U 9%eE@srddlmZddlmZddlmZddlmZddlm Z m Z ddl m Z ddl mZddlmZdd lmZdd lmZdd lmZdd lmZmZmZmZdd lmZddlm Z m!Z!m"Z"GdddeZ#Gddde#e Z$Gdddee#Z%Gdddee#Z&Gdddee#Z'Gddde#Z(Gddde Z)ddZ*dd Z+e#e#_,e&e#_-e%e#_.e'e#_/e$e#_0e'e#_1d!S)") annotations)product)Add) StdFactKB) AtomicExprExpr)Pow)S)default_sort_key)sympifysqrt)ImmutableDenseMatrix)BasisDependentZeroBasisDependentBasisDependentMulBasisDependentAdd) CoordSys3D)Dyadic BaseDyadic DyadicAddc@seZdZUdZdZdZdZded<ded<ded<ded <ded <d ed <ed dZ ddZ ddZ ddZ ddZ e je _ddZddZeje_ddZd*ddZedd Zd!d"Zeje_d#d$Zd%d&Zd'd(Zd)S)+Vectorz Super class for all Vector classes. Ideally, neither this class nor any of its subclasses should be instantiated by the user. FTg(@z type[Vector] _expr_type _mul_func _add_func _zero_func _base_func VectorZerozerocCs|jS)a Returns the components of this vector in the form of a Python dictionary mapping BaseVector instances to the corresponding measure numbers. Examples ======== >>> from sympy.vector import CoordSys3D >>> C = CoordSys3D('C') >>> v = 3*C.i + 4*C.j + 5*C.k >>> v.components {C.i: 3, C.j: 4, C.k: 5} ) _componentsselfr"R/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/vector/vector.py components%szVector.componentscCs t||@S)z7 Returns the magnitude of this vector. r r r"r"r# magnitude:szVector.magnitudecCs ||S)z@ Returns the normalized version of this vector. )r%r r"r"r# normalize@szVector.normalizecst|tr^ttrtjStj}|jD].\}}|jd}||||jd7}q*|Sddl m }t||tfst t |ddt||rfdd}|St|S)aN Returns the dot product of this Vector, either with another Vector, or a Dyadic, or a Del operator. If 'other' is a Vector, returns the dot product scalar (SymPy expression). If 'other' is a Dyadic, the dot product is returned as a Vector. If 'other' is an instance of Del, returns the directional derivative operator as a Python function. If this function is applied to a scalar expression, it returns the directional derivative of the scalar field wrt this Vector. Parameters ========== other: Vector/Dyadic/Del The Vector or Dyadic we are dotting with, or a Del operator . Examples ======== >>> from sympy.vector import CoordSys3D, Del >>> C = CoordSys3D('C') >>> delop = Del() >>> C.i.dot(C.j) 0 >>> C.i & C.i 1 >>> v = 3*C.i + 4*C.j + 5*C.k >>> v.dot(C.k) 5 >>> (C.i & delop)(C.x*C.y*C.z) C.y*C.z >>> d = C.i.outer(C.i) >>> C.i.dot(d) C.i r)Delz is not a vector, dyadic or z del operatorcsddlm}||S)Nr)directional_derivative)Zsympy.vector.functionsr))fieldr)r r"r#r)}s z*Vector.dot..directional_derivative) isinstancerrrrr$itemsargsdotZsympy.vector.deloperatorr( TypeErrorstr)r!otherZoutveckvZvect_dotr(r)r"r r#r.Fs"(      z Vector.dotcCs ||SNr.r!r1r"r"r#__and__szVector.__and__cCsnt|trdt|trtjStj}|jD]4\}}||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+rrrr$r,crossr-outer)r!r1Zoutdyadr2r3Z cross_productr9r"r"r#r8s  z Vector.crosscCs ||Sr4r8r6r"r"r#__xor__szVector.