Chapter 6-The Change of Variables Formula and Applications of Integration

Chapter 6 :
The Change of Variables Formula and Applications of Integration

6. 1 :
The Geometry of Maps I R2 to 13
from


T(x *, y )
*
(x , y) :




Y Y
N N




*
T
D D



3 X 3 X




eX :
(r , 8) =
(0 , 17 x 10 , 2n)


find T(r , 0) =
(rcos(0) usin(0) ,




los" (0) + r sin (0) =
R = /



... D =
X + y =




6 2:
.
The change of variables theorem



Jacobian Determinant :
now a transformation distorts a region


X= X(u , v) y =
Y(4 , V)



:-
OX , Y) :
OXIOU OXIOv
0(u , v)
oylou Oyldu



change of variables formula :




(f(x(undxau /x
=




( ,
f(x y)axdy
,
=



((y +
f(x(y v) y(4 v))d(X , 3) dud
, , ,



O(u , v)




Integrals in Polar Coordinates :




// ,
f(x y)axdy,
=



(1 +
f(rcos(0) usin(8))rarde ,




Change of Variables Formula for Triple Integrals
:




J :
d(X , Y 7) ,
: OXIOU OXOr OXOW
O(u v , w)
,
Oy lou dyldu bylow
dzidu Ozldv Glow


(((y +(x , y, z)axdydz =



((( xf(X(u , v ,
w) y(y , V w) z(u
, , , ,
v, w))d(x , y,
O(u , V , w)
z) dudvaw




Cylindrical coordinates
:




X rcos(0) y rsin(0) , z z
= = =
,




& (x , Y , z) =
COSIO) -



rsin(t) 0 :
r

d (r , 0 , z)
Sin (0) rcos(n) 0

00 I



(1/w + (x , y , z)dxdydz =



(((wx f(vcs(t) , usin(t) , z) rarddz