MATH 316-001 - Advanced Calculus II - Spring 2009

Announcements:

 

Final Exam.  The final exam for this class will be held on Monday May 11, 1:30pm-4:15pm in ST1, 242.  The exam is closed book.  The exam questions will be similar in style to homework problems and examples given in class. Coverage on the final exam will be as follows.

You will not be asked to state the theorems below but you will be required to use them appropriately:

 

Theorem 11.13 (A differentiable function is continuous)

Theorem 11.15 (Continuity of first order partials implies differentiability)

Theorem 11.28 (Chain Rule)

Theorem 11.31 (Mean Value Theorem)

Corollary 11.33 (The MVT for functions of several variables.  We proved a generalization of this in class which you will also be expected to know.)

Theorem 11.41 (Inverse Function Theorem)

Theorem 11.47 (Implicit Function Theorem)

Theorem 12.24 (Integration over regions of volume zero)

Theorem 12.46 (Change of Variables Formula)

 

You may be required to state some of the definitions listed below.

 

Definition 9.13 (Limits of functions in several variables)

Definition 9.22 (Continuity.  I know this we had no exercises from 9.3 but you need to know it anyway.)

Definition 11.12 (Differentiability)

Section 11.6 (the Jacobian, p. 359, and the partial Jacobian, p. 364)

Definition 12.3 (Jordan Region, also Theorem 12.14 gives a characterization of Jordan regions.

Definition 12.4 (Jordan volume or Jordan content)

 

Homework #9.  Homework #9 will be due by 5pm on Wednesday May 6.

 

End of Semester Business.  There will be an extra office hour on Thursday 4/23, 430-530pm, and on 4/30, 430-530pm.  There will be a makeup class on Tuesday May 5 from 4pm-5:15pm.  All of the above will be held in room 242, S&T 1.

 

Coverage 3/23/09.   In class today we covered the material in Section 8.2 relevant to the exercises in HW #5 on hyperplanes and linear operators from R^n to R^n.

 

Extension on Homework #4.  Homework #4 is now due on Monday March 23.

 

Exam 1, Wednesday March 4.  The midterm exam will be given in class on Wednesday March 4.  The exam is closed book and no calculator is permitted.  The exam questions will be similar in style to homework problems and to examples given in the lecture.  You will be required to know and be able to apply the following convergence tests.  You will not be asked to state the tests, but will be required to use them correctly and appropriately.  You will not be asked to state the theorems below but you will be required to use them appropriately.  You may be required to state some of the definitions listed below.  Technically the exam will cover Sections 6.4, 6.6, 7.1-7.3 but you will be expected to be able to apply the convergence tests discussed in Sections 6.1-6.3 and to know some basic definitions and results from those sections (listed below).

 

Convergence Tests:

p-Series Test (Coro. 6.13)

Comparison Test (Thm 6.14)

Limit Comparison Test (Thm 6.16)

Root Test (Thm. 6.23)

Ratio Test (Thm. 6.24)

Dirichlet's Test (Thm. 6.31)

Alternating Series Test (Coro. 6.32)

Raabe's Test (Thm. 6.43)

 

Definitions:

Absolute and Conditional Convergence (Def. 6.18)

Pointwise and Uniform Convergence of Sequence of Functions (Def. 7.1 and 7.7)

Pointwise and Uniform Convergence of Series of Functions (Def. 7.13)

Radius and Interval of Convergence of Power Series (Def. 7.20 and 7.23)

 

Theorems:

Cauchy Criterion for Convergence of Series (Thm. 6.8)

The Uniform Limit of Continuous Functions is Continuous (Thm. 7.10)

The Uniform Limit of Riemann Integrable Functions is Riemann Integrable (Thm. 7.10)

Theorems on Term-by-Term Integration and Differentiation (Thm. 7.14)

Weierstrass M-Test (Thm. 7.15)

Abel's Theorem on Uniform Convergence of Power Series (Thm. 2.19)

Term-by-Term Differentiability and Integrability of Power Series (Coro. 7.31 and Thm. 7.32)

 

Class cancellation/reschedule:  Class is cancelled on Wednesday February 4.  I have reserved our classroom (ST1, 242) on Thursday from 4:15--5:30 for a makeup class.  I am very sorry for the inconvenience.  Some notes (such as they are) for this class are posted here.

 

Problem Sessions:  A weekly problem session for Math 316 along the same lines as the one last semester for Math 315 has been set up for 7-8pm on Fridays. The room is not yet decided but interested students should look around at that time in the usual places, that is, room 242, 220, etc.  They are being run by Lars Aiken.

 

Extra office hour.  I will be holding an additional ``office hour'' on certain Thursday afternoons, 430pm-530pm in room 242, ST1 (the same room where we have class).  There will be one this Thursday Feb. 12.  Additional such hours are tentatively scheduled for Feb. 26, March 19, April 2, April 16, April 30, and May 7.  This is a chance to get some hints on homework problems, to see old homework problems worked out, to get questions from lecture answered, etc. all in a group setting.

 

Deadlines.  Please be aware of all relevant deadlines.

Course syllabus:
pdf, html.

Homework #1 (due 02-09-09): Section 6.1, #5, #7, #9, Section 6.2, #1, #2, #10, Section 6.3, #2(a)-(d), #6(a)-(c).  Solutions
Homework #2 (due 02-16-09):  Section 6.4, #3, #4, #8, Section 6.6, #3.  Solutions
Homework #3 (due 02-25-09):  Section 7.1, #1, #5, #6, #10, Section 7.2, #1, #2.   Solutions
Homework #4 (due 03-23-09):  Section 7.3, #1, #2, Section 7.4, #7, #8.  Solutions   (Note change in due date.)
Homework #5 (due 03-30-09):  Section 8.1, #6, Section 8.2, #1, #3, #5.  Solutions 
Homework #6 (due 04-06-09):  Section 9.1, #3, #5(a), (b), #10, Section 9.2 #2, #3   Solutions
Homework #7 (due 04-20-09):  Section 11.2, #8, #9, Section 11.4 #3, #9   Solutions
Homework #8 (due 04-27-09):  Section 11.5, #1, #5, Section 11.6 #1 (a), (b), #2 (a), (b), #3 Solutions 
Homework #9 (due 05-06-09):  Section 12.1 #4(a), (c), #5(a), (b), (c), Section 12.4 #4(a), #6 Solutions 

Exam Solutions:
Midterm:  pdf, Solutions: pdf
 

Math Department.
To contact me, send mail to: dwalnut@gmu.edu.