MATH 260 Class notes/questions February 19, 2013
Functions of several variables
R
Now were going to turn our attention to (not necessarily linear) functions from (subsets of) n
to m . So far in calculus youve considered functions from 1 1 and 1 m these ar
MATH 260 Exam 3
April 11, 2013
1. Calculate
xy dV over the solid S in the rst octant bounded by the surfaces z
and x 3.
S
y4 , y 2
The solid can be described as: For each x between 0 and 3, and each y between 0 and 2, z goes
from 0 to y 4 . So
xy dV
S
3
MATH 260 Exam 2
March 19, 2013
1. Suppose A is a real skew-symmetric n-by-n matrix.
(a) Show that iA is a Hermitian matrix (where i
c
1).
(b) Based on what you know about the eigenvalues of Hermitian matrices, what can you say
about the eigenvalues of A?
MATH 260 Review for Final Exam
April 27, 2013
1. Let M pn,
Rq be the space of n-by-n matrices with real entries.
(a) Show that (with the operations of matrix addition and scalar multiplication), M pn,
vector space.
Rq is a
Rq? Exhibit a basis of M p3, Rq.
1
Math 260 Practice Problems for Midterm 2
kx2 2xy 4y2 have a local minimum at (0,0)?
(B) k 0
(C) k 1
4
1. For which values of k does the function z
(A) k 0
(D) k 4
(E) for no values of k
Need the determinant of the Hessian to be positive and fxx p0, 0q
1
Math 260 Kazdan HW 9 solutions
1. (a) FInd the local maxima and minima of f px, y q : 3x 4y for x2 y 2
1.
(b) Find the maxima and minima of f px, y q : 3x 4y for x2 y 2
1.
(c) Find the global maxima and minima of f px, y q : 3x 4y for x2 y 2 1.
r3, 4
MATH 260 Exam 1
February 12, 2013
1. Let V and W be vector spaces, and let T : V W be a linear map. Suppose T v a, T w b
(with a $ b). and T x a b. Must x v w? Give a proof or a counterexample (in which you
get to pick the spaces, the map T and the vector
MATH 260 Final Exam
April 30, 2013
1. Let M pn,
Rq be the space of n-by-n matrices with real entries.
(a) We know that (with the operations of matrix addition and scalar multiplication), M pn,
is a vector space. What is the dimension of M pn, q? Exhibit a
University of Pennsylvania Department of Mathematics
Math 260 Honors Calculus II Spring Semester 2009
Prof. Grassi, T.A. Asher Auel
Midterm exam 2, April 7, 2009 (solutions)
1. Write a basis for the space of pairs (u, v ) of smooth functions u, v : R R th
University of Pennsylvania Department of Mathematics
Math 260 Honors Calculus II Spring Semester 2009
Prof. Grassi, T.A. Asher Auel
Practice nal exam solutions
1. Let F : R2 R2 be dened by F (x, y ) = (x + y, x y ).
a) If F denotes a force eld, then show
MATH 260 Class notes/questions February 5, 2013
Eigenvalues and eigenvectors in the presence of an inner product
Were still going to be considering linear maps from a vector space to itself, i.e., T : V V . But
now we will assume that V has a real or Herm
MATH 260 Class notes/questions January 29, 2013
Eigenvalues and eigenvectors
In this part of the course, were going to be considering linear maps from a vector space to itself,
i.e., T : V V . Therefore, when we consider matrix representations of our line
MATH 260 Homework assignment 6 February 19, 2013
1. Sketch level curves for
(a) f px, y q xy
(b) f px, y q exy
(c) f px, y q x2 y 4
(d) f px, y, z q cospx2 y 2 z 2 q
(e) f px, y, z q lnp1 x2 2y 2 q.
2. Describe the interior, exterior and boundary of each
MATH 260 Homework assignment 7 February 28, 2013
1. Consider the surface 2x2 5y 2 6z 2
16.
(a) Find a vector orthogonal to the tangent plane to the surface at the point p1, 2, 1q.
(b) Find the equation of the tangent plane to the surface at this point.
2
MATH 260 Homework assignment 8 March 21, 2013
1. Evaluate the line integral
C
x2 y dx px 2y q dy
(a) Where C is the part of the parabola y
y t2 for 0 t 1.
(b) Where C is the part of the parabola y
y sin2 t for 0 t cfw_2.
(c) Where C is the part of the lin
MATH 260 Homework assignment 9 March 28, 2013
1. Let be the rectangle with vertices at
(a)
R
p1, 0q, p1, 0q, p1, 2q and p1, 2q. Calculate:
px2 y2 q dA
(b)
xp1 2y q dA.
R
2. Suppose f : 2 is a continuous function. Find the exact volume of the region bounde
MATH 260 Homework assignment 10 April 4, 2013
1. Let S be the solid bounded by the paraboloid z
5 x2 y2 and the plane z 1.
Calculate
2
z dV .
S
2. Calculate the average value of the function g px, y q 3xy 2 over the box dened by
tpx, y, zq | 1 x 4, 1 y 1
MATH 260 Homework assignment 11 April 16, 2013
1. (Just for practice) Evaluate the line integral
F dr where F
z2 i psin yqj 2xzk, and C
is the curve with endpoints A p0, 0, 0q and B p1, 1, 1q which is obtained by intersecting the
surfaces x2 2y 2 z 2 0 a
MATH 260 January 17 weekend challenge
Forward dierences
At the end of class on Thursday we talked about the concept of forward dierences of functions.
This is useful in many numerical approximations and discrete dynamical systems.
All this is based on the
MATH 260 Notes on systems of ODEs
We talked a bit in class last week about constant-coecient systems of ordinary equations, which
turned out to be a good application of diagonalizing matrices.
The theory goes like this: Let
xptq
R
x1 ptq
x2 ptq
.
.
.
xn
MATH 260 Class notes/questions January 10, 2013
Linear transformations
Last semester, you studied vector spaces (linear spaces) their bases, dimension, the ideas of
linear dependence and linear independence. Now were going to study linear maps (or transfo
MATH 260 Class notes/questions January 22, 2013
Determinants
Determinants are important both from a theoretical and a practical point of view. They provide
a way of summarizing and simplifying the information contained in a square matrix.
You have already
U NIVERSITY OF P ENNSYLVANIA D EPARTMENT OF M ATHEMATICS
Math 260 Honors Calculus II Spring Semester 2009
Prof. Antonella Grassi, T.A. Asher Auel
Selected solutions
HW 1, due January 22, 2009
Exercises from Apostol Vol. II
(January 28, 2009)
2.4 #25 Let V
Exam 2
Math 260
March 13, 2012
Jerry L. Kazdan
12:00 1:20
Directions This exam has two parts. Part A has 6 short answer questions (7 points each, so 42
points) whilePart B has 4 traditional problems (15 points each, so 60 points). Total: 102 points.
Neatn
University of Pennsylvania Department of Mathematics
Math 260 Honors Calculus II Spring Semester 2009
Prof. Grassi, T.A. Asher Auel
Midterm exam 1, February 24, 2009 (solutions)
1.
Let V = M22 (R) be the vector space of 2 2 matrices and let A =
11
. Let T