Ed<ddlmZddlmZddlmZddlmZmZddl m Z m Z m Z ddl mZddlmZddlmZdd lmZdd lmZdd lmZddZdZdZdZdZdZdZdZ dZ!d ddZ"d S)) CoordSys3D)Del) BaseScalar)Vector BaseVector)gradientcurl divergence)diff)S) integrate)simplify)sympify)DyadicNFc |dtjfvr|St|tst dt|tr|t d|rrd|ttD|hz }i}|D]*}| | |+| |}tj}| }|D]X} | |krE| | || | z} |t| |z }M||| z }Y|St|t r||}t|tst dt j} |} |jD]V\} }| t'||| t'| jd|| t'| jd|| zzz } W| S|t d|rt+}t-|}|tD]'} | j|kr|| j(i}|D]*}| | |+| |S|S) aK Global function for 'express' functionality. Re-expresses a Vector, Dyadic or scalar(sympyfiable) in the given coordinate system. If 'variables' is True, then the coordinate variables (base scalars) of other coordinate systems present in the vector/scalar field or dyadic are also substituted in terms of the base scalars of the given system. Parameters ========== expr : Vector/Dyadic/scalar(sympyfiable) The expression to re-express in CoordSys3D 'system' system: CoordSys3D The coordinate system the expr is to be expressed in system2: CoordSys3D The other coordinate system required for re-expression (only for a Dyadic Expr) variables : boolean Specifies whether to substitute the coordinate variables present in expr, in terms of those of parameter system Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy import Symbol, cos, sin >>> N = CoordSys3D('N') >>> q = Symbol('q') >>> B = N.orient_new_axis('B', q, N.k) >>> from sympy.vector import express >>> express(B.i, N) (cos(q))*N.i + (sin(q))*N.j >>> express(N.x, B, variables=True) B.x*cos(q) - B.y*sin(q) >>> d = N.i.outer(N.i) >>> express(d, B, N) == (cos(q))*(B.i|N.i) + (-sin(q))*(B.j|N.i) True rz>system should be a CoordSys3D instanceNzJsystem2 should not be provided for Vectorsch|] }|j S)system).0xs 6lib/python3.11/site-packages/sympy/vector/functions.py zexpress..MsPPP18PPPzCsystem2 should be a CoordSys3D instance variables)rzero isinstancer TypeError ValueErroratomsrrupdate scalar_mapsubsseparaterotation_matrix to_matrixmatrix_to_vectorr componentsitemsexpressargssetrradd)exprrsystem2r system_list subs_dictfoutvecpartsrtempoutdyadvarkv system_sets rr+r+s ` 6; fj ) )#"## #$7  *)** *  (QPTZZ J-O-OPPPTZS[[KI  7 7  f!5!5666699Y''D  # #AF{ #--a00583E3Ea3H3HH*4888%(" D& ! !  G':.. '&'' '+O))++ F FDAq 6S999 FcBBB GsCCCDE FGG  *)** *  (J4==DZZ ++ - -8v%-NN18,,,I 7 7  f!5!5666699Y'' ' rcddlm}||}t|dkrtt |}t ||d}|\}}}|\}}} tj ||t||z} | tj ||t||zz } | tj ||t|| zz } | dkr!t|tr tj } | St|tr tj StjS)a Returns the directional derivative of a scalar or vector field computed along a given vector in coordinate system which parameters are expressed. Parameters ========== field : Vector or Scalar The scalar or vector field to compute the directional derivative of direction_vector : Vector The vector to calculated directional derivative along them. Examples ======== >>> from sympy.vector import CoordSys3D, directional_derivative >>> R = CoordSys3D('R') >>> f1 = R.x*R.y*R.z >>> v1 = 3*R.i + 4*R.j + R.k >>> directional_derivative(f1, v1) R.x*R.y + 4*R.x*R.z + 3*R.y*R.z >>> f2 = 5*R.x**2*R.z >>> directional_derivative(f2, v1) 5*R.x**2 + 30*R.x*R.z r)_get_coord_systemsTr)sympy.vector.operatorsr=lennextiterr+ base_vectors base_scalarsrdotr rrr Zero) fielddirection_vectorr= coord_sysijr9ryzouts rdirectional_derivativerNs?::99999""5))I 9~~i)) yD999((**1a((**1aj)1--UA> vz*A..eQ?? vz*A..eQ?? !8  