Math 53 - Multivariable Calculus
Homework # 4
July 19th
Due: July 24th, 2012
Exercise 1 (10 points). Let F (x, y ) = rn (x + y ), where r =
x2 + y 2
(a) For which values of n do the components P = rn x and Q = rn y of F satisfy P/y = Q/x? (Hint: start by
Mathematics 105, Spring 2004 - M. Christ Final Exam Review Guide The final exam will primarily emphasize the portion of the course concerned with Lebesgue integration, in which we followed Stroock's Chapters 2, 3, 4.1 and 5.0,2,3. From the first part
Alternate Characterization of Measurability.
A. Excision Property.
Proposition 0.1 Let E R. Given
and m (O) < m (E) + .
> 0, there exists an open set O such that E O
Proposition 0.2 A set E R is measurable if and only if given > 0, there exists an open
se
Math 53 - Multivariable Calculus
Quiz # 1
June 20th, 2012
Its common knowledge that two points in R3 determines a line segment joining them, and hence a vector stretching
between the points, while three points determines a plane which contains the points.
Math 53 - Multivariable Calculus
Homework # 5
July 26th
Due: August 1st, 2012
Exercise 1 (5 points).
Find the ux of the vector eld F =
y
x
x 2 +y 2 , x 2 +y 2
outwards through any circle centered at (1, 0) of radius r = 1. To do so,
consider the cases r <
Math 53 - Multivariable Calculus
Homework # 3
July 12th
Due: July 17th, 2012
Exercise 1 (5 points).
(a) Describe and sketch the image of the disk of radius 1, D = u2 + v 2 1, under the transformation x = au, y = bv , where
a, b R and a = 0, b = 0.
(b) Use
Math 53 - Multivariable Calculus
Quiz # 2
June 27th, 2012
Exercise 1. Suppose you have two particles, one is traveling along the space curve r 1 (t) = t, t2 , t3 while the other particle is
traveling along r 2 (r) = 1 + 2s, 1 + 6s, 1 + 14s . Do the partic
Math 53 - Multivariable Calculus
Quiz # 3
July 18th, 2012
Exercise 1. Let F = ex sin(y ), ex cos(y ) . Find a potential function for F and use it to evaluate
F dr , for all curves
from (a, b) to (c, d).
Exercise 2. Use Greens theorem to nd the work done
Lebesgue Outer Measure and Lebesgue Measure.
A. Basic notions of measure.
Our goal is to dene a set function m dened on some collection of sets and taking
values in the nonnegative extended real numbers that generalizes and formalizes the notion
of length
Sigma Algebras and Borel Sets.
A. Algebras.
Denition 0.1 A collection A of subsets of a set X is a -algebra provided that (1) A,
(2) if A A then its complement is in A, and (3) a countable union of sets in A is also in
A.
Remark 0.1 It follows from the de
3/15/2011
It is about the Real Analysis. please sol
Home > All Categories > Science & Mathematics > Mathematics > Open Question
It is about the Real Analysis. please solve these three problems?
For A measureable and B non-measurable with m^* (AB)<
prove
Math 53 - Multivariable Calculus
Quiz # 5
August 1st, 2012
Exercise 1. Let and be any two surfaces with equivalent boundaries = C = . Is it true that
F dS =
3
F dS for any vector eld F which is dierentiable on R ? Justify your answer.
Exercise 2. Let F
Math 53 - Multivariable Calculus
Quiz # 4
July 25th, 2012
Exercise 1. Let C be the ellipse dened by
x2
+
a
y2
b
= 1 and let F = 2x, 3y . Compute the ux of F through C ,
C
F nds.
(Hint: the area enclosed by the ellipse is ab .)
Exercise 2. Use a triple int
Math 53 - Multivariable Calculus
Homework # 2
June 28th
Due: July 3rd, 2012
Exercise 1. (a) Show that if a particle moves with constant speed then the velocity and acceleration vectors are orthogonal.
(b) If a particle with mass m moves with position vect
Math 53 - Multivariable Calculus
Homework # 1
June 21st
Due: June 26th, 2012
Exercise 1 (Stewart 12.3 # 49). Use a scalar projection to show that the distance from a point P1 = (x1 , y1 ) R2 to the line
ax + by + c = 0 is
|ax1 + by1 + c|
.
a 2 + b2
Use th
Mathematics 105 - Spring 2004 - M. Christ1 Solutions to selecta from problem set #2, with problem set #3 2-5. Let f (x, y) = x|y|/ x2 + y 2 for (x, y) = (0, 0), and let f (0, 0) = 0. Show that f is not differentiable at 0. (Note: It is customary to w
Mathematics 105 - Spring 2004 - M. Christ1 Solutions to selecta from problem set #1 No number. Let T : Rn Rm be a linear transformation. Show that there exists some finite constant M such that |T (v)| M |v| for every v Rn . (This was not assigned,
Mathematics 105 - Spring 2004 - M. Christ Problem Set 9 - Solutions to Selecta IX.A Consider the measure space (R1 , B R1 , ) where denotes Lebesgue measure. 1 Consider the measurable functions fn (x) = n [0,n] . Show that fn 0 uniformly on R. Show
Mathematics 105, Spring 2004 - Problem Set IV Solutions
1
IV.A Let {In } be any finite set of open intervals that covers [0, 1] Q. Show that n |In | 1. Explain why this does not prove that |[0, 1] Q|e 1. Solution. This does not prove that |[0,
Mathematics 105, Spring 2004 - Midterm Exam #1 Solutions (1b) Give an example of a function f : R3 R1 such that f is not differentiable at a = (0, 0, 0), but all partial derivatives Di f (a) do exist. Solution: Define f (0, 0, 0) = 0, and for all x