a lhbO@sddlmZddlmZddlmZmZddlmZddl m Z m Z ddl m Z ddlmZddlmZdd lmZdd lmZdd lmZdd lmZmZmZmZdd lmZddl m!Z!m"Z"m#Z#ddl$m%Z%GdddeZ&ddZ'de'egiej(e&<Gddde&e Z)Gdddee&Z*Gdddee&Z+Gdddee&Z,Gddde&Z-Gdd d e Z.d!d"Z/d#d$Z0e&e&_1e+e&_2e*e&_3e,e&_4e)e&_5e,e&_6d%S)&) annotations)product)AddBasic) StdFactKB) AtomicExprExpr)Pow)S)default_sort_key)sympifysqrt)ImmutableDenseMatrix)BasisDependentZeroBasisDependentBasisDependentMulBasisDependentAdd) CoordSys3D)Dyadic BaseDyadic DyadicAdd) VectorKindc@seZdZUdZdZdZdZded<ded<ded<ded <ded <d ed <eZ d ed<e ddZ ddZ ddZ ddZddZddZeje_ddZddZeje_dd Zd.d!d"Ze d#d$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 VectorZerozerorkindcCs|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%A/var/www/auris/lib/python3.9/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& normalizeCszVector.normalizecCs||}||}|dS)aM 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$otherdiffZ diff_mag2r%r%r&r+Is? z Vector.equalscst|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.functionsr0)fieldr0r#r%r&r0s z*Vector.dot..directional_derivative) isinstancerrrr r'itemsargsr*Zsympy.vector.deloperatorr/ TypeErrorstr)r$r,ZoutveckvZvect_dotr/r0r%r#r&r*s"(      z Vector.dotcCs ||SNr*r$r,r%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.) r2rrr r'r3crossr4outer)r$r,Zoutdyadr7r8Z cross_productr>r%r%r&r=s  z Vector.crosscCs ||Sr9r=r;r%r%r&__xor__szVector.__xor__cCsVt|tstdnt|ts(t|tr.tjSddt|j|jD}t |S)a Returns the outer product of this vector with another, in the form of a Dyadic instance. Parameters ========== other : Vector The Vector with respect to which the outer product is to be computed. Examples ======== >>> from sympy.vector import CoordSys3D >>> N = CoordSys3D('N') >>> N.i.outer(N.j) (N.i|N.j) z!Invalid operand for outer productcSs*g|]"\\}}\}}||t||qSr%)r).0Zk1v1Zk2v2r%r%r& z Vector.outer..) r2rr5rrr rr'r3r)r$r,r4r%r%r&r>s   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)r+rr r Zeror*)r$r,Zscalarr%r%r& projection$s  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%r:rAir#r%r&rDXrEz'Vector._projections..) Zsympy.vector.operatorsrHr2rr rFnextiter base_vectorstuple)r$rHZbase_vecr%r#r& _projections>s   zVector._projectionscCs ||Sr9)r>r;r%r%r&__or__Zsz Vector.__or__cstfdd|DS)a Returns the matrix form of this vector with respect to the specified coordinate system. Parameters ========== system : CoordSys3D The system wrt which the matrix form is to be computed Examples ======== >>> from sympy.vector import CoordSys3D >>> C = CoordSys3D('C') >>> from sympy.abc import a, b, c >>> v = a*C.i + b*C.j + c*C.k >>> v.to_matrix(C) Matrix([ [a], [b], [c]]) csg|]}|qSr%r:)rAZunit_vecr#r%r&rDyrEz$Vector.to_matrix..)MatrixrM)r$systemr%r#r& to_matrix_s zVector.to_matrixcCs:i}|jD]&\}}||jtj||||j<q|S)a The constituents of this vector in different coordinate systems, as per its definition. Returns a dict mapping each CoordSys3D to the corresponding constituent Vector. Examples ======== >>> from sympy.vector import CoordSys3D >>> R1 = CoordSys3D('R1') >>> R2 = CoordSys3D('R2') >>> v = R1.i + R2.i >>> v.separate() == {R1: R1.i, R2: R2.i} True )r'r3getrRrr )r$partsvectZmeasurer%r%r&separate|s  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) r2rr5r rF ValueError VectorMulr Z NegativeOne)Zoner,r%r%r& _div_helpers   zVector._div_helperN)F)__name__ __module__ __qualname____doc__Z is_scalarZ is_Vector _op_priority__annotations__rr!propertyr'r(r)r+r*r<r=r@r>rGrOrPrSrWrZr%r%r%r&rs>  C>,$  rcsfdd}|S)NcsNtti}g}|jD]}t|jtr||q|tkrJt|jddSdS)NF)deep)r VectorAddr4r2r!rappenddoit)exprZ vec_classZvectorsZtermclsr%r&_postprocessors    z)get_postprocessor.._postprocessorr%)rhrir%rgr&get_postprocessors rjrcsReZdZdZdfdd ZeddZddZd d Zed d Z d dZ Z S) BaseVectorz) Class to denote a base vector. Ncs|durd|}|dur$d|}t|}t|}|tddvrJtdt|ts\td|j|}t |t ||}||_ |t j i|_ t j |_|jd||_d||_||_||_||f|_d d i}t||_||_|S) Nzx{}zx_{}rzindex must be 0, 1 or 2zsystem should be a CoordSys3D.Z commutativeT)formatr6rangerXr2rr5 _vector_namessuper__new__r _base_instanceOner"_measure_number_name _pretty_form _latex_form_system_idrZ _assumptions_sys)rhindexrRZ pretty_strZ latex_strnameobjZ assumptions __class__r%r&rss0        zBaseVector.