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

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

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

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

View Full Document
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
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

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

View Full Document
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
5. (a) [2 marks] Is the following statement always true? “ If ∫ ∫ = 1 0 1 0 1 ] )

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

View Full Document
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
Ask a homework question - tutors are online