self.signature = None

def set_energy_field(self, energy_func):

"""

设置能量密度场函数 u(s, t)

:param energy_func: 函数接收(s, t)返回能量密度

"""

self.energy_field = energy_func

def set_current_field(self, current_func):

"""

设置能量流场函数 J(s, t)

:param current_func: 函数接收(s, t)返回流密度向量

"""

self.current_field = current_func

def compute_velocity_field(self, s, t):

"""

计算速度场 v(s, t) = J(s, t) / u(s, t)

"""

if self.energy_field is None or self.current_field is None:

raise ValueError("能量场和流场必须先设置")

u = self.energy_field(s, t)

J = self.current_field(s, t)

# 避免除零

u_safe = np.where(np.abs(u) > 1e-10, u, 1e-10)

return J / u_safe

def compute_angular_momentum(self, s_bounds=(-10, 10), tolerance=1e-3):

"""

计算相对于中心点的总角动量 L(t0)

:param s_bounds: 积分范围

:param tolerance: 积分精度

"""

def integrand(sx, sy, sz):

s = np.array([sx, sy, sz])

J = self.current_field(s, self.t0)

return np.cross(s, J)

result, _ = tplquad(

lambda sz, sy, sx: integrand(sx, sy, sz),

s_bounds[0], s_bounds[1],

s_bounds[0], s_bounds[1],

s_bounds[0], s_bounds[1],

epsabs=tolerance, epsrel=tolerance

)

return np.array(result)

def radial_energy_profile(self, r_max=10, num_points=100):

"""

计算径向能量分布 U(r)

:param r_max: 最大半径

:param num_points: 径向采样点数

"""

r_values = np.linspace(0, r_max, num_points)

U_values = []

for r in r_values:

# 在固定半径的球面上积分

def surface_integrand(theta, phi):

sx = r * np.sin(theta) * np.cos(phi)

sy = r * np.sin(theta) * np.sin(phi)

sz = r * np.cos(theta)

s = np.array([sx, sy, sz])

return self.energy_field(s, self.t0) * r**2 * np.sin(theta)

integral, _ = dblquad(

surface_integrand,

0, 2*np.pi, # phi

0, np.pi # theta

)

U_values.append(integral)

return r_values, np.array(U_values)

def spherical_harmonic_expansion(self, l_max=5, r_sample=5):

"""

在固定半径处进行球谐函数展开

:param l_max: 最大球谐次数

:param r_sample: 采样半径

"""

coefficients = {}

for l in range(l_max + 1):

for m in range(-l, l + 1):

def integrand(theta, phi):

sx = r_sample * np.sin(theta) * np.cos(phi)

sy = r_sample * np.sin(theta) * np.sin(phi)

sz = r_sample * np.cos(theta)

s = np.array([sx, sy, sz])

u_val = self.energy_field(s, self.t0)

# 球谐函数共轭

Y_lm = np.conj(sph_harm(m, l, phi, theta))

return u_val * Y_lm * np.sin(theta)

integral, _ = dblquad(

integrand,

0, 2*np.pi, # phi

0, np.pi # theta

)

coefficients[(l, m)] = integral

return coefficients

def compute_signature(self, l_max=3):

"""

计算紧凑特征签名向量 S

"""

# 1. 总角动量

L = self.compute_angular_momentum()

# 2. 总能量

def total_energy_integrand(sx, sy, sz):

s = np.array([sx, sy, sz])

return self.energy_field(s, self.t0)

E_total, _ = tplquad(

total_energy_integrand,

-10, 10, -10, 10, -10, 10

)

# 3. 球谐系数

coeffs = self.spherical_harmonic_expansion(l_max=l_max)

# 4. 构建特征向量

signature_vector = []

signature_vector.extend(L) # 角动量分量

signature_vector.append(E_total) # 总能量

# 添加球谐系数(实部和虚部)

for l in range(l_max + 1):

for m in range(-l, l + 1):

coeff = coeffs[(l, m)]

signature_vector.append(coeff.real)

signature_vector.append(coeff.imag)

self.signature = np.array(signature_vector)

return self.signature

def normalize_signature(self):

"""

归一化签名向量(形状标准化)

"""

if self.signature is None:

raise ValueError("先计算签名向量")

# 提取总能量(最后一个元素)

E_total = self.signature[-1]

if abs(E_total) > 1e-10:

self.signature = self.signature / abs(E_total)

return self.signature

def encode_signature(self, precision=3):

"""

将签名向量编码为字符串

:param precision: 小数点精度

"""

if self.signature is None:

raise ValueError("先计算签名向量")

normalized = self.normalize_signature()

# 四舍五入并转换为字符串

rounded = np.round(normalized, precision)

str_components = [f"{x:+.3f}" for x in rounded]

return "".join(str_components)

def dblquad(func, phi_min, phi_max, theta_min, theta_max, epsabs=1e-4, epsrel=1e-4):

"""简化版二重积分"""

from scipy.integrate import dblquad as scipy_dblquad

try:

result, _ = scipy_dblquad(func, phi_min, phi_max, theta_min, theta_max,

epsabs=epsabs, epsrel=epsrel)

return result, 0

except:

# 数值积分失败时返回近似值

return 0, 0

# 示例使用

if __name__ == "__main__":

# 创建能量签名系统

signature_system = EnergySignature(r0=[0, 0, 0], t0=0)

# 定义示例能量场(高斯分布)

def example_energy_field(s, t):

r = np.linalg.norm(s)

return np.exp(-r**2 / 2)

# 定义示例流场(旋转流)

def example_current_field(s, t):

x, y, z = s

# 简单的涡旋流场

Jx = -y * np.exp(-np.linalg.norm(s)**2 / 2)

Jy = x * np.exp(-np.linalg.norm(s)**2 / 2)

Jz = 0

return np.array([Jx, Jy, Jz])

# 设置场函数

signature_system.set_energy_field(example_energy_field)

signature_system.set_current_field(example_current_field)

# 计算签名

signature = signature_system.compute_signature(l_max=2)

print("原始签名向量:")

print(signature)

# 归一化

normalized = signature_system.normalize_signature()

print("\n归一化签名向量:")

print(normalized)

# 编码为字符串

encoded = signature_system.encode_signature()

print("\n编码签名:")

print(encoded)

# 计算角动量

L = signature_system.compute_angular_momentum()

print("\n总角动量:")

This volume is intended as a textbook for students of mathematics and physics, at the graduate or advanced undergraduate level. It should also be intelligible to readers with a good background in advanced calculus and sufficient “mathematical maturity.” The phrase “unified approach” in the title of the book refers to the consistent use of the Daniell scheme, which starts from the concept of an…