\hbOSSKJr SSKJr SSKJrJr SSKJr SSK J r J r SSK J r SSKJr SSKJr SS KJr SS KJr SS KJr SS KJrJrJrJr SS KJr SSK J!r!J"r"J#r# SSK$J%r% "SS\5r&Sr'S\'"\5/0\RP\&'"SS\&\ 5r)"SS\\&5r*"SS\\&5r+"SS\\&5r,"SS\&5r-"SS\ 5r.S r/S!r0\&\&l1\+\&l2\*\&l3\,\&l4\)\&l5\,"5\&l6g")#) annotations)product)AddBasic) StdFactKB) AtomicExprExpr)Pow)S)default_sort_key)sympifysqrt)ImmutableDenseMatrix)BasisDependentZeroBasisDependentBasisDependentMulBasisDependentAdd) CoordSys3D)Dyadic BaseDyadic DyadicAdd) VectorKindcR\rSrSr%SrSrSrSrS\S'S\S'S\S 'S\S 'S\S 'S \S '\ "5r S\S'\ S5r Sr SrSrSrSr\R\lSrSr\R\lSrS!Sjr\ S5rSr\R\lSrSrSrSrg )"Vectorz Super class for all Vector classes. Ideally, neither this class nor any of its subclasses should be instantiated by the user. FTg(@z type[Vector] _expr_type _mul_func _add_func _zero_func _base_func VectorZerozerorkindcUR$)a* Returns the components of this vector in the form of a Python dictionary mapping BaseVector instances to the corresponding measure numbers. Examples ======== >>> from sympy.vector import CoordSys3D >>> C = CoordSys3D('C') >>> v = 3*C.i + 4*C.j + 5*C.k >>> v.components {C.i: 3, C.j: 4, C.k: 5} ) _componentsselfs K/var/www/auris/envauris/lib/python3.13/site-packages/sympy/vector/vector.py componentsVector.components(s&c[X-5$)z' Returns the magnitude of this vector. rr's r) magnitudeVector.magnitude=sDK  r,c&XR5- $)z0 Returns the normalized version of this vector. )r.r's r) normalizeVector.normalizeCsnn&&&r,cNX- nURU5nURS5$)a Check if ``self`` and ``other`` are identically equal vectors. Explanation =========== Checks if two vector expressions are equal for all possible values of the symbols present in the expressions. Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy.abc import x, y >>> from sympy import pi >>> C = CoordSys3D('C') Compare vectors that are equal or not: >>> C.i.equals(C.j) False >>> C.i.equals(C.i) True These two vectors are equal if `x = y` but are not identically equal as expressions since for some values of `x` and `y` they are unequal: >>> v1 = x*C.i + C.j >>> v2 = y*C.i + C.j >>> v1.equals(v1) True >>> v1.equals(v2) False Vectors from different coordinate systems can be compared: >>> D = C.orient_new_axis('D', pi/2, C.i) >>> D.j.equals(C.j) False >>> D.j.equals(C.k) True Parameters ========== other: Vector The other vector expression to compare with. Returns ======= ``True``, ``False`` or ``None``. A return value of ``True`` indicates that the two vectors are identically equal. A return value of ``False`` indicates that they are not. In some cases it is not possible to determine if the two vectors are identically equal and ``None`` is returned. See Also ======== sympy.core.expr.Expr.equals r)dotequals)r(otherdiff diff_mag2s r)r5 Vector.equalsIs*~|HHTN ""r,c^[U[5(a[T[5(a[R$[RnUR R 5H:up4URSRT5nX%U-URS-- nM< U$SSK J n [X[45(d[[U5S-S-5e[X5(aU4SjnU$[TU5$)aV Returns the dot product of this Vector, either with another Vector, or a Dyadic, or a Del operator. If 'other' is a Vector, returns the dot product scalar (SymPy expression). If 'other' is a Dyadic, the dot product is returned as a Vector. If 'other' is an instance of Del, returns the directional derivative operator as a Python function. If this function is applied to a scalar expression, it returns the directional derivative of the scalar field wrt this Vector. Parameters ========== other: Vector/Dyadic/Del The Vector or Dyadic we are dotting with, or a Del operator . Examples ======== >>> from sympy.vector import CoordSys3D, Del >>> C = CoordSys3D('C') >>> delop = Del() >>> C.i.dot(C.j) 0 >>> C.i & C.i 1 >>> v = 3*C.i + 4*C.j + 5*C.k >>> v.dot(C.k) 5 >>> (C.i & delop)(C.x*C.y*C.z) C.y*C.z >>> d = C.i.outer(C.i) >>> C.i.dot(d) C.i r)Delz is not a vector, dyadic or z del operatorc">SSKJn U"UT5$)Nr)directional_derivative)sympy.vector.functionsr>)fieldr>r(s r)r>*Vector.dot..directional_derivativesI-eT::r,) isinstancerr"rr#r*itemsargsr4sympy.vector.deloperatorr< TypeErrorstr)r(r6outveckvvect_dotr<r>s` r)r4 Vector.dotsP eV $ $$ ++{{"[[F((..066!9==.Q,221M0%v//CJ)GG*+, , e ! ! ;* )4r,c$URU5$Nr4r(r6s r)__and__Vector.__and__sxxr,c|[U[5(a[U[5(a[R$[RnURR 5HHup4UR URS5nURURS5nX$U-- nMJ U$[ X5$)a  Returns the cross product of this Vector with another Vector or Dyadic instance. The cross product is a Vector, if 'other' is a Vector. If 'other' is a Dyadic, this returns a Dyadic instance. Parameters ========== other: Vector/Dyadic The Vector or Dyadic we are crossing with. Examples ======== >>> from sympy.vector import CoordSys3D >>> C = CoordSys3D('C') >>> C.i.cross(C.j) C.k >>> C.i ^ C.i 0 >>> v = 3*C.i + 4*C.j + 5*C.k >>> v ^ C.i 5*C.j + (-4)*C.k >>> d = C.i.outer(C.i) >>> C.j.cross(d) (-1)*(C.k|C.i) rr;) rBrr"r#r*rCcrossrDouter)r(r6outdyadrIrJ cross_productrUs r)rT Vector.crosss@ eV $ $$ ++{{"kkG((..0 $ 166!9 5 %++AFF1I6u9$1NT!!r,c$URU5$rNrTrPs r)__xor__Vector.__xor__zz%  r,c [U[5(d [S5e[U[5(d[U[5(a[R $[ URR5URR55VVVVs/sHuup#upEX5-[X$5-PM nnnnn[U6$s snnnnf)aA Returns the outer product of this vector with another, in the form of a Dyadic instance. Parameters ========== other : Vector The Vector with respect to which the outer product is to be computed. Examples ======== >>> from sympy.vector import CoordSys3D >>> N = CoordSys3D('N') >>> N.i.outer(N.j) (N.i|N.j) z!Invalid operand for outer product) rBrrFr"rr#rr*rCrr)r(r6k1v1k2v2rDs r)rU Vector.outers.