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
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
