
    sh7                     .   S SK Jr  S SKJr  S SKJr  S SKJr  S SKJ	r	  S SK
Jr  S SKJrJr  S SKJr  S S	KJrJrJrJrJrJr  S S
KJr  S SKJrJrJrJrJrJr  S SK J!r!  \" S5      r"\"RG                  5       u  r$r%r&\"RO                  5       u  r(r)r*\" 5       r+\	" S5      u  r,r-r.r/S r0S r1\"Re                  S\/\"RL                  5      r3S\(S-  -  \)-  \*-  r4\" \45      r5\)S-  \$-  S\(-  \%-  -   S\)-  \*-  \&-  -   r6\" \65      r7S r8S r9S r:S r;S r<S r=S r>g)    )
Derivative)Vector)
CoordSys3D)simplify)symbols)S)cossin)Dot)curl
divergencegradientGradient
DivergenceCross)Del)is_conservativeis_solenoidalscalar_potentialdirectional_derivative	laplacianscalar_potential_difference)raisesCza b c qc                  d#   [         [        R                  -  [        R                  :X  d   e[         [        R                  -  R                  5       [        R                  s=:X  a  [	        [        R                  5      :X  d   e   e[         R                  [        R                  5      [         [        R                  -  :X  d   e[         [        -  R                  5       [        R                  :X  d   e[         R                  S[        S-  -  [        -  SS9[        R                  :X  d   e[         R                  S[        S-  -  [        -  5      [         S[        S-  -  [        -  -  :X  d   e[        [        -  [        -  [        [        -   [        -   -  n [         U -  R                  5       [        * [        -  [        [        -  -   [        -  [        [        -  [        [        -  -
  [        -  -   [        * [        -  [        [        -  -   [        -  -   s=:X  a  [	        U 5      :X  d   e   e[         U -  [         R                  U 5      :X  d   e[         R                  S[        S-  -  [        -  5      [        S[        R                  5      [        S[        R                  S-  -  [        R                  5      -
  [        R                  -  [        S[        R                  5      * [        S[        R                  5      -   [        R                  -  -   [        S[        R                  5      * [        S[        R                  S-  -  [        R                  5      -   [        R                  -  -   :X  d   e[         R                  S[        S-  -  [        -  SS9S[        -  [        -  s=:X  a!  [	        S[        S-  -  [        -  5      :X  d   e   e[         [        R                  -  [        R                  s=L a  [!        [        R                  5      :X  d   e   e[         [        R                  -  R                  5       [        R                  L d   e[         R#                  [        R                  5      [         [        R                  -  :X  d   e[         [        -  R                  5       [        R                  L d   e[         [        S-  [        -  -  R                  5       S[        -  s=:X  a  [!        [        S-  [        -  5      :X  d   e   e[         R#                  U SS9[        [        -  [        [        -  -   [        [        -  -   s=:X  a  [!        U 5      :X  d   e   e[         U -  [         R#                  U 5      :X  d   e[         R#                  S[        [        -  [        -  -  [        [        -   [        -   -  SS9S[        [        -  [        S-  -  -  S[        [        S-  -  [        -  -  -
  S[        S-  [        -  [        -  -  -
