# Essays/Gauss-Bonnet/Bonnet02

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

##### 5.2 Cartesian Coordinates

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

```

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

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

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

```

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

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

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

```

```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

... McCONNELL Chapter XI Section 1 ...
... Sokolnikoff Section 19 ...

...

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

...

### 7 Metric Tensor

... McCONNELL Chapter XIV Section 3 ...
... Sokolnikoff Section 53 ...

##### 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
```

##### 7.2.2 V-Coordinate System

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

m=:det@mcv
```

... Sokolnikoff Section 41 ...

```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
```

... McCONNELL Chapter XIV Section 3 ...
... Sokolnikoff Section 62 ...

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

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

##### 7.3.2 V-Coordinate System

... Sokolnikoff Section 53 ...

```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
```