ó ¡¼™\c@sddlmZddlmZddlmZddlmZmZddl m Z m Z m Z ddl mZmZmZmZddlmZddlmZded „Zd „Zd „Zd „Zd „Zd„Zd„Zd„Zd„Z d„Z!dS(iÿÿÿÿ(t CoordSys3D(tDel(t BaseScalar(tVectort BaseVector(tgradienttcurlt divergence(tdifft integratetStsimplify(tsympify(tDyadicc Csð|dks|tjkr|St|tƒs=tdƒ‚nt|tƒrw|dk rgtdƒ‚n|rýg}x<|jtt ƒD](}|j |kr†|j |j ƒq†q†Wt |ƒ}i}x$|D]}|j |j|ƒƒqËW|j|ƒ}ntj}|jƒ} x^| D]V}||kra|j|ƒ| |j|ƒ} |t| |ƒ7}q|| |7}qW|St|tƒr9|dkr›|}nt|tƒs¹tdƒ‚ntj} |} xj|jjƒD]Y\} }| t||d| ƒt| jd|d| ƒt| jd|d| ƒB7} qØW| S|dk rTtdƒ‚n|rèt gƒ}t|ƒ}x9|jtƒD](}|j |kr‚|j|j ƒq‚q‚Wi}x$|D]}|j |j|ƒƒq»W|j|ƒS|SdS(sK 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 is>system should be a CoordSys3D instancesJsystem2 should not be provided for VectorssCsystem2 should be a CoordSys3D instancet variablesiN(Rtzerot isinstanceRt TypeErrortNonet ValueErrortatomsRRtsystemtappendtsettupdatet scalar_maptsubstseparatetrotation_matrixt to_matrixtmatrix_to_vectorR t componentstitemstexpresstargsR tadd(texprRtsystem2Rt system_listtxt subs_dicttftoutvectpartsttemptoutdyadtvartktvt system_set((s5lib/python2.7/site-packages/sympy/vector/functions.pyR! sl0              #      c Cs3ddlm}||ƒ}t|ƒdkrtt|ƒƒ}t||dtƒ}|jƒ\}}}|jƒ\}}} t j ||ƒt ||ƒ} | t j ||ƒt ||ƒ7} | t j ||ƒt || ƒ7} | dkr t |t ƒr t j } n| St |t ƒr%t j StdƒSdS(sé 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 iÿÿÿÿ(t_get_coord_sys_from_expriRN(tsympy.vector.operatorsR2tlentnexttiterR!tTruet base_vectorst base_scalarsRtdotRRRR ( tfieldtdirection_vectorR2t coord_systitjR/R'tytztout((s5lib/python2.7/site-packages/sympy/vector/functions.pytdirectional_derivative€s  ## cCsQtƒ}|jr8tt|ƒƒtt|ƒƒjƒS|j||ƒƒjƒS(s! 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 (Rt is_VectorRRRtdoitR:(R$tdelop((s5lib/python2.7/site-packages/sympy/vector/functions.pyt laplacian±s  &cCsJt|tƒstdƒ‚n|tjkr1tSt|ƒjƒtjkS(sŸ 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 sfield should be a Vector(RRRRR7RR (R;((s5lib/python2.7/site-packages/sympy/vector/functions.pytis_conservativeÐs cCsMt|tƒstdƒ‚n|tjkr1tSt|ƒjƒtdƒkS(s— 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 sfield should be a Vectori(RRRRR7RR R (R;((s5lib/python2.7/site-packages/sympy/vector/functions.pyt is_solenoidalñs cCs t|ƒstdƒ‚n|tjkr4tdƒSt|tƒsRtdƒ‚nt||dt ƒ}|j ƒ}|j ƒ}t |j |dƒ|dƒ}xct|dƒD]Q\}}t|||dƒ}|j |ƒ|}|t |||dƒ7}q°W|S(sà 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 sField is not conservativeiscoord_sys must be a CoordSys3DRi(RHRRRR RRRR!R7R8R9R R:t enumerateR(R;R=t dimensionstscalarst temp_functionR>tdimt partial_diff((s5lib/python2.7/site-packages/sympy/vector/functions.pytscalar_potentials     c Cs t|tƒstdƒ‚nt|tƒr?t||ƒ}n|}|j}t|j|ƒ|dtƒ}t|j|ƒ|dtƒ}i}i} |j ƒ} xNt |j ƒƒD]:\} } | j |ƒ|| | <| j |ƒ| | | >> from sympy.vector import CoordSys3D, Point >>> 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 scoord_sys must be a CoordSys3DR(RRRRRPtoriginR!t position_wrtR7R9RJR8R:R( R;R=tpoint1tpoint2t scalar_fnRQt position1t position2t subs_dict1t subs_dict2RLR>R'((s5lib/python2.7/site-packages/sympy/vector/functions.pytscalar_potential_differenceEs"-    cCsHtj}|jƒ}x,t|ƒD]\}}||||7}q"W|S(sà 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 (RRR8RJ(tmatrixRR*tvectsR>R'((s5lib/python2.7/site-packages/sympy/vector/functions.pyRŠs  cCs|j|jkr9tdt|ƒdt|ƒƒ‚ng}|}x)|jdk rp|j|ƒ|j}qHW|j|ƒt|ƒ}g}|}x&||kr¾|j|ƒ|j}q™Wt|ƒ}|j|ƒ}x+|dkr|j||ƒ|d8}qÝW||fS(s¿ 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. s!No connecting path found between s and iiN( t_rootRtstrt_parentRRRR4tindex(t from_objectt to_objectt other_pathtobjt object_sett from_pathR`R>((s5lib/python2.7/site-packages/sympy/vector/functions.pyt_path±s* $       cOsì|jdtƒ}td„|Dƒƒs7tdƒ‚ng}x€t|ƒD]r\}}x/t|ƒD]!}|||j||ƒ8}qcWt|ƒjt j ƒr¯t dƒ‚n|j |ƒqJW|règ|D]}|j ƒ^qÍ}n|S(sŠ 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.vector import Vector, BaseVector >>> 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 t orthonormalcss|]}t|tƒVqdS(N(RR(t.0tvec((s5lib/python2.7/site-packages/sympy/vector/functions.pys ÷ss#Each element must be of Type Vectors#Vector set not linearly independent(tgettFalsetallRRJtranget projectionR tequalsRRRRt normalize(tvlisttkwargsRht ortho_vlistR>ttermR?Rj((s5lib/python2.7/site-packages/sympy/vector/functions.pyt orthogonalizeÒs#"N("tsympy.vector.coordsysrectRtsympy.vector.deloperatorRtsympy.vector.scalarRtsympy.vector.vectorRRR3RRRtsympyRR R R t sympy.coreR tsympy.vector.dyadicR RRlR!RCRGRHRIRPRZRRgRv(((s5lib/python2.7/site-packages/sympy/vector/functions.pyts""u 1  ! ! 3 E ' !