  :X  d   e[        [        -  [        [        -  -   [        [        -  -   n [         U -  [        [        R                  [        R                  5      [        [        R                  [        R                  5      -   [        [        R                  [        R                  5      -   :X  d   e[         R#                  U SS9Ss=:X  a  [!        U 5      :X  d   e   e[         U -  [         R#                  U 5      :X  d   e[%        [         U -  R                  5       5      S:X  d   e[         R'                  SSS9[        R                  s=:X  a  ['        S5      :X  d   e   e[         R'                  S5      [        S5      :X  d   e[        [        R                  5      R                  5       [        R                  :X  d   e[        [        5      [        [        R                  [        R                  5      [        R                  -  [        [        R                  [        R                  5      [        R                  -  -   [        [        R                  [        R                  5      [        R                  -  -   :X  d   e[        [        5      R                  5       [        s=:X  a  ['        [        5      :X  d   e   e[        [        [        -  [        -  5      [        [        R                  [        R                  -  [        R                  -  [        R                  5      [        R                  -  [        [        R                  [        R                  -  [        R                  -  [        R                  5      [        R                  -  -   [        [        R                  [        R                  -  [        R                  -  [        R                  5      [        R                  -  -   :X  d   e[         R'                  [        [        -  [        -  SS9[        [        -  [        -  [        [        -  [        -  -   [        [        -  [        -  -   s=:X  a"  ['        [        [        -  [        -  5      :X  d   e   e[        [        [        -  [        -  5      [         R'                  [        [        -  [        -  5      :X  d   e[        S[        S-  -  5      R                  5       S[        -  [        -  :X  d   e[        [(        [+        [        5      -  [        -  5      R                  5       [(        * [+        [        5      -  [        S-  -  [        -  [(        [-        [        5      -  [        -  [        -  -   :X  d   e[        R                  [         -  " [(        5      [        R                  L d   e[        R                  [         -  " [(        5      R                  5       [        R                  L d   eU [         -  " [        R                  5      R                  5       [        R                  :X  d   eU [         -  " [        R                  5      R                  5       [        R                  L d   e[        [         -  " [        5      R                  5       S:X  d   e[        [         -  " [        5      R                  5       S:X  d   e[        [         -  " [        5      R                  5       S:X  d   e[        [         -  " [        [        -  [        -  5      R                  5       [        [        -  :X  d   eU [         -  " [        5      R                  5       [        :X  d   eU [         -  " [        [        -  [        -  5      R                  5       S[        -  [        -  [        -  :X  d   eU [         -  " [        [        -   [        -   5      [        R                  [        R                  -   [        R                  -   :X  d   eU [         -  " [        [        -   [        -   5      R                  5       [        [        -   [        -   :X  d   eU [         -  " U 5      R                  5       U :X  d   e[        [         -  " U 5      R                  5       [        :X  d   e[        [         -  " U 5      R                  5       [        :X  d   e[        [         -  " U 5      R                  5       [        :X  d   eU [         -  " [        R                  5      R                  5       [        R                  :X  d   e[/        [        [        -  [        -  5      [        R                  L d   e[/        [        S-  5      [        S5      :X  d   e[/        [        S-  [        S-  -  [        S-  -  5      S[        S-  -  [        S-  -  S[        S-  -  [        S-  -  -   S[        S-  -  [        S-  -  -   :X  d   e[1        S	S
