\hSSKJrJr SSKJr SSKJr SSKJr SSK J r J r SSK J r SSKJr SSKJr SS KJrJrJrJrJr SS KJr "S S \5rS rg))Basicdiff)S)default_sort_key)Matrix)Integral integrate)GeometryEntity)simplify)topological_sort) CoordSys3DVectorParametricRegionparametric_region_listImplicitRegion)_get_coord_systemsc\^\rSrSrSrU4Sjr\S5r\S5r \S5r Sr U=r $)ParametricIntegrala Represents integral of a scalar or vector field over a Parametric Region Examples ======== >>> from sympy import cos, sin, pi >>> from sympy.vector import CoordSys3D, ParametricRegion, ParametricIntegral >>> from sympy.abc import r, t, theta, phi >>> C = CoordSys3D('C') >>> curve = ParametricRegion((3*t - 2, t + 1), (t, 1, 2)) >>> ParametricIntegral(C.x, curve) 5*sqrt(10)/2 >>> length = ParametricIntegral(1, curve) >>> length sqrt(10) >>> semisphere = ParametricRegion((2*sin(phi)*cos(theta), 2*sin(phi)*sin(theta), 2*cos(phi)), (theta, 0, 2*pi), (phi, 0, pi/2)) >>> ParametricIntegral(C.z, semisphere) 8*pi >>> ParametricIntegral(C.j + C.k, ParametricRegion((r*cos(theta), r*sin(theta)), r, theta)) 0 cv>[U5n[U5S:Xa [S5nO)[U5S:a[e[ [ U55nUR S:Xa[R$UR5nUR5nUn[Rn[[UR55Hn XU URU -- nM [U5S:waF[[UR55H$n URXiURU 5nM& UR S:XaUR Sn [#X5n UR$U SUR$U Sp['U[5(a[)U R+U55nO[)U R-5U-5n[/XX45nGOUR S:XaUR1UR UR$5unn[#UU5n[#UU5n[)UR3U55n['U[5(aUR+U5nOUUR-5-n[)U5nUR$USUR$USnnUR$USUR$USnn[/UUUU4UUU45nOUR1UR UR$5n[5UR5R7U5R95n[)UU-5nUVs/sH(nUUR$USUR$US4PM* nn[/U/UQ76n['U[:5(dU$[<TU]}XU5$s snf)NrC) rlenr ValueErrornextiter dimensionsrZero base_vectors base_scalarsrzerorange definitionsubs parametersrlimits isinstancer dot magnituder _bounds_casecrossrjacobiandetrsuper__new__)clsfieldparametricregion coord_set coord_sysr r!parametricfieldri parameterr_difflowerupper integrandresultuvr_ur_v normal_vectorlower_uupper_ulower_vupper_v variablescoeffvarl __class__s N/var/www/auris/envauris/lib/python3.13/site-packages/sympy/vector/integrals.pyr0ParametricIntegral.__new__+sy&u- y>Q "3I ^a  T)_-I  & &! +66M --/  --/  KKs+6678A a!1!Q 3/::;<"1"6"6|HXHcHcdeHf"g=  & &! +(33A6I!'F+229=a@BRBYBYZcBdefBg5/622$VZZ%@A $V%5%5%7%GH ye*CDF  ( (A -##$4$?$?AQAXAXYDAqq!*Cq!*C$SYYs^4M/622+// > +M,C,C,EE  +I/66q9!<>N>U>UVW>XYZ>[WG/66q9!<>N>U>UVW>XYZ>[WGy1gw*?!WgAVWF(()9)D)DFVF]F]^I+667@@KOOQE !67Idmndm]`#'..s3A68H8O8OPS8TUV8WXdmAny-1-F&(++M7?3/?@ @ os/N6c^^^[UR55n/nUHPmUTSmUTSmTR5mTR5mURUUU4SjU55 MR U(dU$[ X44[ S9$)Nrrc3># UHAnTU:wdM TRU15(dTRU15(dM;TU4v MC g7f)N) issuperset).0qlower_ppupper_ps rM 2ParametricIntegral._bounds_case..sGKQ!q&V((!--1C1CQC1HaVQs A ,A  A )key)listkeysatomsextendr r)r1r&r'VErTrUrVs @@@rMr+ParametricIntegral._bounds_casess   AQilGQilGmmoGmmoG HHKQK K  #QF0@A Ac URS$)Nrargsselfs rMr2ParametricIntegral.fieldyy|rac URS$)Nrrcres rMr3#ParametricIntegral.parametricregionrhra) __name__ __module__ __qualname____firstlineno____doc__r0 classmethodr+propertyr2r3__static_attributes__ __classcell__)rLs@rMrrsN8FAPBB&rarcn[U5S:Xa[US[5(a[XS5$[US[5(a[ US5Sn[ X5$[US[5(a)[ US5nSnUHnU[ X5- nM U$[U/UQ76$)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) rr(rrrrvector_integrater r )r2region regions_listr>regs rMrvrvs\ 6{a fQi!1 2 2%eAY7 7 fQi 0 0+F1I6q9F#E2 2 fQi 0 01&)rs@""/!/0,6@@5D>%ra