%((?@ @z**5*--;;  4??002E4D4D4J4J4LMOM4F8BXbJr..M O$Os!C c8UR[R5(a'U(a[R$[R$U(a#UR U5UR U5- $UR U5UR U5- U-$)aC Returns the vector or scalar projection of the 'other' on 'self'. Examples ======== >>> from sympy.vector.coordsysrect import CoordSys3D >>> C = CoordSys3D('C') >>> i, j, k = C.base_vectors() >>> v1 = i + j + k >>> v2 = 3*i + 4*j >>> v1.projection(v2) 7/3*C.i + 7/3*C.j + 7/3*C.k >>> v1.projection(v2, scalar=True) 7/3 )r5rr#r Zeror4)r(r6scalars r) projectionVector.projection$sj$ ;;v{{ # ##166 4 4 88E?TXXd^3 388E?TXXd^3d: :r,cDSSKJn [U[5(a/[R [R [R 4$[ [U"U555R5n[UVs/sHo0RU5PM sn5$s snf)ai Returns the components of this vector but the output includes also zero values components. Examples ======== >>> from sympy.vector import CoordSys3D, Vector >>> C = CoordSys3D('C') >>> v1 = 3*C.i + 4*C.j + 5*C.k >>> v1._projections (3, 4, 5) >>> v2 = C.x*C.y*C.z*C.i >>> v2._projections (C.x*C.y*C.z, 0, 0) >>> v3 = Vector.zero >>> v3._projections (0, 0, 0) r)_get_coord_systems) sympy.vector.operatorsrjrBr"r renextiter base_vectorstupler4)r(rjbase_vecis r) _projectionsVector._projections>sn, > dJ ' 'FFAFFAFF+ +/567DDF848ahhqk84554s<Bc$URU5$rN)rUrPs r)__or__ Vector.__or__Zr]r,c|[UR5Vs/sHo RU5PM sn5$s snf)a Returns the matrix form of this vector with respect to the specified coordinate system. Parameters ========== system : CoordSys3D The system wrt which the matrix form is to be computed Examples ======== >>> from sympy.vector import CoordSys3D >>> C = CoordSys3D('C') >>> from sympy.abc import a, b, c >>> v = a*C.i + b*C.j + c*C.k >>> v.to_matrix(C) Matrix([ [a], [b], [c]]) )Matrixrnr4)r(systemunit_vecs r) to_matrixVector.to_matrix_sA4**,.,/7xx),./ /.s9c0nURR5H@up#URUR[R 5X#--XR'MB U$)a] The constituents of this vector in different coordinate systems, as per its definition. Returns a dict mapping each CoordSys3D to the corresponding constituent Vector. Examples ======== >>> from sympy.vector import CoordSys3D >>> R1 = CoordSys3D('R1') >>> R2 = CoordSys3D('R2') >>> v = R1.i + R2.i >>> v.separate() == {R1: R1.i, R2: R2.i} True )r*rCgetryrr#)r(partsvectmeasures r)separateVector.separate|sR(!__224MD"'))DKK"E"&.#1E++ 5 r,c2[U[5(a [U[5(a [S5e[U[5(aCU[R:Xa [ S5e[ U[U[R55$[S5e)z'Helper for division involving vectors. zCannot divide two vectorszCannot divide a vector by zeroz#Invalid division involving a vector) rBrrFr re ValueError VectorMulr NegativeOne)oner6s r) _div_helperVector._div_helperso c6 " "z%'@'@78 8 V $ $ !ABBS#eQ]]";< <AB Br,N)F)__name__ __module__ __qualname____firstlineno____doc__ is_scalar is_Vector _op_priority__annotations__rr$propertyr*r.r1r5r4rQrTr[rUrgrrrur{rr__static_attributes__rr,r)rrs IIL !