ó ¡¼™\c@sÉddlmZddlmZmZmZddlmZmZddl m Z m Z ddl m Z mZmZddlmZddlmZmZmZmZddlmZmZmZd efd „ƒYZd eefd „ƒYZd eefd„ƒYZdeefd„ƒYZdeefd„ƒYZ defd„ƒYZ!defd„ƒYZ"d„Z#d„Z$d„Z%ee_&ee_'ee_(e e_)ee_*e%e_+e ƒe_,dS(iÿÿÿÿ(t StdFactKB(tStPowtsympify(t AtomicExprtExpr(trangetdefault_sort_key(tsqrttImmutableMatrixtAdd(t CoordSys3D(tBasisDependenttBasisDependentAddtBasisDependentMultBasisDependentZero(t BaseDyadictDyadict DyadicAddtVectorcBsÂeZdZeZdZed„ƒZd„Zd„Z d„Z d„Z e je _d„Z d„Z e je _d „Zed „Zed „ƒZd „Zeje_d „Zd„ZRS(s Super class for all Vector classes. Ideally, neither this class nor any of its subclasses should be instantiated by the user. g(@cCs|jS(s‚ 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} (t _components(tself((s2lib/python2.7/site-packages/sympy/vector/vector.pyt componentsscCst||@ƒS(s7 Returns the magnitude of this vector. (R(R((s2lib/python2.7/site-packages/sympy/vector/vector.pyt magnitude+scCs||jƒS(s@ Returns the normalized version of this vector. (R(R((s2lib/python2.7/site-packages/sympy/vector/vector.pyt normalize1scsýt|tƒrtˆtƒr%tjStj}xL|jjƒD];\}}|jdjˆƒ}||||jd7}q>W|Sddl m }t|tƒ rÎt||ƒ rÎt t |ƒddƒ‚nt||ƒrð‡fd†}|Stˆ|ƒS(sN 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 iiiÿÿÿÿ(tDels is not a vector, dyadic or s del operatorcsddlm}||ˆƒS(Niÿÿÿÿ(tdirectional_derivative(tsympy.vector.functionsR(tfieldR(R(s2lib/python2.7/site-packages/sympy/vector/vector.pyRns( t isinstanceRt VectorZeroRtzeroRtitemstargstdottsympy.vector.deloperatorRt TypeErrortstr(Rtothertoutvectktvtvect_dotRR((Rs2lib/python2.7/site-packages/sympy/vector/vector.pyR"7s (   cCs |j|ƒS(N(R"(RR&((s2lib/python2.7/site-packages/sympy/vector/vector.pyt__and__uscCs™t|tƒrŒt|tƒr%tjStj}xW|jjƒD]F\}}|j|jdƒ}|j|jdƒ}|||7}q>W|St||ƒS(sà 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) ii( RRRRRR tcrossR!touter(RR&toutdyadR(R)t cross_productR-((s2lib/python2.7/site-packages/sympy/vector/vector.pyR,zs  cCs |j|ƒS(N(R,(RR&((s2lib/python2.7/site-packages/sympy/vector/vector.pyt__xor__¦scCs±t|tƒstdƒ‚n%t|tƒs<t|tƒrCtjSg}x[|jjƒD]J\}}x;|jjƒD]*\}}|j||t ||ƒƒquWqYWt |ŒS(s± 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) s!Invalid operand for outer product( RRR$RRRRR tappendRR(RR&R!tk1tv1tk2tv2((s2lib/python2.7/site-packages/sympy/vector/vector.pyR-«s&cCsh|jtjƒr&|rtjStjS|rF|j|ƒ|j|ƒS|j|ƒ|j|ƒ|SdS(sê Returns the vector or scalar projection of the 'other' on 'self'. Examples ======== >>> from sympy.vector.coordsysrect import CoordSys3D >>> from sympy.vector.vector import Vector, BaseVector >>> 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 N(tequalsRRRR"(RR&tscalar((s2lib/python2.7/site-packages/sympy/vector/vector.pyt projectionÑs cCs‚ddlm}t|tƒr>tdƒtdƒtdƒfStt||ƒƒƒjƒ}tg|D]}|j |ƒ^qfƒS(sé 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) iÿÿÿÿ(t_get_coord_sys_from_expri( tsympy.vector.operatorsR9RRRtnexttitert