# Essays/Gauss-Bonnet/Bonnet04

### 11 Geodesic Curvature

... McCONNELL Chapter XIV Section 10 ...
... Sokolnikoff Section 63 ...

##### 11.1 V-Coordinate System

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

tgnt=:(0,mcv22^_0.5"_)"1
```

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

tgntdrv1=:((0{1{1{ch2k)%mcv22)"1
tgntdrv2=:((_0.5*(mcv22^_2:)*mcv22v2)+(1{1{1{ch2k)%mcv22)"1
tgntdrv =:(tgntdrv1,tgntdrv2)"1
```

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

*./@(*./)@(0=])@ts0@:(+/@((<0 1)|:axs@((1;mcv),1;tgnt*/tgntdrv))"1) vpgen''
1
```

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

geoc=:+/@((<0 1)|:axs@((1;ecv),1;tgnt*/tgntdrv))"1
```

### 12 Gauss-Bonnet Theorem

... McCONNELL Chapter XIV EXAMPLES XIV Problem 14 ...
... Sokolnikoff Section 74 ...

##### 12.1 U-Coordinate System
 Figure 2: Tangent to Curve

##### 12.2 V-Coordinate System

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

dS=:%:@m*((f1-s1)%i1)*(f2-s2)%i2
ds=:%:@mcv22*(f2-s2)%i2
```

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

KSurface=:(+/%2:)@:(+/@(+/)@:((K*dS)"1)"3)@(}.,:}:)@:(}:"2)
```

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

geocCurve=:-/@:(+/"1)@:((geoc*ds)"1)@(}:"2@({.,:{:))
```

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

KSurface vpgen''
1.31588

geocCurve vpgen''
1.31783
```