o 3IhbO@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$Zd%d&Zeje_d'd(Zd)d*Zd+d,Zd-S)/Vectorz Super class for all Vector classes. Ideally, neither this class nor any of its subclasses should be instantiated by the user. FTg(@z type[Vector] _expr_type _mul_func _add_func _zero_func _base_func VectorZerozerorkindcC|jS)a Returns the components of this vector in the form of a Python dictionary mapping BaseVector instances to the corresponding measure numbers. Examples ======== >>> from sympy.vector import CoordSys3D >>> C = CoordSys3D('C') >>> v = 3*C.i + 4*C.j + 5*C.k >>> v.components {C.i: 3, C.j: 4, C.k: 5} ) _componentsselfr&S/home/air/sanwanet/gpt-api/venv/lib/python3.10/site-packages/sympy/vector/vector.py components(szVector.componentscCs t||@S)z7 Returns the magnitude of this vector. r r$r&r&r' magnitude= zVector.magnitudecCs ||S)z@ Returns the normalized version of this vector. )r)r$r&r&r' normalizeCr*zVector.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%otherdiff diff_mag2r&r&r'r-Is?  z Vector.equalscst|tr/ttr tjStj}|jD]\}}|jd}||||jd7}q|Sddl m }t||tfsFt t |ddt||rSfdd}|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)sympy.vector.functionsr3)fieldr3r$r&r'r3s  z*Vector.dot..directional_derivative) isinstancerrrr r(itemsargsr,sympy.vector.deloperatorr2 TypeErrorstr)r%r.outveckvvect_dotr2r3r&r$r'r,s" (      z Vector.dotcC ||SNr,r%r.r&r&r'__and__ zVector.__and__cCsnt|tr2t|tr tjStj}|jD]\}}||jd}||jd}|||7}q|St||S)a Returns the cross product of this Vector with another Vector or Dyadic instance. The cross product is a Vector, if 'other' is a Vector. If 'other' is a Dyadic, this returns a Dyadic instance. Parameters ========== other: Vector/Dyadic The Vector or Dyadic we are crossing with. Examples ======== >>> from sympy.vector import CoordSys3D >>> C = CoordSys3D('C') >>> C.i.cross(C.j) C.k >>> C.i ^ C.i 0 >>> v = 3*C.i + 4*C.j + 5*C.k >>> v ^ C.i 5*C.j + (-4)*C.k >>> d = C.i.outer(C.i) >>> C.j.cross(d) (-1)*(C.k|C.i) rr1) r6rrr r(r7crossr8outer)r%r.outdyadr=r> cross_productrGr&r&r'rFs  z Vector.crosscCr@rArFrCr&r&r'__xor__rEzVector.__xor__cCsTt|ts tdt|tst|trtjSddt|j|jD}t |S)a Returns the outer product of this vector with another, in the form of a Dyadic instance. Parameters ========== other : Vector The Vector with respect to which the outer product is to be computed. Examples ======== >>> from sympy.vector import CoordSys3D >>> N = CoordSys3D('N') >>> N.i.outer(N.j) (N.i|N.j) z!Invalid operand for outer productcSs*g|]\\}}\}}||t||qSr&)r).0k1v1k2v2r&r&r' s*z Vector.outer..) r6rr:rrr rr(r7r)r%r.r8r&r&r'rGs  z Vector.outercCsL|tjr|r tjStjS|r||||S|||||S)a Returns the vector or scalar projection of the 'other' on 'self'. Examples ======== >>> from sympy.vector.coordsysrect import CoordSys3D >>> C = CoordSys3D('C') >>> i, j, k = C.base_vectors() >>> v1 = i + j + k >>> v2 = 3*i + 4*j >>> v1.projection(v2) 7/3*C.i + 7/3*C.j + 7/3*C.k >>> v1.projection(v2, scalar=True) 7/3 )r-rr r Zeror,)r%r.scalarr&r&r' projection$s 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_systemscg|]}|qSr&rBrLir$r&r'rQXz'Vector._projections..) sympy.vector.operatorsrUr6rr rRnextiter base_vectorstuple)r%rUbase_vecr&r$r' _projections>s  zVector._projectionscCr@rA)rGrCr&r&r'__or__ZrEz 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]]) crVr&rB)rLunit_vecr$r&r'rQyrYz$Vector.to_matrix..)Matrixr])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(r7getrdrr )r%partsvectmeasurer&r&r'separate|s  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) r6rr:r rR ValueError VectorMulr NegativeOne)oner.r&r&r' _div_helpers  zVector._div_helperN)F)__name__ __module__ __qualname____doc__ is_scalar is_Vector _op_priority__annotations__rr!propertyr(r)r+r-r,rDrFrKrGrTr`rarerjror&r&r&r'rs@  C>, $  rcsfdd}|S)NcsNtti}g}|jD] }t|jtr||q |tkr%t|jddSdS)NF)deep)r VectorAddr8r6r!rappenddoit)expr vec_classvectorstermclsr&r'_postprocessors    z)get_postprocessor.._postprocessorr&)rrr&rr'get_postprocessors  rrcsReZdZdZdfdd ZeddZddZd d Zed d Z d dZ Z S) BaseVectorz) Class to denote a base vector. 