Lecture Notes for 120 - UCLA Department of Mathematics

6.1. CALCULATING CHRISTOFFEL SYMBOLS AND THE GAUSS CURVATURE 136
With this observation and the fact that a matrix and its transpose have the same
determinant we can calculate the products that appear in our 2 ⇥ 2 determinant
✓ @ 2 q
@u 2 ·
h
= det
h
= det
✓ @q
@u ⇥ @q ◆◆ ✓ @ 2 q
@v @v 2 ·
i h
@ 2 q
@u 2
@ 2 q
@u 2
h
= det✓
@ 2 q
@u 2
2
6
= det4
2
= det4
@ 2 q
@u 2
@ 2 q
@u 2
@ 2 q
@u 2
@ 2 q
@u 2
@q
@u
@q
@u
@q
@u
@q
@v
@q
@v
@q
@v
det
i t h
det
i t h
@ 2 q
@v 2
@ 2 q
@v 2
✓ @q
@u ⇥ @q ◆◆
@v
i
@ 2 q
@v 2
@q
@u
@q
@u
@q
@u
· @2 q @q
@v 2 @u · @2 q @q
@v 2 @v · @2 q
@v 2
· @q @q
@u @u · @q @q
@u @v · @q
@v
· @q @q
@v @u · @q @q
@v @v · @q
@v
3
· @2 q
@v2 vvu vvv
uuu g uu g uv
5
uuv g vu g vv
2
= @2 q
@u 2 · @2 q
@v 2 det [I] + det 4
@q
@v
@q
@v
3
7
5
@q
@v
i
i ◆
3
0 vvu vvv
uuu g uu g uv
5
uuv g vu g vv
and similarly
✓ @ 2 q
@u@v ·
2
6
= det4
2
= det4
✓ @q
@u ⇥ @q ◆◆ ✓ @ 2 ✓
q @q
@v @v@u ·
@ 2 q
@u@v · @ 2 q
@u@v
@ 2 q
@u@v · @q
@u
@ 2 q
@u@v · @q
@v
@ 2 q
@u@v ·
@q
@u · @ 2 q
@u@v
@q
@u · @q
@u
@q
@u · @q
@v
@ 2 q
3
5
@u@v uvu uvv
uvu g uu g uv
@u ⇥ @q ◆◆
@v
3
@q
@v · @ 2 q
@u@v
@q
@v · @q
@v
@q
@v · @q
@v
2
= @2 q
@u@v · @ 2 q
@u@v det [I] + det 4 0 3
uvu uvv
uvu g uu g uv
5
uvv g vu g vv
7
5
We need to subtract these quantities but now only need to check the difference
@ 2 q
@u 2 · @2 q @ 2 q
@v 2 @u@v · @ 2 q
@u@v
= @ ✓ @ 2 q
@v @u 2 · @q ◆
@ 3 q
@v @v@u 2 · @q
@v
✓
@ @ 2 q
@u @u@v · @q ◆
+ @3 q
@v @ 2 u@v · @q
@v
= @ @
@v uuv @u uvv