Lecture notes were made while watching the recorded lectures assigned to watch. These notes include theory (theorems) and worked out examples from the lecturer.
These specific notes cover Transformations.
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
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