__new__cCs|jSr9)rzr#r%r%r&rRszBaseVector.systemcCs|jSr9)rw)r$printerr%r%r& _sympystrszBaseVector._sympystrcCs"|j\}}||d|j|S)Nrm)r{_printrq)r$rr}rRr%r%r& _sympyreprs zBaseVector._sympyreprcCs|hSr9r%r#r%r%r& free_symbolsszBaseVector.free_symbolscCs|Sr9r%r#r%r%r&_eval_conjugateszBaseVector._eval_conjugate)NN) r[r\r]r^rsrarRrrrr __classcell__r%r%rr&rks#  rkc@s eZdZdZddZddZdS)rcz2 Class to denote sum of Vector instances. cOstj|g|Ri|}|Sr9)rrsrhr4optionsrr%r%r&rsszVectorAdd.__new__c Cszd}t|}|jddd|D]D\}}|}|D].}||jvr<|j||}|||d7}qrEz%VectorAdd._sympystr..keyz + )listrWr3sortrMr'r) r$rZret_strr3rRrVZ base_vectsrZ temp_vectr%r%r&rs  zVectorAdd._sympystrN)r[r\r]r^rsrr%r%r%r&rcsrcc@s0eZdZdZddZeddZeddZdS) rYz> Class to denote products of scalars and BaseVectors. cOstj|g|Ri|}|Sr9)rrsrr%r%r&rs szVectorMul.__new__cCs|jS)z) The BaseVector involved in the product. )rtr#r%r%r& base_vectorszVectorMul.base_vectorcCs|jS)zU The scalar expression involved in the definition of this VectorMul. )rvr#r%r%r&measure_numberszVectorMul.measure_numberN)r[r\r]r^rsrarrr%r%r%r&rYs  rYc@s$eZdZdZdZdZdZddZdS)rz' Class to denote a zero vector g333333(@0z\mathbf{\hat{0}}cCst|}|Sr9)rrs)rhrr%r%r&rs%s zVectorZero.__new__N)r[r\r]r^r_rxryrsr%r%r%r&rs rc@s eZdZdZddZddZdS)Crossa Represents unevaluated Cross product. Examples ======== >>> from sympy.vector import CoordSys3D, Cross >>> R = CoordSys3D('R') >>> v1 = R.i + R.j + R.k >>> v2 = R.x * R.i + R.y * R.j + R.z * R.k >>> Cross(v1, v2) Cross(R.i + R.j + R.k, R.x*R.i + R.y*R.j + R.z*R.k) >>> Cross(v1, v2).doit() (-R.y + R.z)*R.i + (R.x - R.z)*R.j + (-R.x + R.y)*R.k cCsJt|}t|}t|t|kr,t|| St|||}||_||_|Sr9)r r rrrs_expr1_expr2rhexpr1Zexpr2rr%r%r&rs<s z Cross.__new__cKst|j|jSr9)r=rrr$hintsr%r%r&reFsz Cross.doitNr[r\r]r^rsrer%r%r%r&r*s 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 rrsrrrr%r%r&rs^sz Dot.__new__cKst|j|jSr9)r*rrrr%r%r&regszDot.doitNrr%r%r%r&rJs rc sttr$tfddjDSttrHtfddjDSttrttrjjkrΈjd}jd}||krtjShd ||h }|dd|krdnd}|j |Sdd l m }z|j}WntytYS0t|Stts,ttr2tjSttrbttj\}} | t|Sttrttj\} } | t| StS) a^ Returns cross product of two vectors. Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy.vector.vector import cross >>> R = CoordSys3D('R') >>> v1 = R.i + R.j + R.k >>> v2 = R.x * R.i + R.y * R.j + R.z * R.k >>> cross(v1, v2) (-R.y + R.z)*R.i + (R.x - R.z)*R.j + (-R.x + R.y)*R.k c3s|]}t|VqdSr9r?rIvect2r%r& |rEzcross..c3s|]}t|VqdSr9r?rIvect1r%r&r~rEr>rr.r.rlexpress)r2rrcfromiterr4rkr|rr differencepoprM functionsrrXrr=rrYrKrLr'r3) rrZn1Zn2Zn3signrr8rBm1rCm2r%rrr&r=ks8         r=csDttr$tfddjDSttrHtfddjDSttrttrjjkr|krvtjStjSddl m }z|j}Wnt yt YS0t |SttsttrtjSttr ttj\}}|t |Sttr:ttj\}}|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 c3s|]}t|VqdSr9r:rIrr%r&rrEzdot..c3s|]}t|VqdSr9r:rIrr%r&rrEr.r)r2rrr4rkr|r rurFrrrXrr*rrYrKrLr'r3)rrrr8rBrrCrr%rr&r*s,        r*N)7 __future__r itertoolsrZ sympy.corerrZsympy.core.assumptionsrZsympy.core.exprrrZsympy.core.powerr Zsympy.core.singletonr Zsympy.core.sortingr Zsympy.core.sympifyr Z(sympy.functions.elementary.miscellaneousrZsympy.matrices.immutablerrQZsympy.vector.basisdependentrrrrZsympy.vector.coordsysrectrZsympy.vector.dyadicrrrZsympy.vector.kindrrrjZ"_constructor_postprocessor_mappingrkrcrYrrrr=r*rrrrrr r%r%r%r&sF            < !0*