Mathematics 218: Analysis in Several Variables Professor Robert Gunning Fall Term 2009 Seventh Problem Set
Rik Sengupta December 4, 2009
1
Group 1
1. In R4 consider the differential forms
1 = x1 dx2 + x3 dx4 2 = (x2 + x2 )dx1 dx2 + (x2 + x2 )dx3 dx4 1 3 2
MATHEMATICS 218
ANALYSIS IN SEVERAL VARIABLES
Fall Term, 2008
PROBLEM SET 9: due Wednesday December 3, 2008
READING:
The special cases of integrals of dierential forms over curves and surfaces in R3
are discussed in Folland, Sections 5.1 and 5.3. This wee
MATHEMATICS 218
ANALYSIS IN SEVERAL VARIABLES
Fall Term, 2008
PROBLEM SET 10: due Wednesday December 10, 2007
This is the last weekly assignment of the term. An in-class closed-book quiz
will take the place of the usual 10am lecture on Monday December 8.
MATHEMATICS 218
ANALYSIS IN SEVERAL VARIABLES
Fall Term, 2008
PROBLEM SET 6: due Wednesday November 12, 2008
READING:
The remainder of the course will be devoted to integration, beginning with
integration over open subsets of Rn . The basic denition of th
Mathematics 218: Analysis in Several Variables Professor Robert Gunning Fall Term 2009 Fifth Problem Set
Rik Sengupta November 18, 2009
Evaluate the following integrals:
1
1.
E
Group 1
x2 where E is the set x1 0, 1 x2 + x2 2. 1 1 2
Solution. 3 . Consider
MATHEMATICS 218
ANALYSIS IN SEVERAL VARIABLES
Fall Term, 2008
PROBLEM SET 5: due Wednesday November 5, 2008
This problem set gives a small taste of ODE problems you can now solve. These
exercises should not take much time, which will allow you to look ove
MATHEMATICS 218
ANALYSIS IN SEVERAL VARIABLES
Fall Term, 2008
PROBLEM SET 2: due Wednesday October 1, 2008
READING:
Derivatives in n dimensions are discussed in Spivak on pages 15 - 33 and in
Folland on pages 53 - 77 and 106 - 111. Spivak treats the gener
MATHEMATICS 218
ANALYSIS IN SEVERAL VARIABLES
Fall Term, 2008
PROBLEM SET 3: due Wednesday October 8, 2008
READING:
This week continues to focus on derivatives, and in particular the intricacies of
the chain rule, for which sections 2.2, 2.3, 2.5, and 2.1
MATHEMATICS 218
ANALYSIS IN SEVERAL VARIABLES
Fall Term, 2008
PROBLEM SET 4: due Wednesday October 15, 2008
READING:
Some quite basic and important results about dierentiable mappings in several
variables are the inverse mapping theorem, the implicit func
MATHEMATICS 218
ANALYSIS IN SEVERAL VARIABLES
Fall Term, 2008
PROBLEM SET 8: due Wednesday November 26, 2008
READING:
Integration over submanifolds requires the development of a more careful description of standard subsets of Rn . The denitions of singula
MATHEMATICS 218
ANALYSIS IN SEVERAL VARIABLES
Fall Term, 2008
PROBLEM SET 7: due Wednesday November 19, 2008
READING:
The machinery of dierential forms provides the most convenient tool for handling integration over submanifolds of Rn in general, and clar
Mathematics 218: Analysis in Several Variables Professor Robert Gunning Fall Term 2009 Sixth Problem Set
Rik Sengupta November 25, 2009
1
Group 1
Evaluate the following integrals, using a change of variables to simplify the calculation. 1.
E
(x1 + x2 )2 +
Mathematics 218: Analysis in Several Variables Professor Robert Gunning Fall Term 2009 Eighth Problem Set
Rik Sengupta December 15, 2009
1
Group 1
1. Evaluate x1 dx1 + x2 dx2 + x2 dx3 along the curve x1 = x2 = x3 from 1 (0, 0, 0) to (1, 1, 1). Solution. I
Mathematics 218: Analysis in Several Variables Professor Robert Gunning Fall Term 2009 Fourth Problem Set
Rik Sengupta October 26, 2009
1
Group 1
1. (a) Sketch the plane curve defined by the equation f (x) = x2 - 3x2 - 3 = 1 2 0. (b) At which points can t
MATHEMATICS 218 ANALYSIS IN SEVERAL VARIABLES FALL TERM, 2009 COURSE SYLLABUS
Instructors: R. C. Gunning (902 Fine Hall) and K. Hughes (311 Fine Hall) Classes: MWF 10-10:50am, Fine 214 and Th 7:30 -7:50pm, Fine 214 Course Description The course covers dif
Mathematics 218: Analysis in Several Variables Fall Term 2009 First Problem Set
Rik Sengupta September 30, 2009
1
Group 1
1. Sketch the following subsets of R2 = cfw_x = (x1 , x2 ); and for each subset indicate whether it is open, closed, or compact, and
Mathematics 218: Analysis in Several Variables Professor Robert Gunning Fall Term 2009 Second Problem Set
Rik Sengupta October 7, 2009
1
Group 1
1. f : R2 R1 , where f (x) = x2 + x3 + x1 x2 . 1 2 Solution. The derivative is the 1x2 matrix 2x1 + x2 3x2 + x
Mathematics 218: Analysis in Several Variables Professor Robert Gunning Fall Term 2009 Third Problem Set
Rik Sengupta October 14, 2009
1
Group 1
2 1. Find 1 2 sin (x1 x2 ) and 1 sin (x1 x2 ).
Solution. We have
1 2 sin (x1 x2 ) = 1 cfw_x1 cos (x1 x2 ) = x1
MATHEMATICS 218
ANALYSIS IN SEVERAL VARIABLES
Fall Term, 2008
PROBLEM SET 1: due Wednesday September 24, 2008
READING:
Standard mathematical notation and the basic topological results that will be
needed in the course are covered in Chapter 1 of Folland,