Unformatted text preview: F-1 (1 . 1 , . 9). 3. Invent a linear transformation F : R 2 → R 2 that maps the ellipse x 2 + 4 xy + 5 y 2 = 4 onto the unit circle. Use F to find the area inside the ellipse. [See the Remark on page 157 of the Course Notes.] 4. Invent a linear transformation F : R 3 → R 3 that maps the ellipsoid x 2 + 8 y 2 + 6 z 2 + 4 xy-2 xz +4 yz = 9 onto the unit sphere. Use F to find the volume inside the ellipsoid. 5. Let ( u,v ) = F ( x,y ) = ( x + y 2 ,y ). (a) Show that F is invertible on R 2 by finding F-1 . (b) Let D xy = { ( x,y ) : 0 ≤ x ≤ 1 , ≤ y ≤ 1 } . Find the image of D xy , D uv . Verify that A xy = A uv , where A xy is the area of D xy and A uv is the area of D uv . Give an explanation of this by looking at the Jacobian of F ....
View
Full Document
- Spring '08
- WOLCZUK
- Math, Unit Circle, Inverse function, Injective function, D1 xy
-
Click to edit the document details