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## 5 Derivatives (... continued ...)

### 5.2 Cartesian Coordinates

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

y1dvdvdv=:gadvdvdv

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

y2dv    =:gbdv
y2dvdv  =:gbdvdv
y2dvdvdv=:gbdvdvdv

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

y3dv=:((ga*/gxdv)+gadv*/gx)"1

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

*./@(*./)@:(((2^_17)&tsz@y3dv((2^_11)teq[;])(2^_17)&tsz@(y3 D.1))"1) vpgen''
1

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

y3dvdv=:((ga*/gxdvdv)+(0|:gadv*/gxdv)+(gadv*/gxdv)+gadvdv*/gx)"1

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

*./@(*./)@:(((2^_18)&tsz@y3dvdv((2^_11)teq[;])(2^_18)&tsz@(0|:y3dv D.1))"1) vpgen''
1

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

y3dvdvdv  =:(y3dvdvdvt1+y3dvdvdvt2)"1

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

*./@(*./)@:(((2^_13)&tsz@y3dvdvdv((2^_10)teq[;])(2^_13)&tsz@(0|:y3dvdv D.1))"1) vpgen''
1

## 6 Transformation of Coordinates

Using an admissible transformation, the coordinates of Figure 1 can be transformed into orthogonal coordinates.

## 7 Metric Tensor

### 7.1 Covariant Metric Tensor

#### 7.1.2 V-Coordinate System

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

mcv=:((y1dv*/y1dv)+(y2dv*/y2dv)+y3dv*/y3dv)"1

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

mcvdvt1=:(y1dv*/y1dvdv)+1|:y1dvdv*/y1dv
mcvdvt2=:(y2dv*/y2dvdv)+1|:y2dvdv*/y2dv
mcvdvt3=:(y3dv*/y3dvdv)+1|:y3dvdv*/y3dv
mcvdv  =:(mcvdvt1+mcvdvt2+mcvdvt3)"1

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

*./@(*./)@:(((2^_12)&tsz@mcvdv((2^_10)teq[;])(2^_12)&tsz@(0|:mcv D.1))"1) vpgen''
1

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

mcvdvdvt1=:(y1dv*/y1dvdvdv)+(1|:y1dvdv*/y1dvdv)+(2|:1|:y1dvdv*/y1dvdv)+1|:1|:y1dvdvdv*/y1dv
mcvdvdvt2=:(y2dv*/y2dvdvdv)+(1|:y2dvdv*/y2dvdv)+(2|:1|:y2dvdv*/y2dvdv)+1|:1|:y2dvdvdv*/y2dv
mcvdvdvt3=:(y3dv*/y3dvdvdv)+(1|:y3dvdv*/y3dvdv)+(2|:1|:y3dvdv*/y3dvdv)+1|:1|:y3dvdvdv*/y3dv
mcvdvdv  =:(mcvdvdvt1+mcvdvdvt2+mcvdvdvt3)"1

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

*./@(*./)@:(((2^_12)&tsz@mcvdvdv((2^_10)teq[;])(2^_12)&tsz@(0|:mcvdv D.1))"1) vpgen''
1

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

mcv22  =:(1{1{mcv)"1
mcv22v2=:(1{1{1{mcvdv)"1

### 7.2 Determinant of the Covariant Metric Tensor

#### 7.2.2 V-Coordinate System

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

m=:det@mcv

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

mdv=:(mcv detdv 0|:0|:mcvdv)"1

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

*./@(*./)@:(((2^_12)&tsz@mdv((2^_14)teq[;])(2^_12)&tsz@(m D.1))"1) vpgen''
1

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

mmn  =:(m^_1:)"1
mmndv=:(-@(m^_2:)*mdv)"1

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

*./@(*./)@:(((2^_17)&tsz@mmndv((2^_6)teq[;])(2^_17)&tsz@(mmn D.1))"1) vpgen''
1

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

ecv=:(2 2$0 1 _1 0)*%:@m ecn=:(2 2$0 1 _1 0)%%:@m

### 7.3 Contravariant Metric Tensor

#### 7.3.2 V-Coordinate System

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

mcn=:(mmn*/Mcf@mcv)"1

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

*./@(*./)@:((=@i.@#-:ts0@axs@((1;mcv),1;mcn))"1) vpgen''
1

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

mcndv=:((mmn*/0|:mcv Mcfdv 0|:0|:mcvdv)+0|:mmndv*/Mcf@mcv)"1

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

*./@(*./)@:(((2^_16)&tsz@mcndv((2^_9)teq[;])(2^_16)&tsz@(0|:mcn D.1))"1) vpgen''
1

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Contributed by TomAllen

Essays/Gauss-Bonnet/Bonnet02 (last edited 2008-12-08 10:45:44 by anonymous)