/ SQS9n[1        SS/ SQS9n[/        UR2                  UR4                  -   UR6                  -   5      SUR2                  -  [-        UR4                  5      UR2                  S-  [+        UR4                  5      -  -  -   :X  d   e[/        UR2                  UR4                  -   UR                  -   5      SUR2                  -  :X  d   e[/        [        [        -  [        -  [        [        -   [        -   -  5      [        R                  :X  d   e[/        [        [        S-  -  [        -  [        [        -   [        -   -  5      S[        -  [        -  [        -  S[        -  [        -  [        -  -   S[        -  [        -  [        -  -   :X  d   eg )N   Tdoitr            A	sphericalrthetaphitransformationvariable_namesBcylindricalr&   r'   z)delopr   zeror   r   crossiyjxr/   kr   r   r   Zeror   dotr   r   ar
   r	   r   r   r&   r'   r(   )vr#   r,   s      {/Users/tiagomarins/Projetos/claudeai/copy_bank/venv/lib/python3.13/site-packages/sympy/vector/tests/test_field_functions.pytest_del_operatorr=      s    6;;&++---V[[ &&(FKK     ;;v{{#uv{{'::::AI,,,;;qAvaxd;+v{{:::;;qAvax EAadF1H$4444	!AQAQYRTAaCZNacAaCi]*qbdQqSj!^;G     19A&&&KK!Q$q!133*QqssAvXqss";;QSS@ACC  :a#55qss:;ACC  :aQh#<<accABB C B KK!Q$qtK,!A 1a4N     6;;!&&CJv{{,CCCCCCFKK%%'16611199V[[!UV[[%8888AI'''AqDFN  "ac?Z1Q-??????IIadI#qsQqSy1Q3 qM    19		!$$$99Q!AY!a%!),498!A#ad(a1QT6!8n,qAqDF1H~=> > >	!ac	AaCAAIACC-qssACC !#-acc133#78 8 9 899QT9"a8:a=8888819		!$$$UQY$$&'1,,, NN14N(FKK QK    >>!a(((!&&M!V[[000!HACC-qss2QSS!133&'*4QSS!##*>)CD D E D!H??0Xa[00000!A#a%LACCQSS)133.ACCQSS)133./ACCQSS)133.// 0 / NN1Q3q5tN,aCEAaCEMAaCE!QqSUO     1Q<5>>!A#a%0000!AqD&M!QqSU***1SV8a< &&(Bs1vIadNQ3q6!a/0 1 0 KK%#qvv---[[5 !$**,666Y$**,;;;Y%%'166111YN  "a'''YN  "a'''YN  "a'''Y!A$$&!A#---YN  "a'''Y!A$$&!A#a%'111Iq1uqy!QSS133Y_444YA	"((*a!eai777YN  "a'''YN  "a'''YN  "a'''YN  "a'''Y$**,;;; QqSUqvv%%%QT?ad"""QT!Q$Yq!t^$adF1a4K!AqD&A+-!Q$q!t;< < <3{CXYA3}EXYAQSS177]QUU*+qus177|QSS!VCPQPWPWLEX7Y/YYYYQSS177]QSS()QqssU222 QqSUAEAI&'6;;666Qq!tVAXq1uqy)*aCE!Gac!eAg-!Aa78 8 8    c                     S[         -  [        -  [        -  n [         [        -  [        [        -  -   [        [         -  -   n[         S-  [        -  S[        -  -   [        S-  [        -  [
        -  -
  nS[        -  [         [        -  [        -  [
        -  -   n[        X-  SS9nU [        U5      -  U[        U 5      -  -   R                  5       n[        U5      [        U5      :X  d   e[        X#-  5      R                  5       nU[        U-  -  U[        U-  -  -   U[        -  " U5      -   U[        -  " U5      -   R                  5       n[        U5      [        U5      :X  d   e[        X-  -  R                  5       nU [        U-  -  U[        U 5      -  -   R                  5       n[        U5      [        U5      :X  d   e[        X#-  -  R                  5       nU[        U-  -  U[        U-  -  -
  R                  5       n[        U5      [        U5      :X  d   e[        X-  -  R                  5       n[        U 5      U-  U [        U-  -  -   R                  5       n[        U5      [        U5      :X  d   e[        X#-  -  R                  5       nU[        U-  -  U[        U-  -  -
  U[        -  " U5      -   U[        -  " U5      -
  R                  5       n[        U5      [        U5      :X  d   eg)z
Tests the six product rules defined with respect to the Del
operator

References
==========

.. [1] https://en.wikipedia.org/wiki/Del

r   r   Tr   N)	r6   r4   r/   r3   r5   r7   r0   r   r   )fgur;   lhsrhss         r<   test_product_rulesrE   v   s    	
!AaA	!ac	AaCA	1Q1q!tAvaxA	!ac!eAgA D
!CuQx<!eAh,&
,
,
.CC=HSM))) ,


COUQY0Iq> 5y!n./3tv C=HSM))) AC=


 COU1X/
5
5
7CC=HSM))) AE?
 
 
"COUQY0
6
6
8CC=HSM))) AE?
 
 
"C1XNqEAI/
5
5
7CC=HSM))) AE?