5&11 +C E6 " "{v rc"t}|jrKtt|t t |z S|||S)a! Return the laplacian of the given field computed in terms of the base scalars of the given coordinate system. Parameters ========== expr : SymPy Expr or Vector expr denotes a scalar or vector field. Examples ======== >>> from sympy.vector import CoordSys3D, laplacian >>> R = CoordSys3D('R') >>> f = R.x**2*R.y**5*R.z >>> laplacian(f) 20*R.x**2*R.y**3*R.z + 2*R.y**5*R.z >>> f = R.x**2*R.i + R.y**3*R.j + R.z**4*R.k >>> laplacian(f) 2*R.i + 6*R.y*R.j + 12*R.z**2*R.k )r is_Vectorrr r doitrD)r/delops r laplacianrSss2 EEE ~FD))**T$t**-=-==CCEEE 99UU4[[ ! ! & & ( ((rct|tstd|tjkrdSt |tjkS)a Checks if a field is conservative. Parameters ========== field : Vector The field to check for conservative property Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy.vector import is_conservative >>> R = CoordSys3D('R') >>> is_conservative(R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k) True >>> is_conservative(R.z*R.j) False field should be a VectorT)rrrrr rrFs ris_conservativerWsX4 eV $ $42333  t ;;   ! !V[ 00rct|tstd|tjkrdSt |t juS)a Checks if a field is solenoidal. Parameters ========== field : Vector The field to check for solenoidal property Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy.vector import is_solenoidal >>> R = CoordSys3D('R') >>> is_solenoidal(R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k) True >>> is_solenoidal(R.y * R.j) False rUT)rrrrr rr rErVs r is_solenoidalrYsZ4 eV $ $42333  t e   % % ' '16 11rcht|std|tjkr tjSt |tstdt||d}| }| }t| |d|d}t|ddD]R\}}t|||dz}| ||z }|t|||dzz }S|S)a Returns the scalar potential function of a field in a given coordinate system (without the added integration constant). Parameters ========== field : Vector The vector field whose scalar potential function is to be calculated coord_sys : CoordSys3D The coordinate system to do the calculation in Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy.vector import scalar_potential, gradient >>> R = CoordSys3D('R') >>> scalar_potential(R.k, R) == R.z True >>> scalar_field = 2*R.x**2*R.y*R.z >>> grad_field = gradient(scalar_field) >>> scalar_potential(grad_field, R) 2*R.x**2*R.y*R.z zField is not conservativecoord_sys must be a CoordSys3DTrrrN)rWr rrr rErrrr+rBrCr rD enumerater )rFrH dimensionsscalars temp_functionrIdim partial_diffs rscalar_potentialrbs)> 5 ! !64555  v  i , ,:8999 E9 5 5 5E''))J$$&&Geii 1 66 CCMJqrrN++AA3M71q5>:: yy~~ 4 <Q@@@ rct|tstdt|trt ||}n|}|j}t |||d}t |||d}i}i} |} t| D]A\} } | ||| | <| || | | <B| | | |z S)a) Returns the scalar potential difference between two points in a certain coordinate system, wrt a given field. If a scalar field is provided, its values at the two points are considered. If a conservative vector field is provided, the values of its scalar potential function at the two points are used. Returns (potential at point2) - (potential at point1) The position vectors of the two Points are calculated wrt the origin of the coordinate system provided. Parameters ========== field : Vector/Expr The field to calculate wrt coord_sys : CoordSys3D The coordinate system to do the calculations in point1 : Point The initial Point in given coordinate system position2 : Point The second Point in the given coordinate system Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy.vector