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

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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
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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 ....
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