 
 
"C	?Q%!)_,Iq>YN+,0DF C=HSM)))r>   Pr   r"      c                     [        [        R                  5      SL d   e[        [        5      SL d   e[        S[        -  S[        -  -   S[
        -  -   5      SL d   e[        [        [        -  [        -  [        [        -  [        -  -   [        [        -  [
        -  -   5      SL d   e[        [        [        -  5      SL d   e[        [        5      SL d   e[        [        5      SL d   e[        S[        -  [        -  [        -  [        -  S[        S-  -  [        -  [        -  -   5      SL d   e[        [        [        R                  -  [        R                  [
        -  -   5      SL d   eg )NTr   r"   r   F)r   r   r1   r3   r5   r7   r4   r/   r6   
grad_field
curl_fieldrF    r>   r<   test_conservativerL      s2   6;;'4///1%%%1q51q5=1q501T999AaCEAaCEMAaCE12  1q5!U***:&$...:&%///AaCE!GAI!Q$q
23  1QSS5133q5=)T111r>   c                  P   [        [        R                  5      SL d   e[        [        5      SL d   e[        S[        -  S[        -  -   S[
        -  -   5      SL d   e[        [        [        -  [        -  [        [        -  [        -  -   [        [        -  [
        -  -   5      SL d   e[        [        [        -  5      SL d   e[        [        5      SL d   e[        [        5      SL d   e[        S[        -  S-   [
        -  5      SL d   e[        [        [        5      [        -  [        [        5      [        -  -   [        [        5      [        R
                  -  -   5      SL d   e[        [        [        R                  -  [        R                  [
        -  -   5      SL d   eg )NTr   r"   r   F)r   r   r1   r3   r5   r7   r4   r/   r6   rI   rJ   r	   qr
   rF   rK   r>   r<   test_solenoidalrP      sK   %---t###QQQ./4777!A#a%!A#a%-!A#a%/0  Q5((($---$,,,"Q$(A&$...QCF1H,s1vaccz9:dBBB133Q'4///r>   c                     [        [        R                  [        R                  -  [        R                  -  S[        R
                  -  S[        R                  -  -   [        R                  -   5      [        R                  [        R                  -  S[        R                  -  [        R                  -  -   S[        R                  -  [        R                  -  -   :X  d   e[        S[        R                  S-  -  [        R                  -  S[        R
                  -  S[        R                  -  -   [        R                  -   5      S[        R                  S-  -  S[        R                  -  [        R                  -  -   :X  d   e[        S[        R                  S-  -  [        R                  -  S[        R                  -  5      [        R                  L d   e[        SS/ SQ/ S	QS
9n U R                  5       u  pnU R                  5       u  pEn[        US-  U-  U5      SU-  U-  :X  d   e[        SUS-  -  U-  SU-  SU-  -   U-   5      SUS-  -  SU-  U-  -   :X  d   eg )Nr"   r   rG   r      Dr$   r%   )e_re_thetae_phi)r+   vector_names)r   r   r6   r4   r/   r3   r5   r7   r   r8   r   base_scalarsbase_vectors)rS   r&   r'   r(   rT   rU   rV   s          r<   test_directional_derivativerZ      s   !!##acc'!##+ququ}qss/BCqss133wQRSTSVSVQVWXWZWZQZGZ]^_`_b_b]bcdcfcf]fGffff!!ACCF(133,!##!##0CD!##q&SUVWVYVYSYZ[Z]Z]S]H]]]]!!ACCF(133,!##6!&&@@@34I ;	=ANN$MAc..*C%!!Q$s(C0AaCG;;;!!AqD&*aeai.?%.GHAaQRdFUWXYUYZ]U]L]]]]r>   c                     [        [        R                  [        5      S:X  d   e[        [        [        5      [
        :X  d   e[        [        [        5      [        :X  d   e[        [        [        5      [        :X  d   e[        [        [        -  [        -  [
        [        -  [        -  -   [
        [        -  [        -  -   [        5      [
        [        -  [        -  :X  d   e[        [        [        5      [        :X  d   e[        [        [        R                  -  [        R
                  [        -  -   [        5      [
        [        -  [        [        5      -  [        [        -  [        [        5      -  -   :X  d   e[        [        [        R                  -  [        R
                  [        -  -   [        5      [        R
                  [        R                  -  :X  d   e[!        ["        S 5        g )Nr   c                  8    [        [        [        -  [        5      $ )N)r   r6   r5   r   rK   r>   r<   <lambda>'test_scalar_potential.<locals>.<lambda>   s    /!Q7r>   )r   r   r1   r   r3   r6   r5   r4   r7   r/   rI   scalar_fieldrF   r	   rO   r
   r   