|D*#   (! ' A#F< |kkGO*"X!mmGO" H;4666!]]FN/:4 Cr,rc^U4SjnU$)Nc>[[0Tn/nURH5n[UR[ 5(dM$UR U5 M7 U[:Xa[U6RSS9$g)NF)deep)r VectorAddrDrBr$rappenddoit)expr vec_classvectorstermclss r)_postprocessor)get_postprocessor.._postprocessorsj)$S) IID$))Z00t$  !g&+++7 7 "r,r)rrs` r)get_postprocessorrs8 r,rcb^\rSrSrSrS U4Sjjr\S5rSrSr \S5r Sr S r U=r $) BaseVectoriz! Class to denote a base vector. cN>UcSRU5nUcSRU5n[U5n[U5nU[SS5;a [S5e[ U[ 5(d [ S5eURUn[TU]%U[U5U5nXfl U[R0Ul [RUlURS-U-UlSU-UlXFlX&lX4UlS S 0n[)U5UlX&lU$) Nzx{}zx_{}rzindex must be 0, 1 or 2zsystem should be a CoordSys3D. commutativeT)formatrGrangerrBrrF _vector_namessuper__new__r _base_instanceOner&_measure_number_name _pretty_form _latex_form_system_idr _assumptions_sys) rindexry pretty_str latex_strnameobj assumptions __class__s r)rBaseVector.__new__s  e,J   e,I_  N a #67 7&*--;< <##E*goc1U8V4 ,eeLL3&-  ?# /$d+ $[1  r,cUR$rN)rr's r)ryBaseVector.systems ||r,cUR$rN)r)r(printers r) _sympystrBaseVector._sympystrs zzr,cfURup#URU5S-URU-$)Nr)r_printr)r(rrrys r) _sympyreprBaseVector._sympyreprs1 ~~f%+f.B.B5.IIIr,cU1$rNrr's r) free_symbolsBaseVector.free_symbolss v r,cU$rNrr's r)_eval_conjugateBaseVector._eval_conjugates r,r)NN)rrrrrrrryrrrrr __classcell__)rs@r)rrsK !FJr,rc$\rSrSrSrSrSrSrg)riz* Class to denote sum of Vector instances. c:[R"U/UQ70UD6nU$rN)rrrrDoptionsrs r)rVectorAdd.__new__!''>d>g> r,c8Sn[UR5R55nURSS9 UHWupEUR 5nUH<nXuR ;dMUR UU-nX!R U5S-- nM> MY USS$)Nrc(USR5$)Nr)__str__)xs r)%VectorAdd._sympystr..s1r,keyz + )listrrCsortrnr*r) r(rret_strrCryr base_vectsr temp_vects r)rVectorAdd._sympystrsT]]_**,- / 0!LF,,.J' $ 2Q 6I~~i85@@G " s|r,rN)rrrrrrrrrr,r)rrs r,rc>\rSrSrSrSr\S5r\S5rSr g)riz6 Class to denote products of scalars and BaseVectors. c:[R"U/UQ70UD6nU$rN)rrrs r)rVectorMul.__new__ rr,cUR$)z(The BaseVector involved in the product. )rr's r) base_vectorVectorMul.base_vectors"""r,cUR$)zDThe scalar expression involved in the definition of this VectorMul. )rr's r)measure_numberVectorMul.measure_numbers ###r,rN) rrrrrrrrrrrr,r)rrs4##$$r,rc*\rSrSrSrSrSrSrSrSr g) r"iz Class to denote a zero vector g333333(@0z\mathbf{\hat{0}}c2[R"U5nU$rN)rr)rrs r)rVectorZero.__new__%s ((- r,rN) rrrrrrrrrrrr,r)r"r"sLL%Kr,r"c$\rSrSrSrSrSrSrg)Crossi*a\ Represents unevaluated Cross product. Examples ======== >>> from sympy.vector import CoordSys3D, Cross >>> R = CoordSys3D('R') >>> v1 = R.i + R.j + R.k >>> v2 = R.x * R.i + R.y * R.j + R.z * R.k >>> Cross(v1, v2) Cross(R.i + R.j + R.k, R.x*R.i + R.y*R.j + R.z*R.k) >>> Cross(v1, v2).doit() (-R.y + R.z)*R.i + (R.x - R.z)*R.j + (-R.x + R.y)*R.k c[U5n[U5n[U5[U5:a [X!5*$[R"XU5nXlX#lU$rN)r r rr r_expr1_expr2rexpr1expr2rs r)r Cross.