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