ValueErrorrK   r>   r<   test_scalar_potentialra      s3   FKK+q000Aq!Q&&&Aq!Q&&&Aq!Q&&&AaCEAaCEMAaCE1151Q>>>J*l:::AaccEACCEM1-1SVac#a&j1HHHHAaccEACCEM1-QSS888
:78r>   c                  N   [         R                  R                  SS[        -  S[        -  -   S[
        -  -   5      n [         R                  R                  SS[        -  S[        -  -   S[
        -  -   5      n[         R                  R                  S	[        [        -  [        [        -  -   [        [
        -  -   5      n[        R                  R                  S
[        R                  [        R                  -  [        R                  [        R                  -  -   [        R                  [        R
                  -  -   5      n[        [        R                  [         X5      S:X  d   e[        [        [         [         R                  U5      [        :X  d   e[        [        [         [         R                  U5      [        :X  d   e[        [        [         X5      S:X  d   e[        [        [        -  [        -  [        [        -  [        -  -   [        [        -  [
        -  -   [         U U5      [        [        -  [        -  S-
  :X  d   eS[        R                  -  [        R                  [        [         5      -  [        R                  [#        [         5      -  -   -  [        R                  [#        [         5      -  [        R                  [        [         5      -  -
  S-  -  n[        [        [        [        R                  U5      R%                  5       UR%                  5       :X  d   eg )NP1r    r   r"   P2r   rG      RPPPr   i  )r   origin
locate_newr3   r5   r7   r6   r4   r/   rF   r   r   r8   r_   rI   r
   rO   r	   r   )point1point2genericpointCgenericpointPpotential_diff_Ps        r<    test_scalar_potential_differencero      s1   XX  qsQqSy1Q37FXX  qsQqSy1Q37FHH''acAaCi!A#o>MHH''acc!##gACC.?!##acc'.IJM&qvvq&AQFFF'a(57   (
Aqxx(57   'z1fELLL'!A!A()!A)./0&(57 aCEAI   !##qss3q6zACCAJ67SVacc#a&j0145'
Aqxx(577?xz%%'( ) (r>   c                  "   [        SS/ SQS9n [        SS/ SQS9n[        SU R                  -  S	U R                  -  -   5      SU R                  -  S	U R                  -  U R
                  -  -   :X  d   e[        SU R                  -  U R                  -  S	U R                  -  -   5      SU R                  -  U R                  -  S	U R                  -  U R
                  -  -   S[        U R                  5      -  U R                  -  -   :X  d   e[        S
U R                  -  S
U R                  -  -   S
U R                  -  -   5      [        R                  :X  d   e[        U R                  U R                  -  U R                  -  5      U R                  U R                  -  U R                  -  U R                  U R
                  -  -   U R                  [        U R                  5      -  U R                  -  -   :X  d   e[        U R                  U R                  -  U R                  U R
                  -  -   U R                  U R                  -  -   5      [        U R                  5      U R                  -  [        U R                  5      U R                  -  U R                  -  -   [        U R                  5      U R                  S-  -  -  S-   S[        U R                  5      U R                  -  -  -   :X  d   e[        SU R                  -  U R                  -  U R                  -  U R                  U R
                  -  -   U R                  U R                  -  U R                  -  U R                  -  -   5      [        U R                  5      U R                  -  [        U R                  5      U R                  -  U R                  -  -   [        U R                  5      U R                  S-  -  -  SU R                  -  -   U R                  [        U R                  5      -  -   :X  d   e[        [        R                  5      S
:X  d   e[        S
U R                  -  S
U R
                  -  -   S
U R                  -  -   5      S
:X  d   e[        U R                  U R                  -  U R                  U R
                  -  -   U R                  U R                  -  -   5      [        U R                  5      U R                  -  [        U R                  5      U R                  -  -  U R                  -  U R                  * U R                  -  U R
                  -  -   U R                  U R                  -  U R                  -  -   :X  d   e[        U R                  U R
