Math 3010
Quiz 2
Oct. 3, 2012
Student number:
Name:
1. (6 points) Determine if the following mappings T : R3 R3 are one-to-one
and/or onto. Show your proofs.
(i) T (u, v, w ) = (v sin (u) , w cos (v ) , uv )
(ii) T (u, v, w ) = (uv, vw, uw )
2. (4 points)
Math 3010
Quiz 2
Oct. 3, 2012
1. (6 points) Determine if the following mappings T : R3 R3 are one-to-one and/or onto.
Show your proofs.
(i) T (u, v, w ) = (v sin (u) , w cos (v ) , uv )
Solution: This map is not one-to-one because T (0, 0, 0) = (0, 0, 0)
Math 3010
Quiz 4
Nov. 28, 2012
1. (4 points) For a surface S and xed vector v R3 , prove that
2
v n dS =
(v r) dS.
S
S
where r (x, y, z ) = (x, y, z ).
Solution: Let v = (a, b, c) and F = v r . Then
k
abc
xyz
vr =
= (bz cy, cx az, ay bx) .
(v r ) =
k
x
Math 3010
Fall 2012
Final Exam
Dec. 8, 2012
Time Limit: 180 Minutes
Student Name:
Student Number:
This exam contains 14 pages (including this cover page) and 10 problems. Check to see if
any pages are missing.
Enter all the requested information on the to
Math 3010
Quiz 1
Sept. 19, 2012
1. (6 points) Give a path c : [a, b] Rn or c : R Rn that parametrizes each of the
following curves. If the path has the form c : [a, b] Rn , give the values of a and b.
(a) The line passing through (1, 2, 3) and (2, 0, 7)
S
Math 3010
Midterm
Oct. 24, 2012
Student number:
Name:
1. (15 points) Improper integrals
(a) (9 points) Let D = cfw_(x, y ) |x [a, b] , y [1 (x) , 2 (x)] where i are continuous. Let f : R2 R be bounded and continuous in the interior of D . Suppose f
may be
Math 3010
Bonus Quiz
Nov. 16, 2012
Student number:
Name:
Select the best answer to the following questions.
For questions 1-3 let C be an oriented simple curve in Rn and let c : [a, b] Rn be
an orientation-preserving parametrization of C . Let f : Rn R an
Math 3010
Fall 2012
Final Exam
Dec. 8, 2012
Time Limit: 180 Minutes
Student Name:
Student Number:
This exam contains 14 pages (including this cover page) and 10 problems. Check to see if
any pages are missing.
Enter all the requested information on the to
Math 3010
Midterm
Oct. 24, 2012
1. (15 points) Improper integrals
(a) (9 points) Let D = cfw_(x, y ) |x [a, b] , y [1 (x) , 2 (x)] where i are continuous. Let f : R2 R be bounded and continuous in the interior of D . Suppose f
may be discontinuous on poin
Math 3010
Quiz 3
Nov. 14, 2012
1. (5 points) Evaluate the integral of the function f (x, y, z ) = z + 6 over the surface S
given by
v
(u, v ) = u, , v , u [0, 2] , v [0, 3] .
3
1
Solution: Tu = (1, 0, 0) and Tv = 0, 3 , 1
1
0, 1,
3
Tu Tv =
3
,
10
3
Tu Tv
Math 3010
Quiz 1
Sept. 19, 2012
Student number:
Name:
1. (6 points) Give a path c : [a, b] Rn or c : R Rn that parametrizes each of
the following curves. If the path has the form c : [a, b] Rn , give the values of
a and b.
(a) The line passing through (1,
Math 3010
Quiz 4
Nov. 28, 2012
Student number:
Name:
1. (4 points) For a surface S and xed vector v R3 , prove that
2
v n dS =
S
(v r) dS.
S
where as always r (x, y, z ) = (x, y, z ).
2. Vector elds
(a) (2 points) Find f : R2 R such that (y 3 + 1, 3xy 2 +
Math 3010
Quiz 3
Nov. 14, 2012
Student number:
Name:
1. (5 points) Evaluate the integral of the function f (x, y, z ) = z + 6 over the surface S
given by
v
(u, v ) = u, , v , u [0, 2] , v [0, 3] .
3
2. (5 points) Let D be a region for which Greens theore
Math 3010
Bonus Quiz
Nov. 16, 2012
Student number:
Name:
Select the best answer to the following questions.
For questions 1-3 let C be an oriented simple curve in Rn and let c : [a, b] Rn be
an orientation-preserving parametrization of C . Let f : Rn R an