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 surface of the ellipsoid of
2
2
2
equation x + y + z9 = 1
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 any of the variables as functions of the remaining vari
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 such that F (c) = 0. Further assume that
DF (C) is onto
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 assertion.
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 and a continuously dierentiable
function f : V Rnk dened
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) of any theorems you may use in your justication.
2. (
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 you know they are linearly independent.
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 use in your explanation.
2. (5 points) Determine whether
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 your answer.
2. (5 points) Write an iterated integral that
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 coordinates describing the
volume for the region inside the spher
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 the presence of other students.
5
0
1 7
3 4
1. (2
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, and prove that your parameterization is orientation
pre
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 integral of
the mass form, Mf , over the interior of the
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 Chain Rule for Taylor Polynomials: for smooth functions g