                  -  U R                  U R                  -  -   5      [        U R                  5      U R                  -  [        U R                  5      U R                  -  -  U R                  -  U R                  * U R                  -  U R
                  -  -   SU R                  -  -   :X  d   e[        S
UR                  -  S
UR                  -  -   S
UR                  -  -   5      [        R                  :X  d   e[        UR                  UR                  -  UR                  -  5      UR                  UR                  -  UR                  -  UR                  UR
                  -  -   UR                  UR                  -  UR                  -  -   :X  d   e[        SUR                  -  5      SUR                  -  :X  d   e[        SUR                  -  5      SUR                  -  UR
                  -  :X  d   e[        S	UR                  -  5      S	UR                  -  :X  d   e[        UR                  UR                  -  UR                  UR
                  -  -   UR                  UR                  -  -   5      SSUR                  -  -   :X  d   e[        UR                  UR
                  -  UR                  UR                  -  -   5      S:X  d   e[        UR                  UR
                  -  UR                  UR                  -  -   5      SUR                  -  :X  d   e[        SUR                  -  SUR                  -  UR
                  -  -   S	UR                  -  -   5      [        R                  :X  d   eg )Nr#   r$   r%   r)   r,   r-   r.   r"   r   r   r   r    	   )r   r   r&   r'   r3   r5   r(   r
   r7   r   r1   r   r	   r   r/   )r#   r,   s     r<   .test_differential_operators_curvilinear_systemrr      s   3{CXYA3}EXYAAaccEAaggI%&!ACC%!ACC%)*;;;;AaccE!%%K!AGG)+,!%%aeACCi0G1SQRQXQX\>[\[^[^J^0^^^^AaccEAaggI%ag-.&++===ACCK%&!''!%%-*;aeeACCi*G177SVWXW^W^S_K_abadadJd*ddddaccACCi!''ACC-/!%%!##+=>Ls177|ACC/77#agg,qssAv:MNQRRUVX[\]\c\cXdefehehXhUiij j jaeAEEk!##o3acc!''k!%%6G6KKLLs177|ACC/77#agg,qssAv:MNQRSTSXSXQXX[\[b[bcfghgngnco[oop p pfkk"a'''aeaemae+,111ACC!''!##+%acc	12LAGGQSS 011336155&*acc9IIAGGTUTWTWKXYX[X[O[\ \ \ACC!%%)#$QWWaee);S\!##=M)NPQPSPS(SXYX]X]W]^_^a^aWacdcfcfVf(fijklknknin(nnnn AaccEAaggI%ae+,;;;ACCKO$ACC!##acc'(AACCKPQPSPSO(SSSSAaccE?ae###AaggI!ACC%!##+---AaccE?ae###acc!##g+acc!##g56!ae)CCCacc!##gACC'(A---ACC!##acc'!"ae+++!##!##acc	!AaccE)*fkk999r>   c            
      D   [        S5      n [        S5      n[        S5      n[        U R                  UR                  -  5      UR                  U R                  -  U R                  UR
                  -  -   :X  d   e[        S[        [        5      -  U R                  -  UR                  -  U R                  U R                  [        [        5      UR                  -   -   -  -   5      U R                  SUR                  -  [        [        5      -  -   U R                  -  U R                  UR                  -   [        [        5      -   U R
                  -  -   SU R                  -  [        [        5      -  U R                  -   UR                  -  -   :X  d   e[        U R                  UR                  -  5      [        U R                  UR                  -  5      :X  d   e[        [        U R                  UR                  -  5      5      [        [        U R                  UR                  -  5      5      :X  d   e[        S[        [        5      -  U R                  -  UR                  -  U R                  -  U R                  -  UR                  UR                  -  -   [        U R                  U R                  -   5      UR                  -  -   5      SU R                  -  U R                  -  UR                  -  [        [        5      -  UR                  [        U R                  U R                  -   5      -  -
  U R                  -  SU R                  -  U R                  -  UR                  -  [        [        5      -  UR                  [        U R                  U R                  -   5      -  -
  U R
                  -  -   SU R                  -  U R                  -  UR                  -  [        [        5      -  U R                  -  -   SU R                  -  U R                  -  U R                  -  [        [        5      -  UR                  -  -   UR                  UR
