+
    ix                       a  R t=0 t R t^ RIHt ^ RIt^ RIHtHt ^ RIH	t	 ^ RI
HtHtHtHt ^RIHtHtHt . ROt]! R^4      t] ^ k ]! R]4      t] ^k ]! R	R
R7      w  tt]! RR
R7      w  ttR]]3]	! ]^,          ]^,          ,           4      ]! ]]4      3.R]]3]]! ]4      ,          ]]! ]4      ,          3./t]! R]]]3]4      t] ^k ]! R]]]3]4      t] ^k ]P:                  ! 4       ;_uu_ 4        ]P<                  ! R4       ]! R]R7      w  tttt]P?                  ]]].]	! ]^,          ]^,          ,           4      ]! ]]4      .RRR7       ]P?                  ]]].]]! ]4      ,          ]]! ]4      ,          .RRR7       RRR4       ]PA                  4       ;w  ]n        ]n        ;w  ]n        ]n        w  ]n        ]n        ]PA                  4       ;w  ]n        ]n        ;w  ]n        ]n        w  ]n        ]n        ]PC                  4       ;w  ]n"        ]n#        ;w  ]n"        ]n#        w  ]n"        ]n#        ]PC                  4       ;w  ]n$        ]n%        ;w  ]n$        ]n%        w  ]n$        ]n%        ]PM                  4       ;w  ]n'        ]n(        ;w  ]n'        ]n(        w  ]n'        ]n(        ]PM                  4       ;w  ]n)        ]n*        ;w  ]n)        ]n*        w  ]n)        ]n*        ]! R^4      t+] ^k ]! R]+4      t,] ^k ]! RR
R7      w  ttt-]! RR
R7      w  t.t/ttt0R]]]-3]	! ]^,          ]^,          ,           4      ]! ]]4      ]-3.R ].]/]-3].]! ]/4      ,          ].]! ]/4      ,          ]-3.R!]]]-3]	! ]^,          ]^,          ,           ]-^,          ,           4      ]! ]-]	! ]^,          ]^,          ,           ]-^,          ,           4      ,          4      ]! ]]4      3.R"]]]03]]! ]4      ,          ]! ]04      ,          ]]! ]4      ,          ]! ]04      ,          ]]! ]4      ,          3.R#].]/]-3]	! ].^,          ]-^,          ,           4      ]! ]-]	! ].^,          ]-^,          ,           4      ,          4      ]/3.R$]]]03]]! ]4      ,          ]0]]! ]4      ,          3./t1]! R],]]]-3]14      t2] ^k ]! R],].]/]-3]14      t3] ^k ]! R],]]]03]14      t4] ^k ]P:                  ! 4       ;_uu_ 4        ]P<                  ! R4       ]! R]R7      w  ttt-t.t/ttt0]2P?                  ]3]]]-.]	! ]^,          ]^,          ,           4      ]! ]]4      ]-.RRR7       ]3P?                  ]2].]/]-.].]! ]/4      ,          ].]! ]/4      ,          ]-.RRR7       ]2P?                  ]4]]]-.]	! ]^,          ]^,          ,           ]-^,          ,           4      ]! ]-]	! ]^,          ]^,          ,           ]-^,          ,           4      ,          4      ]! ]]4      .RRR7       ]4P?                  ]2]]]0.]]! ]4      ,          ]! ]04      ,          ]]! ]4      ,          ]! ]04      ,          ]]! ]4      ,          .RRR7       ]3P?                  ]4].]/]-.]	! ].^,          ]-^,          ,           4      ]! ]-]	! ].^,          ]-^,          ,           4      ,          4      ]/.RRR7       ]4P?                  ]3]]]0.]]! ]4      ,          ]0]]! ]4      ,          .RRR7       RRR4       ]2PA                  4       w  ]2n        ]2n        ]2n-        ]3PA                  4       w  ]3n.        ]3n/        ]3n-        ]4PA                  4       w  ]4n        ]4n        ]4n0        ]2PC                  4       w  ]2n"        ]2n#        ]2n5        ]3PC                  4       w  ]3n6        ]3n7        ]3n5        ]4PC                  4       w  ]4n$        ]4n%        ]4n8        ]2PM                  4       w  ]2n'        ]2n(        ]2n9        ]3PM                  4       w  ]3n:        ]3n;        ]3n9        ]4PM                  4       w  ]4n)        ]4n*        ]4n<        R#   + '       g   i     EL; i  + '       g   i     ELb; i)%at  Predefined R^n manifolds together with common coord. systems.