__new__<sT E "%5e%< <%'' 'll3u-   r,c B[URUR5$rN)rTrrr(hintss r)r Cross.doitFsT[[$++..r,rNrrrrrrrrrr,r)rr*s"/r,rc$\rSrSrSrSrSrSrg)DotiJaW Represents unevaluated Dot product. Examples ======== >>> from sympy.vector import CoordSys3D, Dot >>> from sympy import symbols >>> R = CoordSys3D('R') >>> a, b, c = symbols('a b c') >>> v1 = R.i + R.j + R.k >>> v2 = a * R.i + b * R.j + c * R.k >>> Dot(v1, v2) Dot(R.i + R.j + R.k, a*R.i + b*R.j + c*R.k) >>> Dot(v1, v2).doit() a + b + c c[U5n[U5n[X/[S9up[R"XU5nXlX#lU$)Nr)r sortedr r rrrrs r)r Dot.__new__^sDun2BC ll3u-   r,c B[URUR5$rN)r4rrrs r)rDot.doitgs4;; ,,r,rNrrr,r)rrJs&-r,rc^^[T[5(a)[RU4SjTR55$[T[5(a)[RU4SjTR55$[T[ 5(a[T[ 5(aTR TR :XaTRSnTRSnX#:Xa[R$1SkRX#15R5nUS-S-U:XaSOSnUTR R5U-$SSK J n U"TTR 5n[UT5$[T["5(d[T["5(a[R$[T[$5(a=['[)TR*R-555upU [UT5-$[T[$5(a=['[)TR*R-555upU [TU 5-$[!TT5$![a [!TT5s$f=f) a2 Returns cross product of two vectors. Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy.vector.vector import cross >>> R = CoordSys3D('R') >>> v1 = R.i + R.j + R.k >>> v2 = R.x * R.i + R.y * R.j + R.z * R.k >>> cross(v1, v2) (-R.y + R.z)*R.i + (R.x - R.z)*R.j + (-R.x + R.y)*R.k c3<># UHn[UT5v M g7frNrZ.0rqvect2s r) cross..|s!F:a%5//:c3<># UHn[TU5v M g7frNrZrrqvect1s r)rr~s!F:a%q//:rr>rr;r;rexpress)rBrrfromiterrDrrrr# differencepoprn functionsrrTrrr"rrlrmr*rC) rrn1n2n3signrrJr`m1rbm2s `` r)rTrTks %!!!F5::!FFF%!!!F5::!FFF%$$E:)F)F :: #ABABx{{"$$bX.335Bq&A+1"D //1"55 5& #uzz*AE? "%$$ 5*(E(E{{%##d5++11345%E"""%##d5++11345%r"""   '& & 'sII10I1c.^^[T[5(a*[R"U4SjTR55$[T[5(a*[R"U4SjTR55$[T[5(a{[T[5(afTR TR :Xa&TT:Xa[ R$[ R$SSK J n U"TTR 5n[TU5$[T[5(d[T[5(a[ R$[T[5(a=[![#TR$R'555upEU[UT5-$[T[5(a=[![#TR$R'555upgU[TU5-$[TT5$![a [TT5s$f=f)a Returns dot product of two vectors. Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy.vector.vector import dot >>> R = CoordSys3D('R') >>> v1 = R.i + R.j + R.k >>> v2 = R.x * R.i + R.y * R.j + R.z * R.k >>> dot(v1, v2) R.x + R.y + R.z c3<># UHn[UT5v M g7frNrOr s r)rdot..s>:aC5MM:rc3<># UHn[TU5v M g7frNrOrs r)rr&s>:aCqMM:rr;r)rBrrrDrrr rrerrr4rrr"rrlrmr*rC)rrrrJr`r"rbr#s`` r)r4r4s %||>5::>>>%||>5::>>>%$$E:)F)F :: #!UN155 6 6& !uzz*Aua= %$$ 5*(E(Evv %##d5++11345#b%.  %##d5++11345#eR.  ue  %ue$ $ %s2G;;HHN)7 __future__r itertoolsr sympy.corerrsympy.core.assumptionsrsympy.core.exprrr sympy.core.powerr sympy.core.singletonr sympy.core.sortingr sympy.core.sympifyr (sympy.functions.elementary.miscellaneousrsympy.matrices.immutablerrxsympy.vector.basisdependentrrrrsympy.vector.coordsysrectrsympy.vector.dyadicrrrsympy.vector.kindrrr"_constructor_postprocessor_mappingrrrr"rrrTr4rrrr r!r#rr,r)r8s "!,, "/&9C::0==(KC^KC^  c " #4((099x!6,$!6$, #V /F/@-$-B-`'Tl r,