Essays/Gauss-Bonnet/Bonnet03

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8 Christoffel Symbols

... McCONNELL Chapter XII Section 1 ...
... Sokolnikoff Section 31 ...


8.1 U-Coordinate System
8.1.1 Christoffel Symbols of the First Kind
GGB0301C.jpg


8.1.2 Christoffel Symbols of the Second Kind
GGB0302C.jpg


8.2 V-Coordinate System
8.2.1 Christoffel Symbols of the First Kind
GGB0303C.jpg


NB. ... script (ijs) ...

ch1k=:(0.5*(1|:])+(0|:0|:])-])@mcvdv


GGB0304C.jpg


NB. ... script (ijs) ...

ch1kdv=:(0.5*(2|:1|:])+(1|:0|:0|:])-])@mcvdvdv


NB. ... execute (ijx) ...

   *./@(*./)@:(((2^_13)&tsz@ch1kdv((2^_10)teq[;])(2^_13)&tsz@(0|:ch1k D.1))"1) vpgen''
1


8.2.2 Christoffel Symbols of the Second Kind
GGB0306C.jpg


NB. ... script (ijs) ...

ch2k=:(0|:axs@((1;mcn),2;ch1k))"1


GGB0307C.jpg


NB. ... script (ijs) ...

ch2kdv=:((2|:0|:axs@((1;mcn),2;ch1kdv))+0|:0|:axs@((1;mcndv),2;ch1k))"1


NB. ... execute (ijx) ...

   *./@(*./)@:(((2^_12)&tsz@ch2kdv((2^_9)teq[;])(2^_12)&tsz@(0|:ch2k D.1))"1) vpgen''
1


9 Riemann-Christoffel Tensor

... McCONNELL Chapter XIV Section 9 ...
... Sokolnikoff Section 62 ...


9.1 Riemann-Christoffel Tensor of the Second Kind
9.1.1 U-Coordinate System
GGB0310C.jpg


9.1.2 V-Coordinate System
NB. ... script (ijs) ...

R2kt1=:((1|:1|:])-2|:])@ch2kdv
R2kt2=:((2|:1|:])-])@axs@((2;ch2k),1;ch2k)"1
R2k  =:(R2kt1+R2kt2)"1


9.2 Tensor Equation
9.2.1 U-Coordinate System
GGB0312C.jpg


9.2.2 V-Coordinate System
NB. ... execute (ijx) ...

   RC1t1=:((0|:2|:0|:])-1|:0|:0|:])@(ch1kdv-axs@((2;ch2k),2;ch1k+0|:1|:ch1k))"1
   RC1t2=:((0|:1|:0|:])-0|:0|:0|:])@axs@((2;ch2k),1;ch1k)"1
   RC1  =:(RC1t1+RC1t2)"1

   *./@(*./)@:((ts0@axs@((1;mcv),3;R2k)((2^_39)teq[;])ts0@RC1)"1) vpgen''
1


9.3 Tensor Equation
9.3.1 U-Coordinate System
GGB0314C.jpg


9.3.2 V-Coordinate System
NB. ... execute (ijx) ...

   RC2=:((0|:2|:0|:])-1|:0|:0|:])@(ch1kdv-axs@((2;ch2k),2;ch1k))"1

   *./@(*./)@:((ts0@axs@((1;mcv),3;R2k)((2^_39)teq[;])ts0@RC2)"1) vpgen''
1


9.4 Tensor Equation
9.4.1 U-Coordinate System
GGB0316C.jpg


9.4.2 V-Coordinate System
NB. ... execute (ijx) ...

   RC3t1=:(0.5*(0|:])+(1|:0|:0|:])-(0|:2|:0|:])+2|:0|:])@mcvdvdv
   RC3t2=:+/"1@((<0 5)|:axs@((1;mcn),5;((1|:0|:2|:0|:0|:])-1|:0|:0|:3|:0|:])@(]*/])@ch1k))"1
   RC3  =:(RC3t1+RC3t2)"1

   *./@(*./)@:((ts0@axs@((1;mcv),3;R2k)((2^_38)teq[;])ts0@RC3)"1) vpgen''
1


9.5 Riemann-Christoffel Tensor of the First Kind
9.5.1 U-Coordinate System
GGB0318C.jpg


9.5.2 V-Coordinate System
NB. ... script (ijs) ...

R1kt1=:(0.5*(((0|:])+1|:0|:0|:])-(0|:2|:0|:])+2|:0|:]))@mcvdvdv"1
R1kt2=:((1|:0|:0|:])-0|:2|:0|:])@axs@((2;ch2k),2;ch1k)"1
R1k  =:(R1kt1+R1kt2)"1


GGB0320C.jpg


NB. ... execute (ijx) ...

   *./@(*./)@:((ts0@R1k((2^_39)teq[;])ts0@axs@((1;mcv),3;R2k))"1) vpgen''
1


10 Gaussian Curvature

... McCONNELL Chapter XIV Section 9 ...
... Sokolnikoff Section 62 ...


... the invariant K is called the Gaussian curvature ...


10.1 U-Coordinate System
GGB0322C.jpg


10.2 V-Coordinate System
NB. ... script (ijs) ...

Kt1=:+/"1@((<0 3)|:axs@((1;ecn),3;R1k))"1
K  =:0.25&*@(+/)@((<0 1)|:axs@((1;ecn),1;Kt1))"1



Download: File:LGB0301C.txt
Download: File:LGB0302C.txt
Download: File:LGB0303C.txt
Download: File:LGB0304C.txt
Download: File:LGB0306C.txt
Download: File:LGB0307C.txt
Download: File:LGB0310C.txt
Download: File:LGB0312C.txt
Download: File:LGB0314C.txt
Download: File:LGB0316C.txt
Download: File:LGB0318C.txt
Download: File:LGB0320C.txt
Download: File:LGB0322C.txt




Download MoinMoin source: File:Bonnet03.ijs




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Contributed by Tom Allen