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       \centerline{\bf Information Sheet}
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     \centerline{\it Math 336, Section 1,   Spring  1998 }

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\noindent
{\bf Instructor:}  

Prof. John Cao,  Room 221 CCMB, 631-8847 (office), 
                                272- 4296 (home).

e-mail: jcao$@$jcao.math.nd.edu

\noindent
{\bf Textbook:} 

M. H.  Protter and C. B. Morrey:  {\it ``A first course in real analysis"}, 
second edition, Springer-Verlag, 1997.

              



                      

\noindent
{\bf Office Hours:} 

Prof. Cao: Wed 2:30-3:30pm, Room 221 CCMB, or by appointment. 

       % Mr. Ilya German: Tues 2:50 - 3:40 pm in Room B7, White Hall, or by %appointment.

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{\bf Exams:} (to be arranged.) 
 
 

\noindent
{\bf How things count:} (to be discussed.)


\noindent
{\bf Homework  Policy:} 

 One third credits will be given each time if you hand in 
your homework  on time.  No later homework 
 will be accepted. 


\noindent
{\bf Syllabus:} (tentative)

   1. Review;

   2. Chapter 6, Metric Spaces;

   3. Chapter 9, (\S 9.1-\S 9.4), Series;

   4. Chapter 7, (\S 7.1-\S 7.3) Differentiation in $R^n$;

   5. Chapter 13, Fixed point theorems

   6. Chapter 14, (\S 14.1 - \S 14.2) Implicit function theorems.






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{\bf Math 336, Section 1      \hfill              Due Wednesday, January 28, 1998
}



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             \centerline{\it              Homework Assignment  1}
                          

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 \noindent
\noindent
Question 1,  Distance functions and metric spaces (Section 6.1)

       p134, \#1, 2, 3, 5.


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Question 2,  More examples of metric spaces (Section 6.1)

   
  p134, \#7, 8.

\noindent
Question 3,  Applications of Schwarz inequality (Section 6.1)

   
  p134, \#10, 12.
     

       
 

 

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{\bf Math 336, Section 1      \hfill              Due Wednesday, February 4, 1998
}



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             \centerline{\it              Homework Assignment  2}
                          

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Question 1,  Closed and open subsets in metric spaces (Section 6.2)

       p144, \#3, 4, 5.


\noindent
Question 2,  Limit points and isolated points (Section 6.2)

   
  p144, \#6, 7

  p145, \#16

\noindent
Question 3,  Geometry and algebra of subsets (Section 6.2)

   
  p144, \#10, 12.
     

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\noindent
{\bf Math 336, Section 1      \hfill              Due Wednesday, February 4, 1998
}



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             \centerline{\it              Homework Assignment  3}
                          

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\noindent
Question 1,  Countable and uncountable subsets (Section 6.3)

       p149, \#1, 3, 5, 12.


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\noindent
{\bf Math 336, Section 1      \hfill              Due Wednesday, February 27, 1998
}



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             \centerline{\it              Homework Assignment  4}
                          

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\noindent
{\bf Important Reminder:} There will be an exam in Friday's class,
at 9:35 am, February 20, Room 300 CCMB. This exam will cover \S 6.1- \S6.4,
p130-157.


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 \noindent
\noindent
Question 1,  Properties of compact sets (Section 6.4)

       p155-156, \#1, 2, 13.


\noindent
Question 2,  Examples of non-compact sets (Section 6.4)

       p156, \#4, 7, 14.



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\noindent
{\bf Math 336, Section 1      \hfill              Due Wednesday, March 3, 1998
}



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             \centerline{\it              Homework Assignment  5}
                          



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 \noindent
\noindent
Question 1,  Lebesgue Theorem and more about compact sets (Section 6.4)

       p156, \#3, 9, 10.


\noindent
Question 2,  Functions on compact sets (Section 6.5)

       p159, \#3, 11, 12, 13, 14.


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\noindent
{\bf Math 336, Section 1      \hfill              Due Wednesday, March 25, 1998
}



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             \centerline{\it              Homework Assignment  7}
                          


\noindent
{\bf Important Reminder:}

    Exam 2 will take place on March 27, Friday, in class. This Exam will
cover Chapter 6 and Chapter 7.


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 \noindent
\noindent
Question 1,  Continuous maps (Section 6.7)

       p172, \#4, 5, 7, 9, 10, 17, 19.


\noindent
Question 2,  Partial derivatives and the chain rule (Section 7.1)

       p178, \#4, 6, 10.


\noindent
Question 3,  Taylor's theorem and critical points (Section 7.2)

       p187, \#1,

       p188, \#2, 8, 9


\noindent
Question 4,  The gradient of a smooth function (Section 7.3)

       p192, \#2, 3.



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\noindent
{\bf Math 336,      \hfill              Due Wednesday, April 8, 1998
}



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             \centerline{\it              Homework Assignment  8}
                          






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\noindent
Question 1,  Taylor's theorem and critical points (Section 7.2)

       

       p188, \#7, 8, 9


\noindent
Question 2,  The gradient of a smooth function (Section 7.3)

       p192, \#2, 3, 4, 7.



\noindent
Question 3,  Infinite sequences and infinite series (Section 9.1)

       

       p221, \#4, 15.


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\noindent
{\bf Math 336,      \hfill              Due Wednesday, April 15, 1998
}



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             \centerline{\it              Homework Assignment  9}
                          






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\noindent
Question 1,  Tests for convergence and divergence (Section 9.1)

       

       p215, \#2, 4, 7, 9, 10


\noindent
Question 2,   Limit comparison Theorem (Section 9.1)

       

       p215, \#16, 17, 18




\noindent
Question 3,  Absolutely convergent and conditional convergent (Section 9.2)

       

       p221, \#5, 10, 14,  16.


\noindent
Question 4, Convergence radius for  power series (Section 9.2)

       

       p221, \#17, 20, 22


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\noindent
{\bf Math 336,      \hfill              Due Wednesday, April 15, 1998
}



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             \centerline{\it              Homework Assignment  9}
                          






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\noindent
Question 1,  Tests for convergence and divergence (Section 9.1)

       

       p215, \#2, 4, 7, 9, 10


\noindent
Question 2,   Limit comparison Theorem (Section 9.1)

       

       p215, \#16, 17, 18




\noindent
Question 3,  Absolutely convergent and conditional convergent (Section 9.2)

       

       p221, \#5, 10, 14,  16.


\noindent
Question 4, Convergence radius for  power series (Section 9.2)

       

       p221, \#17, 20, 22

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\noindent
{\bf Math 336,      \hfill              Due Wednesday, April 29, 1998
}



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             \centerline{\it          Last     Homework Assignment  }
                          






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\noindent
{\bf An Important Remainder:}

     The Final Exam will take place at 4:15 - 6:15 pm, Monday, 
May 4, 1998 in Room 328, CCMB.

      The final exam will cover Ch6, Ch7, \S9.1-9.4, \S13.1-13.2.

==============================================








\noindent
Question 1,  Fixed points of contraction mappings (Section 13.1)

       

       p334, \#10, 11.


\noindent
Question 2,   Applications of the fixed point theorem (Section 13.2)

       

       p339, \#1, 2, 5







      







     

       
 

 

    





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