base_vectorsttupleR"(RR9tbase_vecti((s2lib/python2.7/site-packages/sympy/vector/vector.pyt _projectionsìs cCs |j|ƒS(N(R-(RR&((s2lib/python2.7/site-packages/sympy/vector/vector.pyt__or__scCs,tg|jƒD]}|j|ƒ^qƒS(s 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]]) (tMatrixR=R"(Rtsystemtunit_vec((s2lib/python2.7/site-packages/sympy/vector/vector.pyt to_matrix scCsQi}xD|jjƒD]3\}}|j|jtjƒ||||j>> 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 (RR tgetRDRR(Rtpartstvecttmeasure((s2lib/python2.7/site-packages/sympy/vector/vector.pytseparate*s (t__name__t __module__t__doc__tTruet is_Vectort _op_prioritytpropertyRRRR"R+R,R0R-tFalseR8RARBRFRK(((s2lib/python2.7/site-packages/sympy/vector/vector.pyR s&   >   ,   &    t BaseVectorcBsSeZdZddd„Zed„ƒZdd„Zed„ƒZeZ eZ RS(sl Class to denote a base vector. Unicode pretty forms in Python 2 should use the prefix ``u``. cCs`|dkrdj|ƒ}n|dkr<dj|ƒ}nt|ƒ}t|ƒ}|tddƒkrxtdƒ‚nt|tƒs–tdƒ‚n|j|}t t |ƒj |t |ƒ|ƒ}||_ it dƒ|6|_t dƒ|_|jd||_d ||_||_||_||f|_itd 6}t|ƒ|_||_|S( Nsx{0}sx_{0}iisindex must be 0, 1 or 2ssystem should be a CoordSys3Dit.ut commutative(tNonetformatR%Rt ValueErrorRR R$t _vector_namestsuperRTt__new__Rt_base_instanceRt_measure_numbert_namet _pretty_formt _latex_formt_systemt_idRORt _assumptionst_sys(tclstindexRDt pretty_strt latex_strtnametobjt assumptions((s2lib/python2.7/site-packages/sympy/vector/vector.pyR\Ms0     $      cCs|jS(N(Rb(R((s2lib/python2.7/site-packages/sympy/vector/vector.pyRDpscCs|jS(N(R_(Rtprinter((s2lib/python2.7/site-packages/sympy/vector/vector.pyt__str__tscCs|hS(N((R((s2lib/python2.7/site-packages/sympy/vector/vector.pyt free_symbolswsN( RLRMRNRWR\RRRDRnRot__repr__t _sympystr(((s2lib/python2.7/site-packages/sympy/vector/vector.pyRTEs# t VectorAddcBs/eZdZd„Zdd„ZeZeZRS(s2 Class to denote sum of Vector instances. cOstj|||Ž}|S(N(R R\(RfR!toptionsRk((s2lib/python2.7/site-packages/sympy/vector/vector.pyR\„sc Cs§d}t|jƒjƒƒ}|jdd„ƒxk|D]c\}}|jƒ}xH|D]@}||jkrW|j||}||j|ƒd7}qWqWWq8W|d S(NttkeycSs|djƒS(Ni(Rn(tx((s2lib/python2.7/site-packages/sympy/vector/vector.pyt‹Rts + iýÿÿÿ(tlistRKR tsortR=RRn( RRmtret_strR RDRIt base_vectsRvt temp_vect((s2lib/python2.7/site-packages/sympy/vector/vector.pyRnˆs  "N(RLRMRNR\RWRnRpRq(((s2lib/python2.7/site-packages/sympy/vector/vector.pyRrs   t VectorMulcBs5eZdZd„Zed„ƒZed„ƒZRS(s> Class to denote products of scalars and BaseVectors. cOstj|||Ž}|S(N(RR\(RfR!RsRk((s2lib/python2.7/site-packages/sympy/vector/vector.pyR\scCs|jS(s) The BaseVector involved in the product. (R](R((s2lib/python2.7/site-packages/sympy/vector/vector.pyt base_vector¡scCs|jS(sU The scalar expression involved in the definition of this VectorMul. (R^(R((s2lib/python2.7/site-packages/sympy/vector/vector.pytmeasure_number¦s(RLRMRNR\RRR~R(((s2lib/python2.7/site-packages/sympy/vector/vector.pyR}˜s