+ j^<^RIHt^RIHt^RIHt^RIHtHt^RI H t H t H t ^RI Ht^RIHt^RIHt^R IHt^R IHtRR ltRtRtRtRtRtRtRtRtRR /Rlt R #)) CoordSys3D)Del) BaseScalar)Vector BaseVector)gradientcurl divergence)diff)S) integrate)sympify)DyadicNFc V^\P39dV#\V\4'g \ R4h\V\4'EdVe \ R4hV'dxVP \\4Uu0uFqDPkK upV0, p/pVF#pVPVPV44K% VPV4p\PpVP4p V FYpWA8wdBVPV4W,PV4,p V\!W4, pKJWV,, pK[ V#\V\"4'dVfTp\V\4'g \ R4h\"Pp Tp VP$P'4F_wrV \)WV R7\)V P*^,WR7\)V P*^,W,R7,,, p Ka V #Ve \ R4hV'd\-4p\/V4pVP \4F1pVPV8wgKVP1VP4K3 /pVF#pVPVPV44K% VPV4#V#uupi)a 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 z>system should be a CoordSys3D instancezJsystem2 should not be provided for VectorszCsystem2 should be a CoordSys3D instance variables)rzero isinstancer TypeError ValueErroratomsrrsystemupdate scalar_mapsubsseparaterotation_matrix to_matrixmatrix_to_vectorr componentsitemsexpressargssetradd)exprrsystem2rx system_list subs_dictfoutvecpartstempoutdyadvarkv system_sets&&&& W/home/luca/screen-checking/.venv/lib/python3.14/site-packages/sympy/vector/functions.pyr"r" sw` 6;; fj ) )"# #$  )* * .2ZZ J-OP-O88-OPTZS[[KI   f!56!99Y'D A{--a0583E3Ea3HH*488("   D& ! ! ?G':..&' '++OO))+DA S9 FB GCDE FG,   )* * J4=DZZ +88v%NN188,,I  f!56 99Y' ' _QsK#c^RIHpV!V4p\V4^8d\\ V44p\ WRR7pVP 4wrEpVP4wrxp \P!W4\W4,p V \P!W4\W4,, p V \P!W4\W 4,, p V ^8Xd'\V\4'd\Pp V #\V\4'd\P#\P#)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 )_get_coord_systemsTr)sympy.vector.operatorsr6lennextiterr" base_vectors base_scalarsrdotr rrr Zero) fielddirection_vectorr6 coord_sysijr1r(yzouts && r4directional_derivativerG~s::"5)I 9~i) D9((*a((*ajj)-U> vzz*.e?? vzz*.e?? !8 5&11++C E6 " "{{vv c\4pVP'd<\\V44\ \ V44, P 4#VP V!V44P 4#)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 doitr=)r&delops& r4 laplacianrMsW2 EE ~~~D)*T$t*-==CCEE 99U4[ ! & & ((rHc\V\4'g \R4hV\P8XdR#\ V4P 4\P8H#)ac 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 simplifyr?s&r4is_conservativerRsH4 eV $ $233   ;   !V[[ 00rHc\V\4'g \R4hV\P8XdR#\ V4P 4\ PJ#)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 rOT)rrrrr rPr r>rQs&r4 is_solenoidalrTsI4 eV $ $233   e  % % '166 11rHcN\V4'g \R4hV\P8Xd\P #\ V\4'g \R4h\WRR7pVP4pVP4p\VPV^,4V^,4p\VR,4FVwrV\WCV^,,4pVPV4V, pV\WsV^,,4, pKX V#)ao 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 CoordSys3DTr:NN)rRrrrr r>rrrr"r;r<r r= enumerater )r?rA dimensionsscalars temp_functionrBdim partial_diffs&& r4scalar_potentialr^s> 5 ! !455  vv  i , ,899 E 5E'')J$$&Geii 1 6 CMJrN+M1q5>: yy~ 4 <Q@@ , rHc:\V\4'g \R4h\V\4'd \ W4pMTpVP p\ VPV4VRR7p\ VPV4VRR7p/p/p VP4p \VP44F9wrV PV4WV ,&V PV4WV ,&K; VPV 4VPV4, #)a Returns the scalar potential difference between two points in a certain coordinate system, wrt a given field. If a scalar field is provided, its values at the two points are considered. If a conservative vector field is provided, the values of its scalar potential function at the two points are used. Returns (potential at