A9 - F-1(1 1 9 3 Invent a linear transformation F R 2 → R...

MATH 237 Assignment 9 Due March 23, 10 a.m.
Use the appropriate drop box by MC 4066.
1. Let ( u, v ) = F ( x, y ) = ( e x cos πy, e x sin πy ) and D
1 xy
= { ( x, y ) : | x | ≤ 0 . 1 , | y | ≤ 0 . 1 } ,
D
2 xy
= { ( x, y ) : | x | ≤ 1 , | y | ≤ 1 }
(a) Find the image of each region D
1 xy
and D
2 xy
, D
1 uv
and D
2 uv
. Make a sketch.
(b) Find the derivative matrix DF ( x, y ) and the Jacobian
∂ ( u, v )
∂ ( x, y )
.
(c) Show that F is one-to-one on D
1 xy
by finding F - 1 .
(d) For ( x, y ) in D
1 xy
, verify that DF
- 1
( u, v ) DF ( x, y ) = I , where I is the identity
matrix. Also verify that the Jacobians satisfy
∂ ( x, y )
∂ ( u, v )
∂ ( u, v )
∂ ( x, y )
= 1.
(e) Verify the approximation A
1 uv
≈
∂ ( u,v )
∂ ( x,y )
(0 , 0) A
1 xy
where A
1 xy
is the area of D
1 xy
and A
2 xy
is the area of D
2 xy
.
(f) Show that F is not one-to-one on D
2 xy
. Explain why this does not contradict the
Inverse Mapping Theorem.
2. Let ( u, v ) = F ( x, y ) = ( x 2 + e y , y 2 + e x ).
(a) Show the Jacobian
∂ ( u, v )
∂ ( x, y )
(0 , 0) 6 = 0 .
(b) By the Inverse Mapping Theorem F
- 1
exists in a neighbourhood of ( x, y ) = (0 , 0).
Find the linear approximation to F - 1 at ( u, v ) = F (0 , 0) = (1 , 1) and use it to