RcBs)eZdZdZdZdZd„ZRS(s' Class to denote a zero vector g333333(@u0s\mathbf{\hat{0}}cCstj|ƒ}|S(N(RR\(RfRk((s2lib/python2.7/site-packages/sympy/vector/vector.pyR\·s(RLRMRNRQR`RaR\(((s2lib/python2.7/site-packages/sympy/vector/vector.pyR®s tCrosscBs eZdZd„Zd„ZRS(sŒ 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 cCsit|ƒ}t|ƒ}t|ƒt|ƒkr>t||ƒ Stj|||ƒ}||_||_|S(N(RRR€RR\t_expr1t_expr2(Rftexpr1texpr2Rk((s2lib/python2.7/site-packages/sympy/vector/vector.pyR\Îs    cKst|j|jƒS(N(R,RR‚(Rtkwargs((s2lib/python2.7/site-packages/sympy/vector/vector.pytdoitØs(RLRMRNR\R†(((s2lib/python2.7/site-packages/sympy/vector/vector.pyR€¼s tDotcBs eZdZd„Zd„ZRS(s 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 cCsat|ƒ}t|ƒ}t||gdtƒ\}}tj|||ƒ}||_||_|S(NRu(RtsortedRRR\RR‚(RfRƒR„Rk((s2lib/python2.7/site-packages/sympy/vector/vector.pyR\ðs    cKst|j|jƒS(N(R"RR‚(RR…((s2lib/python2.7/site-packages/sympy/vector/vector.pyR†ùs(RLRMRNR\R†(((s2lib/python2.7/site-packages/sympy/vector/vector.pyR‡Üs c s tˆtƒr/tj‡fd†ˆjDƒƒStˆtƒr^tj‡fd†ˆjDƒƒStˆtƒrXtˆtƒrXˆjˆjkrˆjd}ˆjd}||kr»tjSdddhj ||hƒj ƒ}|dd|krùdnd}|ˆjj ƒ|Sy-ddl m }t|ˆˆjƒˆƒSWqXtˆˆƒSXntˆtƒsvtˆtƒr}tjStˆtƒr¾ttˆjjƒƒƒ\}}|t|ˆƒStˆtƒrÿttˆjjƒƒƒ\} } | tˆ| ƒStˆˆƒS( s^ 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 c3s|]}t|ˆƒVqdS(N(R,(t.0R@(tvect2(s2lib/python2.7/site-packages/sympy/vector/vector.pys sc3s|]}tˆ|ƒVqdS(N(R,(R‰R@(tvect1(s2lib/python2.7/site-packages/sympy/vector/vector.pys siiiiiÿÿÿÿ(texpress(RR RrtfromiterR!RTReRRt differencetpopR=t functionsRŒR,R€RR}R;R<RR ( R‹RŠtn1tn2tn3tsignRŒR3tm1R5tm2((R‹RŠs2lib/python2.7/site-packages/sympy/vector/vector.pyR,ýs6     $ !!cs tˆtƒr/tj‡fd†ˆjDƒƒStˆtƒr^tj‡fd†ˆjDƒƒStˆtƒrìtˆtƒrìˆjˆjkr¨ˆˆkr¡tjStjSy-ddl m }t ˆ|ˆˆjƒƒSWqìt ˆˆƒSXntˆt ƒs tˆt ƒrtjStˆtƒrRttˆjjƒƒƒ\}}|t |ˆƒStˆtƒr“ttˆjjƒƒƒ\}}|t ˆ|ƒSt ˆˆƒS(s2 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 c3s|]}t|ˆƒVqdS(N(R"(R‰R@(RŠ(s2lib/python2.7/site-packages/sympy/vector/vector.pys <sc3s|]}tˆ|ƒVqdS(N(R"(R‰R@(R‹(s2lib/python2.7/site-packages/sympy/vector/vector.pys >si(RŒ(RR RR!RTReRtOnetZeroRRŒR"R‡RR}R;R<RR (R‹RŠRŒR3R•R5R–((R‹RŠs2lib/python2.7/site-packages/sympy/vector/vector.pyR"+s*  !!cCsƒt|tƒr-t|tƒr-tdƒ‚nRt|tƒrs|tjkrZtdƒ‚nt|t|tjƒƒStdƒ‚dS(s( Helper for division involving vectors. sCannot divide two vectorssCannot divide a vector by zeros#Invalid division involving a vectorN( RRR$RR˜RYR}Rt NegativeOne(toneR&((s2lib/python2.7/site-packages/sympy/vector/vector.pyt _vect_divSsN(-tsympy.core.assumptionsRt sympy.coreRRRtsympy.core.exprRRtsympy.core.compatibilityRRtsympyRR RCR tsympy.vector.coordsysrectR tsympy.vector.basisdependentR R RRtsympy.vector.dyadicRRRRRTRrR}RR€R‡R,R"R›t _expr_typet _mul_funct _add_funct _zero_funct _base_funct _div_helperR(((s2lib/python2.7/site-packages/sympy/vector/vector.pyts2"ÿ:: ! . (