Transformations of Coordinate Systems

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Md=Manifold(2,'Md')

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U=Md.open_subset('U'); V=Md.open_subset('V')

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XU.<x,y>=U.chart(); XV.<xp,yp>=V.chart('xp:x&#8217; yp:y&#8217;')

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Md.declare_union(U,V)

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XU_to_XV=XU.transition_map(

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    XV,(2*x/(x^2+y^2),2*y/(x^2+y^2)),

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    intersection_name='W',

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    restrictions1=x^2+y^2!=0,

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    restrictions2=xp^2+yp^2!=0)

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R3=Manifold(3,'R^3',r'\mathbb{R}^3')

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XR3.<X,Y,Z>=R3.chart()

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Delta1=Md.diff_map(

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    R3,{(XU,XR3):[2*x/(1+x^2+y^2),2*y/(1+x^2+y^2),

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                  (x^2+y^2-1)/(1+x^2+y^2)],

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        (XV,XR3):[3*xp/(1+xp^2+yp^2),3*yp/(1+xp^2+yp^2),

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                  (1-xp^2-yp^2)/(1+xp^2+yp^2)]},

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                   name='Delta1',latex_name=r'\Delta_1')

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Delta2=Md.diff_map(

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    R3,{(XU, XR3):[x/(1+x^2+y^2),y/(1+x^2+y^2),

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                   (x^2+y^2-1)/(1+x^2+y^2)],

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        (XV, XR3):[4*xp/(1+xp^2+yp^2),4*yp/(1+xp^2+yp^2),

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                   (1-xp^2-yp^2)/(1+xp^2+yp^2)]},

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                   name='Delta2',latex_name=r'\Delta_2')

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cols=['darkblue','#ff3636','darkred','#3636ff']

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elements=[(XU,Delta1),(XV,Delta1),(XU,Delta2),(XV,Delta2)]

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p=sum([elements[i][0].plot(

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           chart=XR3,mapping=elements[i][1],

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           number_values=50,color=cols[i],label_axes=False)

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       for i in range(4)])

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p.show(frame=False)

Language: Sage Gap GP HTML Macaulay2 Maxima Octave Python R Singular

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