WTW258: LU 2.5: TRANSFORMATIONS Lecture notes

2. 5. Transformations
A transformation in R2 is a function
TCU ,V ) I ≥iy ) with hluiv)
gcu ,v) and y
"

✗ =
-




÷÷~
a function

example :
polar coordinates

Tlr , a) =
inly ) with ✗
=
rcoso

Y' rsino-




Tdcobidnota transformation Jacobian of
-1
inverse transf .7


214,4 I
2111¥ =
2k an "




au au acxiy ) 24191
•
(4-11) acuiu )

ay ay
↳ partial 24 ◦✓
derivatives
Tcp , ∅, a) =
( x ,y,t ) with :



=p sin
* calculate x ¢ cost
psinosino
y
=




determinant
µ
2- =pcos¢



For coordinates : for spherical coordinates :

polar
mmmm mmmm

ahoy ) 2049171

/
= '" • -
Hino
a,
=
P2 sin ¢
air ,o , # acp , ¢ , ^
↓
o=⑤#
since rcoso

Note ]
rustlers / n'
'


=

•




plsino is positive
p2 ) 0
.




sing > o
'




-
o ≤ ¢≤ IT
i.
lplsin ¢ / =p 's in ¢

, Change of variables ( transf .
) in double integrals
Suv
} Ray
•




Re↓gion
↓
Region
f- is a continuous function defined on Ray ( Region in ✗ Y plane )



f) tiny)dA
Ray
JJ flglu
Suv
, v1 , hcu , v1 )
/ 201191
2C 4,4
DA




)Use7rdhm→ region of integration
is difficult to
describe and / or th⑤function to be

integrated is difficult to integrate
↓
Determine transformation from reason why you have decided to

use a transformation
[IN THIS COURSE IT WILL BE GIVEN ]
example
②
t.gg e "¥-ydA → difficult to integrate
R

① 4- my v x y
-
- -
-




then


can → easier to integrate
② Describe in Suv a. K→ d new region
region
-




→ by rewriting the boundary of the original region Ray