Essays/Christoffel/Christoffel01
1 Reference
'Tensor Analysis' by I. S. Sokolnikoff (Second Edition, 1964).
2 Software
NB. ... execute (ijx) ... 9!:14 '' j601/2006-11-17/17:05
3 Continuous Functions
There is a footnote on page 1 of the book 'Riemannian Geometry' by Luther Pfahler Eisenhart.
'When we consider any function, it is understood that it is real and continuous, as well as its derivatives of
such order as appear in the discussion, in the domain of the variables considered, unless stated otherwise.'
4 Verbs
NB. ... script (ijs) ... NB. ... identify coordinates ... y1=:0{] y2=:1{] y3=:2{] x1=:0{] x2=:1{] x3=:2{] NB. ... open boxed elements ... b0=:>@(0{]) b1=:>@(1{]) b2=:>@(2{]) b3=:>@(3{]) NB. ... tolerant 'set zero' (see 'Essays/Tolerant Comparison') ... tsz=:$@]$[0:`(I.@([>!.0|@]))`]},@] ts0=:(2^_44)&tsz tz =:ts0@: NB. ... tolerant 'equal' (see 'Essays/Tolerant Comparison') ... teq=:*./@,@((b0|@:-b1)<:!.0[*b0>.&:|b1) NB. ... verbs useful for tolerant comparison ... nzmin =:<./@:|@((0<!.0|)#])@, nzmax =:>./@:|@((0<!.0|)#])@, nzcount=:+/@(0<!.0|)@, NB. ... trig verbs ... sin =:1&o. cos =:2&o. arctan=:_3&o. NB. ... axes sum ... axs=:ts0@((b0|:b1)+/@(*"1)"1 _ b2|:b3)
5 Transformation of Coordinates (ISS Section 19)
5.1 General
5.2 Example
I.S.S. Figure 13 on page 114 shows the transformation from Cartesian coordinates (y) to Spherical coordinates (x)
in Euclidean space.
NB. ... script (ijs) ... NB. ... equations to transform from Cartesian coordinates to Spherical coordinates ... cx1=:%:@(*:@y1+*:@y2+*:@y3)"1 cx2=:arctan@(%:@(*:@y1+*:@y2)%y3)"1 cx3=:arctan@(y2%y1)"1 cxx=:(cx1,cx2,cx3)"1 NB. convert y coordinates to x coordinates NB. ... equations to transform from Spherical coordinates to Cartesian coordinates ... cy1=:(x1*sin@x2*cos@x3)"1 cy2=:(x1*sin@x2*sin@x3)"1 cy3=:(x1*cos@x2)"1 cyy=:(cy1,cy2,cy3)"1 NB. convert x coordinates to y coordinates NB. ... from 'numeric' ... steps=:{.+(1&{-{.)*(i.@>:%])@{: NB. ... verbs to generate coordinates ... s1=:steps@(0.5,10,19"_) s2=:steps@((0.5p1%10),(0.5p1-0.5p1%10),19"_) s3=:steps@((0.5p1%10),(0.5p1-0.5p1%10),19"_) NB. ... generate coordinates ... xpgen=:>@,@:(<"1)@(s1,"0 1/s2,"0/s3) ypgen=:cyy@xpgen
6 First Derivatives of Transformation Equations
6.1 General (3 dimensions)
6.2 Example
NB. ... script (ijs) ... dxdy0=:(sin@x2*cos@x3),(sin@x2*sin@x3),cos@x2 dxdy1=:((cos@x2*cos@x3)%x1),((cos@x2*sin@x3)%x1),-@(sin@x2%x1) dxdy2=:-@(sin@x3%x1*sin@x2),(cos@x3%x1*sin@x2),0: dxdy =:(3 3$dxdy0,dxdy1,dxdy2)"1 dydx0=:(sin@x2*cos@x3),(x1*cos@x2*cos@x3),-@(x1*sin@x2*sin@x3) dydx1=:(sin@x2*sin@x3),(x1*cos@x2*sin@x3),x1*sin@x2*cos@x3 dydx2=:cos@x2,-@(x1*sin@x2),0: dydx =:(3 3$dydx0,dydx1,dydx2)"1
7 Second Derivatives of Transformation Equations
7.1 General (3 dimensions)
7.2 Example
NB. ... script (ijs) ... d2xdydx00=:0,(cos@x2*cos@x3),-@(sin@x2*sin@x3) d2xdydx01=:0,(cos@x2*sin@x3),sin@x2*cos@x3 d2xdydx02=:0,-@(sin@x2),0: d2xdydx10=:(-@((cos@x2*cos@x3)%*:@x1)),(-@((sin@x2*cos@x3)%x1)),-@((cos@x2*sin@x3)%x1) d2xdydx11=:(-@((cos@x2*sin@x3)%*:@x1)),(-@((sin@x2*sin@x3)%x1)),(cos@x2*cos@x3)%x1 d2xdydx12=:(sin@x2%*:@x1),-@(cos@x2%x1),0: d2xdydx20=:(sin@x3%(*:@x1)*sin@x2),((sin@x3*cos@x2)%x1**:@(sin@x2)),-@(cos@x3%x1*sin@x2) d2xdydx21=:(-@(cos@x3%(*:@x1)*sin@x2)),(-@((cos@x3*cos@x2)%x1**:@(sin@x2))),(-@(sin@x3%x1*sin@x2)) d2xdydx22=:0,0,0: d2xdydx0 =:d2xdydx00,d2xdydx01,d2xdydx02 d2xdydx1 =:d2xdydx10,d2xdydx11,d2xdydx12 d2xdydx2 =:d2xdydx20,d2xdydx21,d2xdydx22 d2xdydx =:(3 3 3$d2xdydx0,d2xdydx1,d2xdydx2)"1
NB. ... script (ijs) ... d2ydxdx00=:0,(cos@x2*cos@x3),-@(sin@x2*sin@x3) d2ydxdx01=:(cos@x2*cos@x3),-@(x1*sin@x2*cos@x3),-@(x1*cos@x2*sin@x3) d2ydxdx02=:(-@(sin@x2*sin@x3)),(-@(x1*cos@x2*sin@x3)),-@(x1*sin@x2*cos@x3) d2ydxdx10=:0,(cos@x2*sin@x3),sin@x2*cos@x3 d2ydxdx11=:(cos@x2*sin@x3),-@(x1*sin@x2*sin@x3),x1*cos@x2*cos@x3 d2ydxdx12=:(sin@x2*cos@x3),(x1*cos@x2*cos@x3),-@(x1*sin@x2*sin@x3) d2ydxdx20=:0,-@(sin@x2),0: d2ydxdx21=:-@(sin@x2),-@(x1*cos@x2),0: d2ydxdx22=:0,0,0: d2ydxdx0 =:d2ydxdx00,d2ydxdx01,d2ydxdx02 d2ydxdx1 =:d2ydxdx10,d2ydxdx11,d2ydxdx12 d2ydxdx2 =:d2ydxdx20,d2ydxdx21,d2ydxdx22 d2ydxdx =:(3 3 3$d2ydxdx0,d2ydxdx1,d2ydxdx2)"1
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