From bdb591fb062d90cfb3db9b29306ea76ff16fd27b Mon Sep 17 00:00:00 2001 From: Mu_Xin <3044470775@qq.com> Date: Thu, 3 Jul 2025 15:03:37 +0800 Subject: [PATCH] =?UTF-8?q?=E4=BF=AE=E6=94=B9=E4=BB=A3=E7=A0=81?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- multiple_integration.py | 229 +++++++++++++++++++++++++--------------- 1 file changed, 145 insertions(+), 84 deletions(-) diff --git a/multiple_integration.py b/multiple_integration.py index 7569a0a..5d7312f 100644 --- a/multiple_integration.py +++ b/multiple_integration.py @@ -5,76 +5,75 @@ import sympy as sp from convert_formula import * import math -def rectangular_double_integration(region1, region2, function): - """ - 求直角坐标系下二重积分 - :param region1: y的积分区域(以元组或列表形式给出) - :param region2: x的积分区域(同region1) - :param function: 被积函数 - :return: 二重积分值 - """ - lower1, upper1= region1[0], region1[1] - lower2, upper2= region2[0], region2[1] +def rectangular_double_integration(region1, region2, function, integration_order=('y','x')): + var1, var2 = integration_order + lower1, upper1 = region1 # 直接使用符号表达式（可能包含外层变量） + lower2, upper2 = region2 function = sp.sympify(convert_formula(function)) - x, y= sp.symbols('x y') - integration1 = sp.integrate(function, (y, lower1, upper1)) - result = sp.integrate(integration1, (x,lower2,upper2)) + x, y = sp.symbols('x y') + # 根据积分顺序动态选择积分变量 + if var1 == 'y' and var2 == 'x': + integration1 = sp.integrate(function, (y, lower1, upper1)) # 内层积分限可包含x + result = sp.integrate(integration1, (x, lower2, upper2)) + elif var1 == 'x' and var2 == 'y': + integration1 = sp.integrate(function, (x, lower1, upper1)) # 内层积分限可包含y + result = sp.integrate(integration1, (y, lower2, upper2)) + else: + raise ValueError('积分顺序仅支持("y","x")或("x","y")') return result -def polar_double_integration(function, rou, theta): - """ - 求极坐标下的二重积分 - :param function: 用极坐标表示的被积函数 - :param rou: ρ的积分区间(以元组或列表给出) - :param theta: θ的积分区间(同ρ) - :return: 极坐标下的积分值 - """ - rou1, rou2= rou[0], rou[1] - theta1, theta2= theta[0], theta[1] - rou, theta= sp.symbols('rou theta') - function= sp.sympify(convert_formula(function)) - function= function* rou - integration1= sp.integrate(function, (rou, rou1, rou2)) - result= sp.integrate(integration1, (theta, theta1, theta2)) +def polar_double_integration(function, rou_region, theta_region, integration_order=('rou','theta')): + var1, var2 = integration_order + function = sp.sympify(convert_formula(function)) * sp.symbols('rou') # 极坐标雅可比行列式 + rou, theta = sp.symbols('rou theta') + # 根据顺序动态积分 + if var1 == 'rou' and var2 == 'theta': + inner = sp.integrate(function, (rou, *rou_region)) + result = sp.integrate(inner, (theta, *theta_region)) + elif var1 == 'theta' and var2 == 'rou': + inner = sp.integrate(function, (theta, *theta_region)) + result = sp.integrate(inner, (rou, *rou_region)) + else: + raise ValueError('积分顺序仅支持("rou","theta")或("theta","rou")') return result -def rectangular_triple_integration(region1, region2, region3, function): - """ - 求三重积分值 - :param region1: z积分区域(以元组或列表形式给出) - :param region2: y积分区域(同上) - :param region3: x积分区域(同上) - :param function: 被积函数 - :return: 三重积分值 - """ - lower1, upper1= region1[0], region1[1] - lower2, upper2= region2[0], region2[1] - lower3, upper3 = region3[0], region3[1] +def rectangular_triple_integration(region1, region2, region3, function, integration_order=('z','y','x')): + var1, var2, var3 = integration_order function = sp.sympify(convert_formula(function)) - x, y, z= sp.symbols('x y z') - integration1 = sp.integrate(function, (z, lower1, upper1)) - integration2 = sp.integrate(integration1, (y, lower2, upper2)) - result = sp.integrate(integration2, (x, lower3, upper3)) + x, y, z = sp.symbols('x y z') + # 动态构建积分变量元组 + vars = {'x':x, 'y':y, 'z':z} + regions = {var1: region1, var2: region2, var3: region3} + # 按顺序嵌套积分 + inner = sp.integrate(function, (vars[var1], *regions[var1])) + middle = sp.integrate(inner, (vars[var2], *regions[var2])) + result = sp.integrate(middle, (vars[var3], *regions[var3])) return result -def cylindrical_triple_integration(rou, theta, region, function): - """ - 求柱坐标下三重积分值 - :param rou: ρ的积分区域(以元组或列表形式给出) - :param theta: θ的积分区间(以元组或列表形式给出) - :param region: z的积分区间(以元组或列表形式给出) - :param function: 被积函数(以柱坐标形式给出) - :return: 三重积分值 - """ - rou1, rou2 = rou[0], rou[1] - theta1, theta2 = theta[0], theta[1] - z1, z2 = region[0], region[1] - function = sp.sympify(convert_formula(function)) - rou, theta, z= sp.symbols('rou theta z') - function = function * rou - integration1 = sp.integrate(function, (rou, rou1, rou2)) - integration2 = sp.integrate(integration1, (theta, theta1, theta2)) - result = sp.integrate(integration2, (z, z1, z2)) +def cylindrical_triple_integration(function, z_region, rou_region, theta_region, integration_order=('z','rou','theta')): + var1, var2, var3 = integration_order + function = sp.sympify(convert_formula(function)) * sp.symbols('rou') # 柱坐标雅可比行列式rou + z, rou, theta = sp.symbols('z rou theta') + # 动态映射积分变量与区域 + vars = {'z': z, 'rou': rou, 'theta': theta} + regions = {var1: z_region, var2: rou_region, var3: theta_region} + # 按顺序嵌套积分 + inner = sp.integrate(function, (vars[var1], *regions[var1])) + middle = sp.integrate(inner, (vars[var2], *regions[var2])) + result = sp.integrate(middle, (vars[var3], *regions[var3])) + return result + +def spherical_triple_integration(function, r_region, phi_region, theta_region, integration_order=('r','phi','theta')): + var1, var2, var3 = integration_order + function = sp.sympify(convert_formula(function)) * (sp.symbols('r')**2) * sp.sin(sp.symbols('phi')) # 球坐标雅可比行列式r²sinφ + r, phi, theta = sp.symbols('r phi theta') + # 动态映射积分变量与区域 + vars = {'r': r, 'phi': phi, 'theta': theta} + regions = {var1: r_region, var2: phi_region, var3: theta_region} + # 按顺序嵌套积分 + inner = sp.integrate(function, (vars[var1], *regions[var1])) + middle = sp.integrate(inner, (vars[var2], *regions[var2])) + result = sp.integrate(middle, (vars[var3], *regions[var3])) return result def spherical_triple_integration(rou, theta, gamma, function): @@ -201,35 +200,97 @@ class TestIntegralFunctions(unittest.TestCase): def UI(): while True: - command = input('请选择将要进行的操作：1.直角坐标下二重积分 2.极坐标下二重积分 3.直角坐标系下三重积分 4.柱坐标下三重积分 5.球坐标下三重积分 6.退出（所有积分区间请以元组或列表的形式给出）') + command = input('请选择将要进行的操作：1.直角坐标下二重积分 2.极坐标下二重积分 3.直角坐标系下三重积分 4.柱坐标下三重积分 5.球坐标下三重积分 6.退出\n（所有积分区间请不要使用半开半闭区间的形式给出，积分顺序由内到外输入，极坐标、柱坐标、球坐标需在原函数上乘的雅戈比行列式已经自动传入，不要重复输入）') if command=='1': function = input('请输入被积函数：') - region1 = input('请输入第一重积分区间：') - region2 = input('请输入第二重积分区间：') - print('直角坐标系下二重积分值：', rectangular_double_integration(region1, region2, function)) + order = input('请输入积分顺序（格式："y,x"或"x,y"，默认y,x）：') or 'y,x' + var1, var2 = order.split(',') # 解析积分变量顺序 + # 输入符号积分限（支持变量表达式如"x"、"2*y"） + lower1 = sp.sympify(convert_formula(input(f'请输入{var1}的下限：'))) + upper1 = sp.sympify(convert_formula(input(f'请输入{var1}的上限：'))) + lower2 = sp.sympify(convert_formula(input(f'请输入{var2}的下限：'))) + upper2 = sp.sympify(convert_formula(input(f'请输入{var2}的上限：'))) + print('直角坐标系下二重积分值：', rectangular_double_integration( + region1=(lower1, upper1), + region2=(lower2, upper2), + function=function, + integration_order=tuple(order.split(',')) + )) if command=='2': - function = input('请输入用极坐标表示的被积函数：') - rou = input('请输入ρ的积分区间：') - theta = input('请输入θ的积分区间：') - print('极坐标下二重积分值：', polar_double_integration(function, rou, theta)) + function = input('请输入极坐标被积函数：') + order = input('请输入积分顺序（格式："rou,theta"或"theta,rou"，默认rou,theta）：') or 'rou,theta' + var1, var2 = order.split(',') + # 符号积分限输入 + lower1 = sp.sympify(convert_formula(input(f'请输入{var1}的下限：'))) + upper1 = sp.sympify(convert_formula(input(f'请输入{var1}的上限：'))) + lower2 = sp.sympify(convert_formula(input(f'请输入{var2}的下限：'))) + upper2 = sp.sympify(convert_formula(input(f'请输入{var2}的上限：'))) + print('极坐标二重积分值：', polar_double_integration( + function=function, + rou_region=(lower1, upper1) if var1=='rou' else (lower2, upper2), + theta_region=(lower2, upper2) if var1=='rou' else (lower1, upper1), + integration_order=tuple(order.split(',')) + )) if command=='3': function = input('请输入被积函数：') - region1 = input('请输入第一重积分区间：') - region2 = input('请输入第二重积分区间：') - region3 = input('请输入第三重积分区间：') - print('直角坐标系下三重积分值：', rectangular_triple_integration(region1, region2, region3, function)) + order = input('请输入积分顺序（格式："z,y,x"或其他排列，默认z,y,x）：') or 'z,y,x' + vars = order.split(',') + # 输入三个符号积分限 + regions = [ + (sp.sympify(convert_formula(input(f'请输入{vars[0]}的下限：'))), + sp.sympify(convert_formula(input(f'请输入{vars[0]}的上限：'))),), + (sp.sympify(convert_formula(input(f'请输入{vars[1]}的下限：'))), + sp.sympify(convert_formula(input(f'请输入{vars[1]}的上限：'))),), + (sp.sympify(convert_formula(input(f'请输入{vars[2]}的下限：'))), + sp.sympify(convert_formula(input(f'请输入{vars[2]}的上限：'))),) + ] + print('直角坐标三重积分值：', rectangular_triple_integration( + function=function, + region1=regions[0], + region2=regions[1], + region3=regions[2], + integration_order=tuple(vars) + )) if command=='4': - function = input('请输入用柱坐标表示的被积函数：') - rou = input('请输入ρ的积分区间：') - theta = input('请输入θ的积分区间：') - region = input('请输入z的积分区间：') - print('柱坐标系下三重积分值：',cylindrical_triple_integration(rou, theta, region, function)) + function = input('请输入柱坐标被积函数：') + order = input('请输入积分顺序（格式："z,rou,theta"或其他排列，默认z,rou,theta）：') or 'z,rou,theta' + vars = order.split(',') + # 输入符号积分限（支持变量表达式） + regions = [ + (sp.sympify(convert_formula(input(f'请输入{vars[0]}的下限（r/phi/theta）：'))), + sp.sympify(convert_formula(input(f'请输入{vars[0]}的上限（r/phi/theta）：')))), + (sp.sympify(convert_formula(input(f'请输入{vars[1]}的下限（r/phi/theta）：'))), + sp.sympify(convert_formula(input(f'请输入{vars[1]}的上限（r/phi/theta）：')))), + (sp.sympify(convert_formula(input(f'请输入{vars[2]}的下限（r/phi/theta）：'))), + sp.sympify(convert_formula(input(f'请输入{vars[2]}的上限（r/phi/theta）：')))) + ] + print('柱坐标三重积分值：', cylindrical_triple_integration( + function=function, + z_region=regions[0] if vars[0]=='z' else next(r for r in regions if vars.index(r[0].name) == 0), + rou_region=regions[1] if vars[1]=='rou' else next(r for r in regions if vars.index(r[1].name) == 1), + theta_region=regions[2] if vars[2]=='theta' else next(r for r in regions if vars.index(r[2].name) == 2), + integration_order=tuple(vars) + )) if command=='5': - function = input('请输入用球坐标表示的被积函数：') - rou = input('请输入ρ的积分区间：') - theta = input('请输入θ的积分区间：') - gamma = input('请输入γ的积分区间：') - print('球坐标系下三重积分值：',spherical_triple_integration(rou, theta, gamma, function)) + function = input('请输入球坐标被积函数（使用r,phi,theta）：') + order = input('请输入积分顺序（格式："r,phi,theta"或其他排列，默认r,phi,theta）：') or 'r,phi,theta' + vars = order.split(',') + # 输入符号积分限（支持变量表达式） + regions = [ + (sp.sympify(convert_formula(input(f'请输入{vars[0]}的下限（r/phi/theta）：'))), + sp.sympify(convert_formula(input(f'请输入{vars[0]}的上限（r/phi/theta）：')))), + (sp.sympify(convert_formula(input(f'请输入{vars[1]}的下限（r/phi/theta）：'))), + sp.sympify(convert_formula(input(f'请输入{vars[1]}的上限（r/phi/theta）：')))), + (sp.sympify(convert_formula(input(f'请输入{vars[2]}的下限（r/phi/theta）：'))), + sp.sympify(convert_formula(input(f'请输入{vars[2]}的上限（r/phi/theta）：')))) + ] + print('球坐标三重积分值：', spherical_triple_integration( + function=function, + r_region=regions[0] if vars[0]=='r' else next(r for r in regions if vars.index(r[0].name) == 0), + phi_region=regions[1] if vars[1]=='phi' else next(r for r in regions if vars.index(r[1].name) == 1), + theta_region=regions[2] if vars[2]=='theta' else next(r for r in regions if vars.index(r[2].name) == 2), + integration_order=tuple(vars) + )) if command=='6': break