Lecture Notes for 120 - UCLA Department of Mathematics

4.3. THE ABSTRACT FRAMEWORK 84
Before discussing general surfaces it might be instructive to see what happens
if q (u, v) :U ! R 2 is simply a reparametrization of the plane. Thus @q
@u , @q
@v
form
abasisateachpointq. Taking partial derivatives of these fields give us
@ ⇥ @q
⇤ h
i
@q
=
@ 2 q @ 2 q
= ⇥ ⇤
@q @q
@u
@u @v
@u 2 @u@v @u @v [ u] ,
@ ⇥ @q
⇤ h
i
@q
=
@q @ 2 q
= ⇥ ⇤
@q @q
@v
@u @v
@v@u @v 2
@u @v [ v]
or in condensed form
@ ⇥ @q
⇤ h
i
@q
@w
@u @v =
@ 2 q @ 2 q
= ⇥ ⇤
@q @q
@w@u @w@v @u @v [ w] ,w= u, v
The matrices [ w] tell us how the tangent fields change with respect to themselves.
Agoodexamplecomesfromconsideringpolarcoordinatesq (r, ✓) =(r cos ✓, r sin ✓)
apple
@q cos ✓
=
, @q apple
@r sin ✓ @✓ = r sin ✓
r cos ✓
@ 2 apple
apple
q
= @2 q
@r@✓ @✓@r = sin ✓
, @2 q
cos ✓ @r 2 =0, @2 q
@✓ 2 = r cos ✓
r sin ✓
so
@ ⇥ @q
⇤ h
i
@q
=
@ 2 q @ 2 q
= ⇥ ⇤ apple @q @q 0 0
@r
@r @✓
@r@r @r@✓
@r @✓
1
0
r
@ ⇥ @q
⇤ h
i
@q
=
@ 2 q @ 2 q
= ⇥ ⇤ apple @q @q 0 r
@✓
@r @✓
@✓@r @✓@✓
@r @✓ 1
r
0
apple 0 0
[ r] =
1 ,
0
r
apple 0 r
[ ✓] =
1
r
0
The key is that only Cartesian coordinates have the property that its coordinate
fields are constant. When using general coordinates we are naturally
forced to find these quantities. To see why this is, consider a curve q (t) =
(r (t) cos ✓ (t) ,r(t)sin✓ (t)) in the plane. Its velocity is naturally given by
˙q =ṙ @q @q
+ ˙✓
@r @✓
If we wish to calculate its acceleration, then we must compute the derivatives of the
coordinate fields. This involves the chain rule as well as the formulas just developed
¨q = ¨r @q @q
+ ¨✓
@r @✓ +ṙ d @q
dt
= ¨r @q
✓
@q dr
+ ¨✓
@r @✓ +ṙ dt
= ¨r @q @q
+ ¨✓
@r
= ¨r @q
=
⇣
¨r
@r + ˙✓ d @q
dt @✓
@
@r + d✓ @
dt @✓
◆ ✓ @q
@r + ˙✓ dr
dt
@✓ +ṙ2 @2 q
@r 2 +2ṙ ˙✓ @2 q
@r@✓ + ˙✓ 2 @2 q
@✓ 2
@q
+ ¨✓
@r @✓ +2ṙ ˙✓ 1 @q
˙✓ 2 r @q
r @✓
!
@r
r ˙✓ 2⌘ @q
@r + ¨✓ 2ṙ ˙✓ @q
+
r @✓
@
@r + d✓
dt
◆
@ @q
@✓ @✓