Essays/Christoffel/Christoffel02

8 Metric Tensor (ISS Section 29)

8.1 General (3 dimensions)

8.2 Example

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

NB. ... covariant metric tensor in y coordinates ...
g20=:=@i.@#"1

NB. ... contravariant metric tensor in y coordinates ...
g02=:=@i.@#"1

NB. ... first derivatives of covariant metric tensor in y coordinates ...
dg20dy=:(3 3 3$0:)"1

NB. ... covariant metric tensor in x coordinates ...
h20=:(3 3$1,0,0,0,*:@x1,0,0,0,*:@(x1*sin@x2))"1

NB. ... contravariant metric tensor in x coordinates ...
h02=:(3 3$1,0,0,0,%@(*:@x1),0,0,0,%@(*:@(x1*sin@x2)))"1

NB. ... first derivatives of covariant metric tensor in x coordinates ...
dh20dx0=:9$0:
dh20dx1=:0,0,0,(2*x1),0,0,0,0,0:
dh20dx2=:0,0,0,0,0,0,(2*x1**:@(sin@x2)),(2**:@x1*sin@x2*cos@x2),0:
dh20dx =:(3 3 3$dh20dx0,dh20dx1,dh20dx2)"1

9 Christoffel Symbols (ISS Section 31)

9.1 General

9.2 Example

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

NB. ... Christoffel symbols of the first kind in y coordinates ...
gC1k=:(3 3 3$0:)"1

NB. ... Christoffel symbols of the second kind in y coordinates ...
gC2k=:(3 3 3$0:)"1

NB. ... first derivatives of Christoffel symbols of the second kind in y coordinates ...
dgC2kdy=:(3 3 3 3$0:)"1

hCf0=:x1**:@(sin@x2)
hCf1=:*:@x1*sin@x2*cos@x2
hCf2=:cos@x2%sin@x2
hCf3=:sin@x2*cos@x2

hC1k0=:0,0,0,0,x1,0,0,0,hCf0
hC1k1=:0,x1,0,-@x1,0,0,0,0,hCf1
hC1k2=:0,0,hCf0,0,0,hCf1,-@hCf0,-@hCf1,0:

NB. ... Christoffel symbols of the first kind in x coordinates ...
hC1k=:(3 3 3$hC1k0,hC1k1,hC1k2)"1

hC2k0=:0,0,0,0,%@x1,0,0,0,%@x1
hC2k1=:0,%@x1,0,-@x1,0,0,0,0,hCf2
hC2k2=:0,0,%@x1,0,0,hCf2,-@hCf0,-@hCf3,0:

NB. ... Christoffel symbols of the second kind in x coordinates ...
hC2k=:(3 3 3$hC2k0,hC2k1,hC2k2)"1

hC2kf0=:-@(%@(*:@x1))
hC2kf1=:-@(%@(*:@(sin@x2)))
hC2kf2=:-@(*:@(sin@x2))
hC2kf3=:-@(2*x1*sin@x2*cos@x2)
hC2kf4=:*:@(sin@x2)-*:@(cos@x2)

dhC2kdx00=:9$0:
dhC2kdx01=:9$0,0,0,hC2kf0,0,0,0,0,0:
dhC2kdx02=:9$0,0,0,0,0,0,hC2kf0,0,0:

dhC2kdx10=:9$0,0,0,hC2kf0,0,0,0,0,0:
dhC2kdx11=:9$_1,0,0,0,0,0,0,0,0:
dhC2kdx12=:9$0,0,0,0,0,0,0,hC2kf1,0:

dhC2kdx20=:9$0,0,0,0,0,0,hC2kf0,0,0:
dhC2kdx21=:9$0,0,0,0,0,0,0,hC2kf1,0:
dhC2kdx22=:9$hC2kf2,hC2kf3,0,0,hC2kf4,0,0,0,0:

dhC2kdx0=:dhC2kdx00,dhC2kdx01,dhC2kdx02
dhC2kdx1=:dhC2kdx10,dhC2kdx11,dhC2kdx12
dhC2kdx2=:dhC2kdx20,dhC2kdx21,dhC2kdx22

NB. ... first derivatives of Christoffel symbols of the second kind in x coordinates ...
dhC2kdx=:(3 3 3 3$dhC2kdx0,dhC2kdx1,dhC2kdx2)"1

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Contributed by Tom Allen

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