point2) - (potential at point1) The position vectors of the two Points are calculated wrt the origin of the coordinate system provided. Parameters ========== field : Vector/Expr The field to calculate wrt coord_sys : CoordSys3D The coordinate system to do the calculations in point1 : Point The initial Point in given coordinate system position2 : Point The second Point in the given coordinate system Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy.vector import scalar_potential_difference >>> R = CoordSys3D('R') >>> P = R.origin.locate_new('P', R.x*R.i + R.y*R.j + R.z*R.k) >>> vectfield = 4*R.x*R.y*R.i + 2*R.x**2*R.j >>> scalar_potential_difference(vectfield, R, R.origin, P) 2*R.x**2*R.y >>> Q = R.origin.locate_new('O', 3*R.i + R.j + 2*R.k) >>> scalar_potential_difference(vectfield, R, P, Q) -2*R.x**2*R.y + 18 rVTr) rrrrr^originr" position_wrtr<rXr;r=r) r?rApoint1point2 scalar_fnr` position1 position2 subs_dict1 subs_dict2rZrBr(s &&&& r4scalar_potential_differenceriCsZ i , ,899%  $U6    F++F3Y"&(I++F3Y"&(IJJ$$&G)0023!"y!1 1:!"y!1 1:4 >>* % z(B BBrHc\PpVP4p\V4FwrEW%W4,,, pK V#)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 )rrr;rX)matrixrr,vectsrBr(s&& r4rrs@@[[F    !E&!eh," MrHcVPVP8wd3\R\V4,R,\V4,4h.pTpVPe VP V4VPpK-VP V4\ V4p.pTpW49d VP V4VPpK%\ V4pVPW"PV4RR1,4We3#)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 N) _rootrstr_parentappendr$r8extendindex) from_object to_object other_pathobj object_set from_pathrts&& r4_pathr{sIOO+<[)*,3469)nEF FJ C ++ !#kkcZJI C kk  NE Z 0 0 5 9r 9:;  rH orthonormalc\;QJdRV4F 'dK RM RM !RV44'g \R4h.p\V4F|wr4\V4F'pWBV,P W,4,pK) VP \ P4'd \R4hVPV4K~ V'dVUu.uFqfP4NK ppV#uupi)a Takes a sequence of independent vectors and orthogonalizes them using the Gram - Schmidt process. Returns a list of orthogonal or orthonormal vectors. Parameters ========== vlist : sequence of independent vectors to be made orthogonal. orthonormal : Optional parameter Set to True if the vectors returned should be orthonormal. Default: False Examples ======== >>> from sympy.vector.coordsysrect import CoordSys3D >>> from sympy.vector.functions import orthogonalize >>> C = CoordSys3D('C') >>> i, j, k = C.base_vectors() >>> v1 = i + 2*j >>> v2 = 2*i + 3*j >>> orthogonalize(v1, v2) [C.i + 2*C.j, 2/5*C.i + (-1/5)*C.j] References ========== .. [1] https://en.wikipedia.org/wiki/Gram-Schmidt_process c3B"TFp\V\4xK R#5i)N)rr).0vecs& r4 orthogonalize..s8%3z#v&&%sFTz#Each element must be of Type Vectorz#Vector set not linearly independent) allrrXrange projectionequalsrrrrr normalize)r|vlist ortho_vlistrBtermrCrs$* r4 orthogonalizersF 38%83338%8 8 8=>>KU#qA N--eh7 7D ;;v{{ # #BC C4 $2=>+3}}+ > ?sC;)NF)!sympy.vector.coordsysrectrsympy.vector.deloperatorrsympy.vector.scalarrsympy.vector.vectorrrr7rr r sympy.core.functionr sympy.core.singletonr sympy.integrals.integralsr sympy.corersympy.vector.dyadicrr"rGrMrRrTr^rirr{rrHr4rsi0(*2==$"/&nb.b)>1B2B0fBCJ$N<4e4rH