__xor__cCsVt|tstdnt|ts(t|tr.tjSddt|j|jD}t |S)a Returns the outer product of this vector with another, in the form of a Dyadic instance. Parameters ========== other : Vector The Vector with respect to which the outer product is to be computed. Examples ======== >>> from sympy.vector import CoordSys3D >>> N = CoordSys3D('N') >>> N.i.outer(N.j) (N.i|N.j) z!Invalid operand for outer productcSs*g|]"\\}}\}}||t||qSr")r).0Zk1v1Zk2v2r"r"r# sz Vector.outer..) r+rr/rrrrr$r,r)r!r1r-r"r"r#r9s   z Vector.outercCsP|tjr|rtjStjS|r4||||S|||||SdS)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 N)equalsrrr Zeror.)r!r1Zscalarr"r"r# projections  zVector.projectioncsPddlm}ttr&tjtjtjfStt|}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_systemscsg|]}|qSr"r5r<ir r"r#r?sz'Vector._projections..) Zsympy.vector.operatorsrCr+rr rAnextiter base_vectorstuple)r!rCZbase_vecr"r r# _projectionss   zVector._projectionscCs ||Sr4)r9r6r"r"r#__or__sz 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]]) csg|]}|qSr"r5)r<Zunit_vecr r"r#r?3sz$Vector.to_matrix..)MatrixrH)r!systemr"r r# to_matrixs zVector.to_matrixcCs:i}|jD]&\}}||jtj||||j<q|S)a The constituents of this vector in different coordinate systems, as per its definition. Returns a dict mapping each CoordSys3D to the corresponding constituent Vector. Examples ======== >>> from sympy.vector import CoordSys3D >>> R1 = CoordSys3D('R1') >>> R2 = CoordSys3D('R2') >>> v = R1.i + R2.i >>> v.separate() == {R1: R1.i, R2: R2.i} True )r$r,getrMrr)r!partsvectmeasurer"r"r#separate6s  zVector.separatecCsXt|trt|trtdn6t|trL|tjkr:tdt|t|tjStddS)z( Helper for division involving vectors. zCannot divide two vectorszCannot divide a vector by zeroz#Invalid division involving a vectorN) r+rr/r rA ValueError VectorMulrZ NegativeOne)oner1r"r"r# _div_helperPs   zVector._div_helperN)F)__name__ __module__ __qualname____doc__Z is_scalarZ is_Vector _op_priority__annotations__propertyr$r%r&r.r7r8r;r9rBrJrKrNrSrWr"r"r"r#rs:  >,$  rcsJeZdZdZd fdd ZeddZddZd d Zed d Z Z S) BaseVectorz) Class to denote a base vector. Ncs|dkrd|}|dkr$d|}t|}t|}|tddkrJtdt|ts\td|j|}t |t ||}||_ |t j i|_ t j |_|jd||_d||_||_||_||f|_d d i}t||_||_|S) Nzx{}zx_{}rzindex must be 0, 1 or 2zsystem should be a CoordSys3D.Z commutativeT)formatr0rangerTr+rr/ _vector_namessuper__new__r _base_instanceOner_measure_number_name _pretty_form _latex_form_system_idrZ _assumptions_sys)clsindexrM pretty_strZ latex_strnameobjZ assumptions __class__r"r#rgbs0        zBaseVector.__new__cCs|jSr4)rnr r"r"r#rMszBaseVector.systemcCs|jSr4)rk)r!printerr"r"r# _sympystrszBaseVector._sympystrcCs"|j\}}||d|j|S)Nra)ro_printre)r!rxrrrMr"r"r# _sympyreprs zBaseVector._sympyreprcCs|hSr4r"r r"r"r# free_symbolsszBaseVector.free_symbols)NN) rXrYrZr[rgr^rMryr{r| __classcell__r"r"rvr#r_\s# r_c@s eZdZdZddZddZdS) VectorAddz2 Class to denote sum of Vector instances. cOstj|f||}|Sr4)rrgrqr-optionsrur"r"r#rgszVectorAdd.