# Essays/Gauss-Bonnet/Bonnet03

### 8 Christoffel Symbols

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

##### 8.2.1 Christoffel Symbols of the First Kind

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

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

```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

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

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

```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 ...

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

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

##### 9.2.2 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
```

##### 9.3.2 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
```

##### 9.4.2 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
```

##### 9.5.2 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
```

```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 ...

##### 10.2 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
```