o 8VaE@sFddlmZddlmZddlmZmZmZddlm Z m Z ddl m Z ddl mZmZmZddlmZddlmZmZmZmZdd lmZmZmZGd d d eZGd d d ee ZGdddeeZGdddeeZ GdddeeZ!GdddeZ"Gddde Z#ddZ$ddZ%ee_&e e_'ee_(e!e_)ee_*e!e_+dS))Type) StdFactKB)SPowsympify) AtomicExprExpr)default_sort_key)sqrtImmutableMatrixAdd) CoordSys3D)BasisDependentBasisDependentAddBasisDependentMulBasisDependentZero) BaseDyadicDyadic DyadicAddc@seZdZdZdZdZdZdZdZdZ dZ dZ e ddZ ddZd d Zd d Zd dZeje_ddZddZeje_ddZd"ddZe ddZddZeje_ddZddZd d!ZdS)#Vectorz Super class for all Vector classes. Ideally, neither this class nor any of its subclasses should be instantiated by the user. Tg(@NcC|jS)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} ) _componentsselfr5/usr/lib/python3/dist-packages/sympy/vector/vector.py componentsszVector.componentscCs t||@S)z7 Returns the magnitude of this vector. )r rrrr magnitude4 zVector.magnitudecCs ||S)z@ Returns the normalized version of this vector. )rrrrr normalize:rzVector.normalizecst|tr/ttr tjStj}|jD]\}}|jd}||||jd7}q|Sddl m }t|tsIt||sIt t |ddt||rVfdd}|St|S)aN 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 operatorcsddlm}||S)Nr)directional_derivative)Zsympy.vector.functionsr")Zfieldr"rrrr"ws  z*Vector.dot..directional_derivative) isinstancer VectorZerorzeroritemsargsdotZsympy.vector.deloperatorr! TypeErrorstr)rotherZoutveckvZvect_dotr!r"rrrr(@s" (      z Vector.dotcC ||SNr(rr+rrr__and__~ zVector.__and__cCsnt|tr2t|tr tjStj}|jD]\}}||jd}||jd}|||7}q|St||S)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 ) r#rr$r%rr&crossr'outer)rr+Zoutdyadr,r-Z cross_productr5rrrr4s  z Vector.crosscCr.r/r4r1rrr__xor__r3zVector.__xor__cCsxt|ts tdt|tst|trtjSg}|jD]\}}|jD]\}}|||t ||q&qt |S)a 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) r#rr)r$rr%rr&appendrr)rr+r'Zk1v1Zk2v2rrrr5s  z Vector.outerFcCsL|tjr|r tjStjS|r||||S|||||S)a Returns the vector or scalar projection of the 'other' on 'self'. Examples ======== >>> from sympy.vector.coordsysrect import CoordSys3D >>> C = CoordSys3D('C') >>> i, j, k = C.base_vectors() >>> v1 = i + j + k >>> v2 = 3*i + 4*j >>> v1.projection(v2) 7/3*C.i + 7/3*C.j + 7/3*C.k >>> v1.projection(v2, scalar=True) 7/3 )Zequalsrr%rZeror()rr+Zscalarrrr projections zVector.projectioncsPddlm}ttrtjtjtjfStt|}t fdd|DS)a Returns the components of this vector but the output includes also zero values components. Examples ======== >>> from sympy.vector import CoordSys3D, Vector >>> C = CoordSys3D('C') >>> v1 = 3*C.i + 4*C.j + 5*C.k >>> v1._projections (3, 4, 5) >>> v2 = C.x*C.y*C.z*C.i >>> v2._projections (C.x*C.y*C.z, 0, 0) >>> v3 = Vector.zero >>> v3._projections (0, 0, 0) r)_get_coord_sys_from_exprcg|]}|qSrr0.0irrr z'Vector._projections..) Zsympy.vector.operatorsr=r#r$rr;nextiter base_vectorstuple)rr=Zbase_vecrrr _projectionss  zVector._projectionscCr.r/)r5r1rrr__or__r3z Vector.__or__cstfdd|DS)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]]) cr>rr0)r@Zunit_vecrrrrB/rCz$Vector.to_matrix..)MatrixrF)rsystemrrr to_matrixs zVector.to_matrixcCs:i}|jD]\}}||jtj||||j<q|S)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 )rr&getrKrr%)rpartsvectZmeasurerrrseparate2s  zVector.separatecCsRt|trt|trtdt|tr%|tjkrtdt|t|tjStd)z( Helper for division involving vectors. zCannot divide two vectorszCannot divide a vector by zeroz#Invalid division involving a vector) r#rr)rr; ValueError VectorMulrZ NegativeOne)Zoner+rrr _div_helperLs  zVector._div_helper)F)__name__ __module__ __qualname____doc__Z is_Vector _op_priority _expr_type _mul_func _add_func _zero_func _base_funcr%propertyrrrr(r2r4r7r5r<rHrIrLrPrSrrrrrs: >, &  rcsBeZdZdZd fdd ZeddZddZed d ZZ S) BaseVectorzl Class to denote a base vector. Unicode pretty forms in Python 2 should use the prefix ``u``. Ncs|dur d|}|durd|}t|}t|}|tddvr%tdt|ts.td|j|}t |t ||}||_ |t j i|_ t j |_|jd||_d||_||_||_||f|_d d i}t||_||_|S) Nzx{}zx_{}rzindex must be 0, 1 or 2zsystem should be a CoordSys3D.Z commutativeT)formatr*rangerQr#r r)Z _vector_namessuper__new__r_base_instanceOner_measure_number_name _pretty_form _latex_form_systemZ_idrZ _assumptions_sys)clsindexrKZ pretty_strZ latex_strnameobjZ assumptions __class__rrrf`s0        zBaseVector.__new__cCrr/)rmrrrrrKzBaseVector.systemcCrr/)rj)rprinterrrr _sympystrszBaseVector._sympystrcCs|hSr/rrrrr free_symbolsruzBaseVector.free_symbols)NN) rTrUrVrWrfr^rKrwrx __classcell__rrrsrr_Xs# r_c@ eZdZdZddZddZdS) VectorAddz2 Class to denote sum of Vector instances. cOtj|g|Ri|}|Sr/)rrfror'optionsrrrrrrfzVectorAdd.__new__c Cszd}t|}|jddd|D]"\}}|}|D]}||jvr5|j||}|||d7}qq|ddS)NrbcSs |dS)Nr)__str__)xrrrs z%VectorAdd._sympystr..keyz + )listrPr&sortrFrZ_print) rrvZret_strr&rKrOZ base_vectsrZ temp_vectrrrrws   zVectorAdd._sympystrN)rTrUrVrWrfrwrrrrr{s r{c@s0eZdZdZddZeddZeddZdS) rRz> Class to denote products of scalars and BaseVectors. cOr|r/)rrfr}rrrrfrzVectorMul.__new__cCr)z) The BaseVector involved in the product. )rgrrrr base_vectorszVectorMul.base_vectorcCr)zU The scalar expression involved in the definition of this VectorMul. )rirrrrmeasure_numberszVectorMul.measure_numberN)rTrUrVrWrfr^rrrrrrrRs rRc@s$eZdZdZdZdZdZddZdS)r$z' Class to denote a zero vector g333333(@0z\mathbf{\hat{0}}cCst|}|Sr/)rrf)rorrrrrrfs zVectorZero.__new__N)rTrUrVrWrXrkrlrfrrrrr$s  r$c@rz)Crossa Represents unevaluated Cross product. Examples ======== >>> from sympy.vector import CoordSys3D, Cross >>> R = CoordSys3D('R') >>> v1 = R.i + R.j + R.k >>> v2 = R.x * R.i + R.y * R.j + R.z * R.k >>> Cross(v1, v2) Cross(R.i + R.j + R.k, R.x*R.i + R.y*R.j + R.z*R.k) >>> Cross(v1, v2).doit() (-R.y + R.z)*R.i + (R.x - R.z)*R.j + (-R.x + R.y)*R.k cCsJt|}t|}t|t|krt|| St|||}||_||_|Sr/)rr rrrf_expr1_expr2roZexpr1Zexpr2rrrrrrfs z Cross.__new__cKt|j|jSr/)r4rrrkwargsrrrdoitz Cross.doitNrTrUrVrWrfrrrrrrs rc@rz)Dota Represents unevaluated Dot product. Examples ======== >>> from sympy.vector import CoordSys3D, Dot >>> from sympy import symbols >>> R = CoordSys3D('R') >>> a, b, c = symbols('a b c') >>> v1 = R.i + R.j + R.k >>> v2 = a * R.i + b * R.j + c * R.k >>> Dot(v1, v2) Dot(R.i + R.j + R.k, a*R.i + b*R.j + c*R.k) >>> Dot(v1, v2).doit() a + b + c cCsBt|}t|}t||gtd\}}t|||}||_||_|S)Nr)rsortedr rrfrrrrrrrfsz Dot.__new__cKrr/)r(rrrrrrrrzDot.doitNrrrrrrs rc sttrtfddjDSttr$tfddjDSttrttrjjkrejd}jd}||krEtjShd ||h }|dd|krZdnd}|j |Sdd l m }z|j}WntytYSwt|SttsttrtjSttrttj\}} | t|Sttrttj\} } | t| StS) a^ 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|]}t|VqdSr/r6r?vect2rr zcross..c3|]}t|VqdSr/r6r?vect1rrrrr>rr r r`express)r#r r{fromiterr'r_rnrr% differencepoprF functionsrrQrr4r$rRrDrErr&) rrZn1Zn2Zn3signrr-r9m1r:m2rrrrr4 s:           r4cs@ttrtfddjDSttr$tfddjDSttr`ttr`jjkr>kr;tjStjSddl m }z|j}Wnt yZt YSwt |SttsjttrmtjSttrttj\}}|t |Sttrttj\}}|t |St S)a2 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 c3rr/r0r?rrrrKrzdot..c3rr/r0r?rrrrMrr r)r#r rr'r_rnrrhr;rrrQrr(r$rRrDrErr&)rrrr-r9rr:rrrrr(:s.         r(N),typingrZsympy.core.assumptionsrZ sympy.corerrrZsympy.core.exprrrZsympy.core.compatibilityr Zsympyr r rJr Zsympy.vector.coordsysrectr Zsympy.vector.basisdependentrrrrZsympy.vector.dyadicrrrrr_r{rRr$rrr4r(rYrZr[r\r]r%rrrrs4   L7 !0*