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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller lexisparks. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $4.59. You're not tied to anything after your purchase.
