U 9%e<@sddlmZddlmZddlmZddlmZmZddl m Z m Z m Z ddl mZddlmZddlmZdd lmZdd lmZdd lmZd#ddZddZddZddZddZddZddZddZ ddZ!d d d!d"Z"d S)$) CoordSys3D)Del) BaseScalar)Vector BaseVector)gradientcurl divergence)diff)S) integrate)simplify)sympify)DyadicNFc Cs|dtjfkr|St|ts$tdt|tr|dk r>td|rdd|ttD|h}i}|D]}| | |qf| |}tj}| }|D]@} | |kr| | || | } |t| |7}q||| 7}q|St|trh|dkr|}t|ts tdtj} |} |jD]D\} }| t||| dt| jd|| dt| jd || dB7} q| S|dk rztd|rt}t|}|tD]} | j|kr|| jqi}|D]}| | |q| |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 VectorscSsh|] }|jqS)system).0xrrU/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/vector/functions.py Mszexpress..zCsystem2 should be a CoordSys3D instance variables)rzero isinstancer TypeError ValueErrorZatomsrrupdateZ scalar_mapsubsZseparateZrotation_matrixZ to_matrixmatrix_to_vectorr componentsitemsexpressargssetrradd)exprrZsystem2rZ system_listZ subs_dictfoutvecpartsrtempZoutdyadvarkvZ system_setrrrr"s`0         r"c Csddlm}||}t|dkrtt|}t||dd}|\}}}|\}}} t ||t ||} | t ||t ||7} | t ||t || 7} | dkrt |trtj } | St |trtj St jSdS)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_systemsTrN)sympy.vector.operatorsr.lennextiterr" base_vectors base_scalarsrdotr rrr Zero) fieldZdirection_vectorr. coord_sysijr,ryzoutrrrdirectional_derivatives     r>cCs: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 rZdoitr5)r&Zdeloprrr 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 field should be a VectorT)rrrrrr r7rrris_conservatives   rBcCs2t|tstd|tjkr dSt|tjkS)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 r@T)rrrrr r r r6rArrr is_solenoidals   rCcCst|std|tjkr tjSt|ts2tdt ||dd}| }| }t | |d|d}t|ddD]>\}}t|||d}| ||}|t |||d7}qx|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)rBrrrr r6rrrr"r3r4r r5 enumerater )r7r8 dimensionsscalarsZ temp_functionr9dimZ partial_diffrrrscalar_potentials  rIc Cst|tstdt|tr(t||}n|}|j}t|||dd}t|||dd}i}i} |} t | D],\} } | ||| | <| || | | <qv| | | |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 rDTr) rrrrrIoriginr"Z position_wrtr4rEr3r5r) r7r8Zpoint1Zpoint2Z scalar_fnrJZ position1Z position2Z subs_dict1Z subs_dict2rGr9rrrrscalar_potential_differenceDs&-     rKcCs4tj}|}t|D]\}}||||7}q|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 )rrr3rE)matrixrr(Zvectsr9rrrrrs rcCs|j|jkr(tdt|dt|g}|}|jdk rL|||j}q0||t|}g}|}||kr|||j}qft|}||}|dkr||||d8}q||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)_rootrstr_parentappendr$r0index)Z from_objectZ to_object other_pathobjZ object_set from_pathrQr9rrr_paths4       rU) orthonormalcGstdd|Dstdg}t|D]N\}}t|D]}|||||8}q6t|tjrjt d| |q&|rdd|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 css|]}t|tVqdS)N)rrrZvecrrr sz orthogonalize..z#Each element must be of Type Vectorz#Vector set not linearly independentcSsg|] }|qSr) normalizerWrrr sz!orthogonalize..) allrrErangeZ projectionr equalsrrrrP)rVvlistZ ortho_vlistr9termr:rrr orthogonalizes#  r`)NF)#Zsympy.vector.coordsysrectrZsympy.vector.deloperatorrZsympy.vector.scalarrZsympy.vector.vectorrrr/rrr Zsympy.core.functionr Zsympy.core.singletonr Zsympy.integrals.integralsr Zsympy.simplify.simplifyr Z sympy.corerZsympy.vector.dyadicrr"r>r?rBrCrIrKrrUr`rrrrs(          q1!!3E'!