237q2 - , ( [ dy dx y x f , then ∫ ∫ = 1 1 1 ] ) , ( [...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
1. [5 marks] Suppose R R f 3 : is of class ) ( 3 2 R C and 3 2 : R R g is defined by ) 1 2 , 1 , 3 ( ) , ( 2 + + - = y x y x y x g . Compute 2 2 y ϖ where ) , )( ( y x f g ° = in terms of the derivatives of f . To have the same notations in all papers, please consider ) , , ( w v u f where 1 2 , 1 , 3 2 + = + = - = y w x v y x u . 2
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2. [5 marks] Let R R F 2 : be defined by ) 1 ( ) , ( 2 + + = y x x y x F . Near which points of the set } 0 ) , ( : ) , {( = = y x F y x S is S the graph of a 1 C function ) ( x f y = or ) ( y g x = ? Justify your answer. 3
Background image of page 2
3. (a) [2 marks] Under what assumptions on 3 2 : R R f is f im locally a graph of a smooth surface? (b) [6 marks] Let S be the surface generated by revolving the curve y y x - = 2 , 3 0 y around the y-axis. Show that the surface S is smooth in the neighborhood of the point ) 3 , 2 , 1 ( P and find the unit normal vector to S at this point. Is S smooth globally? Why or why not? 4
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
4. [6 marks] Let C be the curve of intersection of the paraboloid 0 2 2 = - + z y x and the plane 0 1 2 = + - z y . Find a parametrization of C. Is the curve C smooth? Justify. 5
Background image of page 4
5. (a) [2 marks] Is the following statement always true? “ If ∫ ∫ = 1 0 1 0 1 ] )
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: , ( [ dy dx y x f , then ∫ ∫ = 1 1 1 ] ) , ( [ dx dy y x f ” Justify shortly your answer. (b) [6 marks] Evaluate dydx y x ∫ ∫ + 4 2 3 1 6 6. (a) [3 marks] Give the precise definition of the notion that the set 2 R Z ⊂ has “zero content” and prove that if 2 1 R Z ⊂ and 2 2 R Z ⊂ have zero content, then 2 1 Z Z ∪ has also zero content. (b) [5 marks] Let = = = otherwise n some for y x if y x f n n 1 ... 3 , 2 , 1 , ) ( ) , ( ) , ( 2 2 1 , 2 1 . Show that f is integrable on the rectangle ] 2 , [ ] 1 , [ × = E and determine the value of I = ∫ ∫ E dx y x f ) , ( . You may use any known theorems/properties without the proof, but you have to formulate them. ← Use the reverse side of the previous page if you need more space 7...
View Full Document

This note was uploaded on 10/21/2010 for the course MATHEMATIC MAT237Y1 taught by Professor Romauldstanczak during the Fall '09 term at University of Toronto.

Page1 / 6

237q2 - , ( [ dy dx y x f , then ∫ ∫ = 1 1 1 ] ) , ( [...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online