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. commutativeT)formatr;rangerkr6rr: _vector_namessuper__new__r _base_instanceOner#_measure_number_name _pretty_form _latex_form_system_idr _assumptions_sys)rindexrd pretty_str latex_strnameobj assumptions __class__r&r'rs0        zBaseVector.__new__cCr"rA)rr$r&r&r'rdzBaseVector.systemcCr"rA)r)r%printerr&r&r' _sympystrszBaseVector._sympystrcCs"|j\}}||d|j|S)Nr)r_printr)r%rrrdr&r&r' _sympyreprs zBaseVector._sympyreprcCs|hSrAr&r$r&r&r' free_symbolsrzBaseVector.free_symbolscCs|SrAr&r$r&r&r'_eval_conjugateszBaseVector._eval_conjugate)NN) rprqrrrsrrxrdrrrr __classcell__r&r&rr'rs#  rc@ eZdZdZddZddZdS)rzz2 Class to denote sum of Vector instances. cOtj|g|Ri|}|SrA)rrrr8optionsrr&r&r'rzVectorAdd.__new__c Cszd}t|}|jddd|D]"\}}|}|D]}||jvr5|j||}|||d7}qq|ddS)NrcSs |dS)Nr)__str__)xr&r&r's z%VectorAdd._sympystr..keyz + )listrjr7sortr]r(r) r%rret_strr7rdrh base_vectsr temp_vectr&r&r'rs   zVectorAdd._sympystrN)rprqrrrsrrr&r&r&r'rzs rzc@s0eZdZdZddZeddZeddZdS) rlz> Class to denote products of scalars and BaseVectors. cOrrA)rrrr&r&r'r rzVectorMul.__new__cCr")z) The BaseVector involved in the product. )rr$r&r&r' base_vectorszVectorMul.base_vectorcCr")zU The scalar expression involved in the definition of this VectorMul. )rr$r&r&r'measure_numberszVectorMul.measure_numberN)rprqrrrsrrxrrr&r&r&r'rls rlc@s$eZdZdZdZdZdZddZdS)rz' Class to denote a zero vector g333333(@0z\mathbf{\hat{0}}cCst|}|SrA)rr)rrr&r&r'r%s zVectorZero.__new__N)rprqrrrsrvrrrr&r&r&r'rs  rc@r)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|||}||_||_|SrA)r r rrr_expr1_expr2rexpr1expr2rr&r&r'r<s z Cross.__new__cKt|j|jSrA)rFrrr%hintsr&r&r'r|Fz Cross.doitNrprqrrrsrr|r&r&r&r'r*s rc@r)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 rrrrrr&r&r'r^sz Dot.__new__cKrrA)r,rrrr&r&r'r|grzDot.doitNrr&r&r&r'rJs rc sttrtfddjDSttr$tfddjDSttrttrjjkrejd}jd}||krEtjShd ||h }|dd|krZdnd}|j |Sdd l m }z|j}WntytYSwt|SttsttrtjSttrttj\}} | t|Sttrttj\} } | t| StS) a^ Returns cross product of two vectors. Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy.vector.vector import cross >>> R = CoordSys3D('R') >>> v1 = R.i + R.j + R.k >>> v2 = R.x * R.i + R.y * R.j + R.z * R.k >>> cross(v1, v2) (-R.y + R.z)*R.i + (R.x - R.z)*R.j + (-R.x + R.y)*R.k c3|]}t|VqdSrArJrWvect2r&r' |zcross..c3|]}t|VqdSrArJrWvect1r&r'r~rr>rr1r1rexpress)r6rrzfromiterr8rrrr differencepopr] functionsrrkrrFrrlr[r\r(r7) rrn1n2n3signrr>rNm1rPm2r&rrr'rFks:           rFcs@ttrtfddjDSttr$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 c3rrArBrWrr&r'rrzdot..c3rrArBrWrr&r'rrr1r)r6rrr8rrr rrRrrrkrr,rrlr[r\r(r7)rrrr>rNrrPrr&rr'r,s.         r,N)7 __future__r itertoolsr sympy.corerrsympy.core.assumptionsrsympy.core.exprrrsympy.core.powerr sympy.core.singletonr sympy.core.sortingr sympy.core.sympifyr (sympy.functions.elementary.miscellaneousrsympy.matrices.immutablerrcsympy.vector.basisdependentrrrrsympy.vector.coordsysrectrsympy.vector.dyadicrrrsympy.vector.kindrrr"_constructor_postprocessor_mappingrrzrlrrrrFr,rrrrrr r&r&r&r'sH            < !0*