PtjddlmZmZddlmZddlmZddlmZddl m Z m Z ddl m Z ddlmZddlmZdd lmZmZmZmZmZdd lmZGd d eZd ZdS))Basicdiff)S)default_sort_key)Matrix)Integral integrate)GeometryEntity)simplify)topological_sort) CoordSys3DVectorParametricRegionparametric_region_listImplicitRegion)_get_coord_systemscdeZdZdZfdZedZedZedZ xZ S)ParametricIntegrala3 Represents integral of a scalar or vector field over a Parametric Region Examples ======== >>> from sympy import cos, sin, pi >>> from sympy.vector import CoordSys3D, ParametricRegion, ParametricIntegral >>> from sympy.abc import r, t, theta, phi >>> C = CoordSys3D('C') >>> curve = ParametricRegion((3*t - 2, t + 1), (t, 1, 2)) >>> ParametricIntegral(C.x, curve) 5*sqrt(10)/2 >>> length = ParametricIntegral(1, curve) >>> length sqrt(10) >>> semisphere = ParametricRegion((2*sin(phi)*cos(theta), 2*sin(phi)*sin(theta), 2*cos(phi)), (theta, 0, 2*pi), (phi, 0, pi/2)) >>> ParametricIntegral(C.z, semisphere) 8*pi >>> ParametricIntegral(C.j + C.k, ParametricRegion((r*cos(theta), r*sin(theta)), r, theta)) 0 ct|}t|dkrtd}n6t|dkrtt t |}jdkr tjS| }| }|}tj }ttjD]} ||| j| zz }t|dkrKttjD])} ||| j| }*jdkrjd} t#|| } j| dj| d} } t'|tr#t)| |}n$t)| |z}t/|| | | f}njdkr|jj\}}t#||}t#||}t)||}t'|tr||}n||z}t)|}j|dj|d}}j|dj|d}}t/||||f|||f}n|jj}t5j|}t)||z}fd|D}t/|g|R}t'|t:s|St=||S)NrCc^g|])}|j|dj|df*S)rr)limits).0varparametricregions T/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/vector/integrals.py z.ParametricIntegral.__new__..ks?nnn]`#'.s3A68H8OPS8TUV8WXnnn) rlenr ValueErrornextiter dimensionsrZero base_vectors base_scalarsrzerorange definitionsubs parametersrr isinstancer dot magnituder _bounds_casecrossrjacobiandetrsuper__new__)clsfieldr coord_set coord_sysr'r(parametricfieldri parameterr_difflowerupper integrandresultuvr_ur_v normal_vectorlower_uupper_ulower_vupper_v variablescoeffl __class__s ` rr6zParametricIntegral.__new__+s&u-- y>>Q  "3II ^^a   T)__--I  &! + +6M --//  --//  Ks+67788 @ @A a!1!>Q  3/:;;<< h h"1"6"6|AHXHcdeHf"g"g  &! + +(3A6I!Y''F+29=a@BRBYZcBdefBg5E/622 I$VZZ%@%@AA $V%5%5%7%7%GHH y9eU*CDDFF  (A - -##$4$?AQAXYYDAqq!**Cq!**C$SYYs^^44M/622 F+// >> +M,C,C,E,EE  ++I/6q9!<>N>UVW>XYZ>[WG/6q9!<>N>UVW>XYZ>[WGy1gw*?!WgAVWWFF(()9)DFVF]^^I+677@@KKOOQQE !677InnnndmnnnAy-1---F&(++ AM77??3/?@@ @r c^t|}g}|D]i|d|d|fd|Dj|s|St ||ft S)Nrrc3K|]:}|k|hs|h4|fV;dS)N) issuperset)rqlower_ppupper_ps r z2ParametricIntegral._bounds_case..sgKK!q&&((!--+1181C1CQC1H1H+1aV&&&&KKr )key)listkeysatomsextendr r)r7r-rVErUrVrWs @@@rr1zParametricIntegral._bounds_casess     K KAQilGQilGmmooGmmooG HHKKKKKKQKKK K K K K B #QF0@AAA Ar c|jdS)Nrargsselfs rr8zParametricIntegral.fieldy|r c|jdS)Nrrarcs rrz#ParametricIntegral.parametricregionrer ) __name__ __module__ __qualname____doc__r6 classmethodr1propertyr8r __classcell__)rPs@rrrs8FAFAFAFAFAPBB[B&XXr rct|dkrt|dtrt||dSt|dtr+t |dd}t ||St|dtr1t |d}d}|D]}|t ||z }|St|g|RS)a Compute the integral of a vector/scalar field over a a region or a set of parameters. Examples ======== >>> from sympy.vector import CoordSys3D, ParametricRegion, vector_integrate >>> from sympy.abc import x, y, t >>> C = CoordSys3D('C') >>> region = ParametricRegion((t, t**2), (t, 1, 5)) >>> vector_integrate(C.x*C.i, region) 12 Integrals over some objects of geometry module can also be calculated. >>> from sympy.geometry import Point, Circle, Triangle >>> c = Circle(Point(0, 2), 5) >>> vector_integrate(C.x**2 + C.y**2, c) 290*pi >>> triangle = Triangle(Point(-2, 3), Point(2, 3), Point(0, 5)) >>> vector_integrate(3*C.x**2*C.y*C.i + C.j, triangle) -8 Integrals over some simple implicit regions can be computed. But in most cases, it takes too long to compute over them. This is due to the expressions of parametric representation becoming large. >>> from sympy.vector import ImplicitRegion >>> c2 = ImplicitRegion((x, y), (x - 2)**2 + (y - 1)**2 - 9) >>> vector_integrate(1, c2) 6*pi Integral of fields with respect to base scalars: >>> vector_integrate(12*C.y**3, (C.y, 1, 3)) 240 >>> vector_integrate(C.x**2*C.z, C.x) C.x**3*C.z/3 >>> vector_integrate(C.x*C.i - C.y*C.k, C.x) (Integral(C.x, C.x))*C.i + (Integral(-C.y, C.x))*C.k >>> _.doit() C.x**2/2*C.i + (-C.x*C.y)*C.k rr) r!r.rrrrvector_integrater r )r8region regions_listrCregs rroros\ 6{{a fQi!1 2 2 8%eVAY77 7 fQi 0 0 3+F1I66q9F#E622 2 fQi 0 0 1&)<r~sS""""""""""""""//////!!!!!!////////000000,,,,,,666666@@@@@@@@@@@@@@555555D>%>%>%>%>%r