Transcription of Laplacian in Spherical Coordinates - Ulyana’s Space
1 Laplacian in Spherical CoordinatesWe want to write the Laplacian functional 2= 2 x2+ 2 y2+ 2 z2(1)in Spherical Coordinates x=rsin cos y=rsin sin z=rcos (2)To do so we need to invert the previous transformation rules and repeatedly use the chain rule x(r, , )= r x r+ x + x y(r, , )= r y r+ y + y z(r, , )= r z r+ z + z (3)and 2 x2(r, , )= x( x)== x( r x r+ x + x )==( 2r x2) r+( r x r x) 2 r2+( r x x) 2 r +( r x x) 2 r ++( 2 x2) +( x x) 2 2+( x r x) 2 r+( x x) 2 ++( 2 x2) +( x x) 2 2+( x x) 2 +( r x x) 2 r (4) 2 y2(r, , )= y( y)== y( r y r+ y + y )==( 2r y2) r+( r y r y) 2 r2+( r y y) 2 r +( r y y) 2 r ++( 2 y2) +( y y) 2 2+( y r y) 2 r+( y y) 2 ++( 2 y2) +( y y) 2 2+( y y) 2 +( r y y) 2 r (5)1 2 z2(r, , )= z( z)== z( r z r+ z + z )==( 2r z2) r+( r z r z) 2 r2+( r z z) 2 r +( r z z) 2 r ++( 2 z2) +( z z) 2 2+( z r z) 2 r+( z z) 2 ++( 2 z2)
2 +( z z) 2 2+( z z) 2 +( r z z) 2 r (6)where we have r= x2+y2+z2 = arctan( x2+y2z2) = arctan(yx)(7)Then we have r ( 2r x2+ 2r y2+ 2r z2) r=1r2 r(8) 2 r2 (( r x)2+( r y)2+( r z)2) 2 r2= 2 r2(9) ( 2 x2+ 2 y2+ 2 z2) =cos r2sin (10) 2 2 (( x)2+( y)2+( z)2) 2 2=1r2 2 2(11) ( 2 x2+ 2 y2+ 2 z2) = 0(12) 2 2 (( x)2+( y)2+( z)2) 2 2=1r2sin2 2 2(13)The mixed derivative terms, 2 r , 2 r and 2 , cancel all the terms together we get 2= 2 r2+2r r+1r2 2 2+cos r2sin +1r2sin2 2 2==1r2sin ( rr2sin r+ sin + 1sin )(14) 2=1r2sin ( rr2sin r+ sin + 1sin )(15)2