__new__c Cszd}t|}|jddd|D]D\}}|}|D].}||jkr<|j||}|||d7}qz%VectorAdd._sympystr..keyz + )listrSr,sortrHr$rz) r!rxZret_strr,rMrQZ base_vectsrZ temp_vectr"r"r#rys  zVectorAdd._sympystrN)rXrYrZr[rgryr"r"r"r#r~sr~c@s0eZdZdZddZeddZeddZdS) rUz> Class to denote products of scalars and BaseVectors. cOstj|f||}|Sr4)rrgrr"r"r#rgszVectorMul.__new__cCs|jS)z) The BaseVector involved in the product. )rhr r"r"r# base_vectorszVectorMul.base_vectorcCs|jS)zU The scalar expression involved in the definition of this VectorMul. )rjr r"r"r#measure_numberszVectorMul.measure_numberN)rXrYrZr[rgr^rrr"r"r"r#rUs  rUc@s$eZdZdZdZdZdZddZdS)rz' Class to denote a zero vector g333333(@0z\mathbf{\hat{0}}cCst|}|Sr4)rrg)rqrur"r"r#rgs zVectorZero.__new__N)rXrYrZr[r\rlrmrgr"r"r"r#rs rc@s eZdZdZddZddZdS)Crossa Represents unevaluated Cross product. Examples ======== >>> from sympy.vector import CoordSys3D, Cross >>> R = CoordSys3D('R') >>> v1 = R.i + R.j + R.k >>> v2 = R.x * R.i + R.y * R.j + R.z * R.k >>> Cross(v1, v2) Cross(R.i + R.j + R.k, R.x*R.i + R.y*R.j + R.z*R.k) >>> Cross(v1, v2).doit() (-R.y + R.z)*R.i + (R.x - R.z)*R.j + (-R.x + R.y)*R.k cCsJt|}t|}t|t|kr,t|| St|||}||_||_|Sr4)r r rrrg_expr1_expr2rqexpr1Zexpr2rur"r"r#rgs z Cross.__new__cKst|j|jSr4)r8rrr!hintsr"r"r#doitsz Cross.doitNrXrYrZr[rgrr"r"r"r#rs rc@s eZdZdZddZddZdS)Dota Represents unevaluated Dot product. Examples ======== >>> from sympy.vector import CoordSys3D, Dot >>> from sympy import symbols >>> R = CoordSys3D('R') >>> a, b, c = symbols('a b c') >>> v1 = R.i + R.j + R.k >>> v2 = a * R.i + b * R.j + c * R.k >>> Dot(v1, v2) Dot(R.i + R.j + R.k, a*R.i + b*R.j + c*R.k) >>> Dot(v1, v2).doit() a + b + c cCsBt|}t|}t||gtd\}}t|||}||_||_|S)Nr)r sortedr rrgrrrr"r"r#rgsz Dot.__new__cKst|j|jSr4)r.rrrr"r"r#r szDot.doitNrr"r"r"r#rs rc sttr$tfddjDSttrHtfddjDSttrttrjjkrЈjd}jd}||krtjSdddh ||h }|dd|krdnd}|j |Sdd l m }z|j}Wn tk r tYSXt|Stts0ttr6tjSttrfttj\}} | 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 c3s|]}t|VqdSr4r:rDvect2r"r# !szcross..c3s|]}t|VqdSr4r:rDvect1r"r#r#srr'r`express)r+rr~fromiterr-r_rprr differencepoprH functionsrrTrr8rrUrFrGr$r,) rrZn1Zn2Zn3signrr3r=m1r>m2r"rrr#r8s8         r8csFttr$tfddjDSttrHtfddjDSttrttrˆjjkr|krvtjStjSddl m }z|j}Wnt k rt YSXt |SttsttrtjSttr ttj\}}|t |Sttr>> 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|VqdSr4r5rDrr"r#rQszdot..c3s|]}t|VqdSr4r5rDrr"r#rSsr'r)r+rrr-r_rpr rirArrrTrr.rrUrFrGr$r,)rrrr3r=rr>rr"rr#r.@s,       r.N)2 __future__r itertoolsrZsympy.core.addrZsympy.core.assumptionsrZsympy.core.exprrrZsympy.core.powerrZsympy.core.singletonr Zsympy.core.sortingr Zsympy.core.sympifyr Z(sympy.functions.elementary.miscellaneousr Zsympy.matrices.immutablerrLZsympy.vector.basisdependentrrrrZsympy.vector.coordsysrectrZsympy.vector.dyadicrrrrr_r~rUrrrr8r.rrrrrrr"r"r"r#s<           K9 !0*