B [@=@sddlmZddlmZddlmZddlmZmZddl m Z m Z m Z ddl mZmZmZmZddlmZddlmZdd d Zd dZddZddZddZddZddZddZddZddZd S) ) CoordSys3D)Del) BaseScalar)Vector BaseVector)gradientcurl divergence)diff integrateSsimplify)sympify)DyadicNFc Cs$|dks|tjkr|St|ts(tdt|tr|dk rDtd|rg}x*|ttD]}|j |krZ| |j qZWt |}i}x|D]}| | |qW||}tj}|} xH| D]@}||kr||| ||} |t| |7}q|| |7}qW|St|tr|dkr"|}t|ts6tdtj} |} xR|jD]D\} }| t||| dt| jd|| dt| jd|| dB7} qLW| S|dk rtd|rt g}t|}x,|tD]}|j |kr||j qWi}x|D]}| | |qW||S|SdS)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 VectorszCsystem2 should be a CoordSys3D instance) variables)rzero isinstancer TypeError ValueErrorZatomsrrsystemappendsetupdateZ scalar_mapsubsZseparateZrotation_matrixZ to_matrixmatrix_to_vectorrZ componentsitemsexpressargsradd)exprrZsystem2rZ system_listxZ subs_dictfoutvecpartsZtempZoutdyadvarkvZ system_setr(5lib/python3.7/site-packages/sympy/vector/functions.pyr sl0              rc Csddlm}||}t|dkrtt|}t||dd}|\}}}|\}}} t ||t ||} | t ||t ||7} | t ||t || 7} | dkrt |trtj } | St |trtj St dSdS)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_sys_from_exprT)rN)sympy.vector.operatorsr*lennextiterr base_vectors base_scalarsrdotr rrr ) fieldZdirection_vectorr* coord_sysijr&r!yzoutr(r(r)directional_derivatives     r9cCs:t}|jr(tt|tt|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 )rZ is_Vectorrr rZdoitr1)r Zdelopr(r(r) laplaciansr:cCs2t|tstd|tjkr dSt|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 zfield should be a VectorT)rrrrrr )r2r(r(r)is_conservatives   r;cCs4t|tstd|tjkr dSt|tdkS)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 zfield should be a VectorTr)rrrrr r r )r2r(r(r) is_solenoidals   r<cCst|std|tjkr"tdSt|ts4tdt||dd}| }| }t | |d|d}xRt |ddD]>\}}t|||d}| ||}|t |||d7}q|W|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 conservativerzcoord_sys must be a CoordSys3DT)rrN)r;rrrr rrrrr/r0r r1 enumerater )r2r3Z dimensionsscalarsZ temp_functionr4ZdimZ partial_diffr(r(r)scalar_potentials  r?c Cst|tstdt|tr(t||}n|}|j}t|||dd}t|||dd}i}i} |} x>> 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 zcoord_sys must be a CoordSys3DT)r) rrrrr?originrZ position_wrtr0r=r/r1r) r2r3Zpoint1Zpoint2Z scalar_fnr@Z position1Z position2Z subs_dict1Z subs_dict2r>r4r!r(r(r)scalar_potential_differenceDs"-     rAcCs8tj}|}x$t|D]\}}||||7}qW|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 )rrr/r=)Zmatrixrr#Zvectsr4r!r(r(r)rs rcCs|j|jkr(tdt|dt|g}|}x|jdk rN|||j}q2W||t|}g}|}x||kr|||j}qlWt|}||}x"|dkr||||d8}qW||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)Z_rootrstrZ_parentrrr,index)Z from_objectZ to_objectZ other_pathobjZ object_setZ from_pathrCr4r(r(r)_paths*           rEcOs|dd}tdd|Ds&tdg}x^t|D]R\}}x&t|D]}|||||8}qFWt|tj r|t d| |q4W|rdd|D}|S) 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.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 orthonormalFcss|]}t|tVqdS)N)rr).0vecr(r(r) sz orthogonalize..z#Each element must be of Type Vectorz#Vector set not linearly independentcSsg|] }|qSr()Z normalize)rGrHr(r(r) sz!orthogonalize..) getallrr=rangeZ projectionr Zequalsrrrr)ZvlistkwargsrFZ ortho_vlistr4Ztermr5r(r(r) orthogonalizes# rO)NF) Zsympy.vector.coordsysrectrZsympy.vector.deloperatorrZsympy.vector.scalarrZsympy.vector.vectorrrr+rrr Zsympyr r r r Z sympy.corerZsympy.vector.dyadicrrr9r:r;r<r?rArrErOr(r(r(r)s"      u1!!3E'!