import scalar_potential_difference >>> R = CoordSys3D('R') >>> P = R.origin.locate_new('P', R.x*R.i + R.y*R.j + R.z*R.k) >>> vectfield = 4*R.x*R.y*R.i + 2*R.x**2*R.j >>> scalar_potential_difference(vectfield, R, R.origin, P) 2*R.x**2*R.y >>> Q = R.origin.locate_new('O', 3*R.i + R.j + 2*R.k) >>> scalar_potential_difference(vectfield, R, P, Q) -2*R.x**2*R.y + 18 r[Tr) rrrrrboriginr+ position_wrtrCr\rBrDr$) rFrHpoint1point2 scalar_fnrd position1 position2 subs_dict1 subs_dict2r^rIrs rscalar_potential_differencermDsHZ i , ,:8999%  $UI66   F++F33Y"&(((I++F33Y"&(((IJJ$$&&G)002233221!"y!1!1 71:!"y!1!1 71: >>* % % z(B(B BBrctj}|}t|D]\}}||||zz }|S)a Converts a vector in matrix form to a Vector instance. It is assumed that the elements of the Matrix represent the measure numbers of the components of the vector along basis vectors of 'system'. Parameters ========== matrix : SymPy Matrix, Dimensions: (3, 1) The matrix to be converted to a vector system : CoordSys3D The coordinate system the vector is to be defined in Examples ======== >>> from sympy import ImmutableMatrix as Matrix >>> m = Matrix([1, 2, 3]) >>> from sympy.vector import CoordSys3D, matrix_to_vector >>> C = CoordSys3D('C') >>> v = matrix_to_vector(m, C) >>> v C.i + 2*C.j + 3*C.k >>> v.to_matrix(C) == m True )rrrBr\)matrixrr4vectsrIrs rr(r(sQ@[F    ! !E&!!1!eAh, Mrc"|j|jkr2tdt|zdzt|zg}|}|j#|||j}|j#||t |}g}|}||vr |||j}||v t |}||}|dkr&||||dz}|dk&||fS)z Calculates the 'path' of objects starting from 'from_object' to 'to_object', along with the index of the first common ancestor in the tree. Returns (index, list) tuple. z!No connecting path found between z and Nrr)_rootr str_parentappendr-r?index) from_object to_object other_pathobj object_set from_pathrvrIs r_pathr}s]IO+F<[))*,3469)nnEFF FJ C +#k +cZJI C Z k Z   NNEA q&A''' Q q& ) r) orthonormalctd|Dstdg}t|D]\}}t|D]&}|||||z}'t |tjrtd| ||r d|D}|S)aO Takes a sequence of independent vectors and orthogonalizes them using the Gram - Schmidt process. Returns a list of orthogonal or orthonormal vectors. Parameters ========== vlist : sequence of independent vectors to be made orthogonal. orthonormal : Optional parameter Set to True if the vectors returned should be orthonormal. Default: False Examples ======== >>> from sympy.vector.coordsysrect import CoordSys3D >>> from sympy.vector.functions import orthogonalize >>> C = CoordSys3D('C') >>> i, j, k = C.base_vectors() >>> v1 = i + 2*j >>> v2 = 2*i + 3*j >>> orthogonalize(v1, v2) [C.i + 2*C.j, 2/5*C.i + (-1/5)*C.j] References ========== .. [1] https://en.wikipedia.org/wiki/Gram-Schmidt_process c3@K|]}t|tVdS)N)rrrvecs r z orthogonalize..s,883z#v&&888888rz#Each element must be of Type Vectorz#Vector set not linearly independentc6g|]}|Sr) normalizers r z!orthogonalize..s >>>3s}}>>>r) allrr\range projectionrequalsrrr ru)r~vlist ortho_vlistrItermrJs r orthogonalizersF 88%888 8 8?=>>>KU##!!4q 8 8A KN--eAh77 7DD D>>  - - DBCC C4    ?>>+>>> r)NF)#sympy.vector.coordsysrectrsympy.vector.deloperatorrsympy.vector.scalarrsympy.vector.vectorrrr>rr r sympy.core.functionr sympy.core.singletonr sympy.integrals.integralsr sympy.simplify.simplifyr sympy.corersympy.vector.dyadicrr+rNrSrWrYrbrmr(r}rrrrrs000000((((((******22222222==========$$$$$$""""""//////,,,,,,&&&&&&nnnnb...b)))>111B222B000fBCBCBCJ$$$NB',4444444r