U 9%e@sddlmZmZddlmZddlmZddlmZddl m Z m Z ddl m Z ddlmZddlmZdd lmZmZmZmZmZdd lmZGd d d eZd dZdS))Basicdiff)S)default_sort_key)Matrix)Integral integrate)GeometryEntity)simplify)topological_sort) CoordSys3DVectorParametricRegionparametric_region_listImplicitRegion)_get_coord_systemscsDeZdZdZfddZeddZeddZedd Z Z 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 cs|t|}t|dkrtd}nt|dkr0tn tt|}jdkrLtjS| }| }|}t j }t tjD]} ||| j| 7}qtt|dkrt tjD]} ||| j| }qjdkrDjd} t|| } j| dj| d} } t|t r t| |}nt| |}t|| | | f}njdkr|jj\}}t||}t||}t||}t|t r||}n ||}t|}j|dj|d}}j|dj|d}}t||||f|||f}nP|jj}tj|}t||}fdd|D}t|f|}t|tsh|St||SdS)NrCcs*g|]"}|j|dj|dfqS)rr)limits).0varparametricregionU/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/vector/integrals.py ksz.ParametricIntegral.__new__..) rlenr ValueErrornextiter dimensionsrZZero base_vectors base_scalarsr zerorangeZ definitionsubs parametersrr isinstancer dotZ magnituder _bounds_casecrossrZjacobianZdetrsuper__new__)clsfieldrZ coord_setZ coord_sysr#r$ZparametricfieldriZ parameterZr_difflowerupperZ integrandresultuvZr_uZr_vZ normal_vectorZlower_uZupper_uZlower_vZupper_v variablesZcoeffl __class__rrr.+sZ                  zParametricIntegral.__new__cszt|}g}|D]H|d|d|fdd|Dq|sf|St||ftdSdS)Nrrc3s6|].}|kr|hs&|hr|fVqdS)N) issuperset)rqZlower_ppZupper_prr s  z2ParametricIntegral._bounds_case..)key)listkeysZatomsextendr r)r/r(rVErr>rr+ss   zParametricIntegral._bounds_casecCs |jdS)Nrargsselfrrrr0szParametricIntegral.fieldcCs |jdS)NrrGrIrrrrsz#ParametricIntegral.parametricregion) __name__ __module__ __qualname____doc__r. classmethodr+propertyr0r __classcell__rrr:rrs H  rcGst|dkrt|dtr(t||dSt|dtrPt|dd}t||St|dtrt|d}d}|D]}|t||7}qr|St|f|S)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)r0regionZ regions_listr5regrrrrRs.   rRN)Z sympy.corerrZsympy.core.singletonrZsympy.core.sortingrZsympy.matricesrZsympy.integralsrrZsympy.geometry.entityr Zsympy.simplify.simplifyr Zsympy.utilities.iterablesr Z sympy.vectorr r rrrZsympy.vector.operatorsrrrRrrrrs