# Essays/SpaceTime2D/SpaceTime2D07

### 6 Christoffel Symbols

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

##### 6.1 Christoffel Symbols of the First Kind

```NB. ... script SpaceTime2D.ijs (continued) ...

ch1k    =:(0.5*(1|:])+(0 1|:])-])"3@mcvdv
ch1kdv  =:(0.5*(1 3|:])+(0 1 3|:])-])"4@mcvdvdv
ch1kdvdv=:(0.5*(1 3 4|:])+(0 1 3 4|:])-])"5@mcvdvdvdv
```

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

p7aXd1=.((aRL'P'),<'P');<(aRL'Q'),<'Q'

aRsetA''
p7a1d1=.p7aXd1 ch1kdv''
mXsetV''
p7a1d2=.p7aXd1(0|:[:(p7aXd1 ch1k])D.1])"1(vGen aRR'P')
p7a1d1((2^_8)gXteq[;])p7a1d2
1
(p7a1d1=.0),p7a1d2=.0
0 0

aRsetA''
p7a2d1=.p7aXd1 ch1kdvdv''
mXsetV''
p7a2d2=.p7aXd1(0|:[:(p7aXd1 ch1kdv])D.1])"1(vGen aRR'P')
p7a2d1((2^_6)gXteq[;])p7a2d2
1
(p7a2d1=.0),p7a2d2=.0
0 0
```

##### 6.2 Christoffel Symbols of the Second Kind

```NB. ... script SpaceTime2D.ijs (continued) ...

ch2k      =:mcn(0|:gXsmx)"2 3 ch1k
ch2kdv    =:(mcn(0 3|:[gXsmx 2|:])"2 4 ch1kdv)+mcndv(0 1|:(1|:[)gXsmx])"3 3 ch1k
ch2kdvdvt1=:(mcn(0 3 4|:[gXsmx 2|:])"2 5 ch1kdvdv)+mcndv(0 4 1|:(1|:[)gXsmx 2|:])"3 4 ch1kdv
ch2kdvdvt2=:(mcndv(0 1 4|:(1|:[)gXsmx 2|:])"3 4 ch1kdv)+mcndvdv(0 1 2|:(1|:[)gXsmx])"4 3 ch1k
ch2kdvdv  =:ch2kdvdvt1+ch2kdvdvt2
```

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

p7bXd1=.((aRL'P'),<'P');<(aRL'Q'),<'Q'

aRsetA''
p7b1d1=.p7bXd1 ch2kdv''
mXsetV''
p7b1d2=.p7bXd1(0|:[:(p7bXd1 ch2k])D.1])"1(vGen aRR'P')
p7b1d1((2^_7)gXteq[;])p7b1d2
1
(p7b1d1=.0),p7b1d2=.0
0 0

aRsetA''
p7b2d1=.p7bXd1 ch2kdvdv''
mXsetV''
p7b2d2=.p7bXd1(0|:[:(p7bXd1 ch2kdv])D.1])"1(vGen aRR'P')
p7b2d1((2^_6)gXteq[;])p7b2d2
1
(p7b2d1=.0),p7b2d2=.0
0 0
```

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

p7cXd1=.((aRL'P'),<'P');<(aRL'Q'),<'Q'

aRsetA''
p7cXd1(ch1k((2^_39)gXteq[;])mcv(0|:gXsmx)"2 3 ch2k)''
1
```