Quiz 11
May 3, 2012
Name:
1. (5 Points) Let be the 1 form on R3 dened by = xyz dy + xey dz + sin(y ) dx.
Compute the exterior derivative of .
2. (5 Points) Let = x2 dx dy + z 3 dx dz . Let S be the su
Quiz 1
January 26, 2012
Name:
1. (5 points) State the implicit function theorem (the short version is ne).
2. (5 points) Determine whether the system
x2 + y + z 3 = 3
xyz = 1
(1)
(2)
implicitly denes
Quiz 1
January 31, 2012
Name:
1. (5 points) State the implicit function theorem (the short version is ne).
Let U be an open subset of Rn+m with F : U Rn a continuously differentiable function and c U
Quiz 2
February 7, 2012
Name:
1. (5 points) Dene a smooth k -dimensional manifold in Rn .
2. (5 points) Determine whether the locus of equation
ex + 2ey + 3ez = 10
is a manifold. Justify your assertio
Quiz 2
February 7, 2012
Name:
1. (5 points) Dene a smooth k -dimensional manifold in Rn .
A subset M Rn is a smooth k -dimensional manifold if for every z M ,
there exists an open set U containing z a
Quiz 3
February 9, 2012
Name:
1. (5 points) A manifold has been parameterized by : R2 R3 where
u
2
2
u
u2 +v . Find the tangent space at the point 9. State the
=
2
v
v2
2
2
content (not just a number
Quiz 4
February 16, 2012
Name:
1. (5 points) Find the degree 3 Taylor polynomial of
origin.
1
1+x2 +y 2
about the
2. (5 points) Write x2 + xy + yz as a sum of linearly independent squares.
Explain why
Quiz 5
March 8, 2012
Name:
1. (5 Points) Explain mathematically what it means for a function
f : Rn R to be integrable (more specically Riemann integrable). Dene
any terminology or notation you may us
Quiz 6
March 26, 2012
Name:
1. (5 Points) Integrate the function1 sin x over the triangle in R2 whose
x
vertices are located at the points (0, 0), (, 0), and (, ). Cite any theorems
you may use in you
Quiz 7
March 29, 2012
Name:
1. (5 points) Write an integral in polar coordinates for the area between
the circles x2 + y 2 = 9 and x2 + y 2 = 4.
2. (5 points) Write an integral in spherical coordinate
Take-home Quiz
Name:
Instructions: Please turn this quiz in at the beginning of discussion on Thursday. You should work alone in the
sense that any answers written on your quiz should not be made in t
Quiz 6
April 18, 2012
Name:
1. (5 Points) Provide a way to orient the unit circle in R2 , and provide an
orientation preserving parameterization for the unit circle. Please be as
precise as possible,
Quiz 10
April 25, 2012
Name:
1. (10 Points) Let T be the torus obtained by rotating a circle of radius 1 centered at
(R, 0, 0) about the z axis where R > 1. Let f (x, y, z ) = x2 + y 2 . Compute the i
Homework Note
February 14, 2012
Name: Derek Olson
3.4.3 Find the Taylor Polynomial of degree 2 of the funtion F (x, y ) =
x + y + xy at the point (2, 3).
This problem is a nice example of using the Ch