review #6: jacobians, etc. calculus 3 / section...
TRANSCRIPT
Review #6: Jacobians, Etc.Calculus 3 / Section 015
Wednesday, December 12
1. Complete each of the following generic integrals.
(a) Line integral over the parameterization ~r(t) :
Zf(~r(t)) d
(b) Surface integral over the parameterization ~r(u, v) :
ZZf(~r(u, v)) d d
(c) Double integral in a change of coordinates from xy to uv:
ZZf(u, v) d d
(d) Double integral in polar coordinates:
ZZf(~r(r, ✓)) d d
(e) Triple integral in a change of coordinates:
ZZZf(u, v, w) d d d
(f) Triple integral in cylindrical coordinates:
ZZZf(~r(r, ✓, z)) d d d
(g) Triple integral in spherical coordinates:
ZZZf(~r(⇢, ✓,�)) d d d
IF ' HH t
I Fax ful u v
134¥41a v
r r f
facxyizsalum
witu v w
f r O Z
p's into p O lo
2. Use #1 on the previous page to answer the following.
(a) Show that (c) is a special case of (b) by writing the change of coordinates as aparameterization.
(b) Show that (d) is a special case of (c) by writing polar coordinates as a transfor-mation and taking the Jacobian.
(c) Convince yourself that each of the answers on the previous page is a reasonablenotion of “size” for either a single, pair, or triple of vectors. Discuss.
(d) Derive the following using (b).Surface integral over a cylinder of radius R:
ZZf(~r(✓, z)) d d
(e) Derive the following using (b).Surface integral over a sphere of radius R:
ZZf(~r(✓,�)) d d
~
We can write the transformation as flu, v ) =L Xiu ,
v ), y Cu ,
v ),
0 ) in whichI J K
case we get rixru =det¥¥¥¥.
oof-detfzI.nu?zfk=3Yui# I
so truxrvl = 1341*1 .
T : FIFI:& so 134%41=1.9::%9d-trcoszo-c-rsin.at- r
so dxdyerdr do.
I vector : trial is length ,makes sense
2 vectors : I Fux Ful and 13%71-1 are both the areas of parallelograms formed by grid lines
in the regionor surface .
3 vectors : 1%471741 is the volume of the parallelepiped formed by Fu,
Tv,
and Fw.
FCO # = l Raso,
Rsino , HR O z
To = f Rs.NO, Roos O ,
O ) Fox fz = ( Ros 0kt ( Rand J
Fz = ( O,
0, D liroxrzl -
- R
R2 sin 0 O 9
NO , 9) = I R since cos O ,Rs in 9 sin O , Roos 9)
to =L - Rs in 9 sin O,
Rs in 9050 , O )The = ( Roos 9 cos O ,
Raskin O ,- Rs in 9)
I Foxtel = It R'since cos of itC- R2 sinks in O) - R' since cos 46in Ot cos 20 ) Kl= R2 EGostsin299-RVE.EE = Rs in 4