Coordinate systems are predefined as well as the transformation laws between
them.

Coordinate functions can be accessed as attributes of the manifold (eg `R2.x`),
as attributes of the coordinate systems (eg `R2_r.x` and `R2_p.theta`), or by
using the usual `coord_sys.coord_function(index, name)` interface.
)AnyN)Dummysymbols)sqrt)acosatan2cossin)ManifoldPatchCoordSystemzR^2originzx yT)realz	rho theta)nonnegativerectangularpolarignorezx y r theta)clsF)inversefill_in_gapszR^3zx y zzrho psi r theta phicylindrical	sphericalzx y z rho psi r theta phic                   V ^8  d   Qh/ ^ \         9   d
   \        ;R&   ^\         9   d
   \        ;R&   ^\         9   d
   \        ;R&   ^\         9   d
   \        ;R&   ^\         9   d
   \        ;R&   ^\         9   d
   \        ;R&   ^\         9   d
   \        ;R&   ^\         9   d
   \        ;R&   ^\         9   d
   \        ;R	&   # )
   R2	R2_originR2_rR2_pR3	R3_originR3_rR3_cR3_s)__conditional_annotations__r   )formats   "q/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/diffgeom/rn.py__annotate__r&      s      4  C 58 % $3 $9L H Gc GMN F Ec EOL  C MP % $3 $QD K Jc JEF O Nc NGH O Nc NI    )r   r   relations_2dr   r   r   r   relations_3dr    r!   r"   )r   r   )r   r   )r   r   )r   r   )r   r   )r   r   )r   r   )r   r   )>r#   __doc__typingr   warningssympy.core.symbolr   r   (sympy.functions.elementary.miscellaneousr   (sympy.functions.elementary.trigonometricr   r   r   r	   diffgeomr
   r   r   __all__r   r   xyrthetar(   r   r   catch_warningssimplefilter
connect_tocoord_functionsbase_vectorse_xe_ye_re_thetabase_oneformsdxdydrdthetar   r   zrhopsiphir)   r    r!   r"   e_ze_rhoe_psie_phidzdrhodpsidphir&   )r#   s   @r%   <module>rP      s=     . 9 L L 2 2 5!
 x$	 $u4 1;D15 1vQTAqD[(95A;'GH5zAc%jL!CJ,+GH
 y1a&,G GQJE E (#]6NAq!UOOD1a&adQTk*E!QK8!  7 	OOD1e*3u:qU|4!  7  :>9M9M9O O
bd O%Y[)+EIEYEYE[ [bh [-io0B
 FJEVEVEX X X-	0B$(QUQbQbQd d 
 d5Y]I$58N$, @D?Q?Q?S Sru S)y|Y\,<DGTWKOK]K]K_ _ ry _19<!14HDGT[
 5!
 x$	 $
'
%1a!"7TJ S!UC #aAY&*1a4!Q$;&7q!a%H%J"c3]&)#c(lCCL!%D%F Aq!9$(A1q!t);$<$(41q!tad0B+C)C$D$)!QK$1#2 !Auc?$%c%jLS$9$%c%jLS$9$%c%jL$2#3 !Ca=$(a!Q$$7$(4QA+>)>$?$'$)#* !Auc?$%c%jL#qU|#D#F#* y1a)\J Jy3Q-N NYE3N N (#'./JPU'V$Aq!S#q%OOD1a)adQTk*E!QK;!  7 	OOD3Q-SXs3s8|Q7!  7 	OOD1a)adQTkAqD014 $QTAqD[1a4%7 899 4:;@A;H!  7 	OOD1eS/3u:c#h.#!3# 1##&s81,-.s5z\;!  7
 	OOD3Q-c1fq!tm,d1T#q&1a4-5H3H.I3O!  7 	OOD1eS/3u:sAc%jL9!  7- 6 --/ !113 $(DF#335 
DH  $002 $(DH#'#4#4#6  
DJ%)%6%6%8 "$,
 !..0 $' $ 2 2 4 	49dg"&"4"4"6 diI z s   8Ba9Ha a	 a1	