                  -  -   UR                  [        U R                  U R                  -   5      -   UR                  -  -   :X  d   e[        U R                  U R                  -  U R
                  U R                  -  -   U R                  U R                  -  -   UR                  UR                  -  -   UR
                  UR                  -  -   UR                  UR                  -  -   UR                  UR                  -  -   UR
                  UR                  -  -   UR                  UR                  -  -   5      [        S5      :X  d   e[        SU R                  -  U R                  -  U R                  -  UR
                  UR                  -  UR                  -  -   SU R
                  -  U R                  -  U R                  -  -   5      SU R                  -  UR                  [        UR
                  UR                  5      -  -   :X  d   e[        S[        [        5      -  U R                  -  UR                  -  UR                  -  UR                  -  5      SU R                  -  UR                  -  [        [        5      -  [        UR                  UR                  5      -  SU R                  -  UR                  -  [        [        5      -  -   SUR                  -  UR                  -  [        [        5      -  [        UR                  U R                  5      -  -   :X  d   e[        U R                  UR                  -  UR                  -  [        U R                  U R                  -  U R                  UR
                  -  5      -  5      U R                  UR                  -  UR                  -  [!        [        U R                  U R                  -  U R                  UR
                  -  5      5      -  UR                  UR                  -  [        [        U R                  U R                  -  U R                  UR
                  -  5      U R                  5      -  -   U R                  UR                  -  [        [        U R                  U R                  -  U R                  UR
                  -  5      UR                  5      -  -   U R                  UR                  -  [        [        U R                  U R                  -  U R                  UR
                  -  5      UR                  5      -  -   :X  d   e[        U R                  UR                  -  UR                  -  U R                  U R                  -  UR                  UR                  -  -   -  5      SU R                  -  UR                  -  UR                  -  U R                  S-  UR                  -  [        U R                  UR                  5      -  -   U R                  S-  UR                  -  [        U R                  UR                  5      -  -   UR                  S-  UR                  -  [        UR                  U R                  5      -  -   U R                  UR                  S-  -  [        UR                  UR                  5      -  -   :X  d   eg )	Nr:   bcr"   rq   re   r   r   )r   r   r6   r4   r3   r5   r	   rO   r   r/   r
   r7   r   r   r   r   r   )r:   rt   ru   s      r<   test_mixed_coordinatesrv     s   3A3A3AACCGACC!##acc' 1111Ac!fHQSSL$QSS!##s1vaccz*:%;;<CC!ACC%A,#qssQSSy3q6'9133&>>!ACC%A,QRQTQTBTVWVYVYAYYZ Z Z
 ACCH!##qss(!3333CACCL!Xc!##acc'l%;;;;Ac!fHQSSL$QSS(,acc!##g5ACCGQSS8HHIaccE!##IaccM#a& 133s1339~#55qss:accE!##IaccM#a& 133s1339~#55qss:;>?eACCimCPQF>RTUTWTW=WXaccE!##IaccM#a& !##%&()ACC03433QSS133Y3G2LMM M M
 acc!##gacc!##goacc!##g-ACC7ACC?ACCG!##acc'QRSRURUVWVYVYRYYZ[Z]Z]^_^a^aZaabfghifjjjjaeACCimacc!##gacck1AaccE!##IaccMABaccEACCACC%%& & &aAhqssl133&qss*133./	!##acc	#a&QSS!##&133qss3q6)99AaccE!##Ic!f<LSQRQTQTVWVYVY]<ZZ[ [ [acc!##gacck%ACCQSS"99:33qss7133;z%ACCQSS"9::33qss73uQSSWacc!##g.44533qss73uQSSWacc!##g.445 33qss73uQSSWacc!##g.4455 5 5
 acc!##gacck133qss7QSSW#456!##acc	!##Qqss
3qssACC=()Qqss
3qssACC=() Qqss
3qssACC=() ACCF
3qssACC=(	)) ) )r>   N)?sympy.core.functionr   sympy.vector.vectorr   sympy.vector.coordsysrectr   sympy.simplifyr   sympy.core.symbolr   
sympy.corer   (sympy.functions.elementary.trigonometricr	   r
   r   sympy.vector.operatorsr   r   r   r   r   r   sympy.vector.deloperatorr   sympy.vector.functionsr   r   r   r   r   r   sympy.testing.pytestr   r   rY   r3   r5   r7   rX   r6   r4   r/   r0   r:   rt   ru   rO   r=   rE   orient_new_axisrF   r_   rI   vector_fieldrJ   rL   rP   rZ   ra   ro   rr   rv   rK   r>   r<   <module>r      s   * & 0 # %  ? # Z Z (L L (sO
..
1a
..
1aY
1a\8~0*f c1acc"Avaxzl#
!tAv!A~!Aa',
20
^	9)0:B#)r>   