國立清華大學數學系
高等微積分一
Math 2149-00 (Spring 2016) Advanced Calculus (I)
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Office: 綜三 609 |
Course
Description:
This year-long, 6-credits
course is an introduction to mathematical analysis intended for students who have
received a standard training on single-variable calculus. We will study
analytical tools, topological and geometric concepts needed for multi-variable
calculus in a rigorous manner. The first semester will be focusing on series,
topology of R^n, and differential calculus. The
second semester will be focusing on implicit and inverse function theorems,
constraint extrema, and integral calculus. Those who
planning on taking more advanced courses about mathematical analysis, such as
real and functional analysis, should seriously consider taking the other
8-credits advanced calculus offered by the math department.
Registered students are
recommended to use the iLMS e-learning system (http://www.nthu.edu.tw/elearning.php),
where your teaching assistants will post contents of every lecture online. It
is only open to registered students. Contact your teaching assistants if you
have any question.
Textbook:
C. Canuto,
A. Tabacco: Mathematical Analysis II, 2nd edition,
Springer, 2015.
This book is vailable in pdf format at the
NTHU library. NTHU students are allowed to download for personal use.
This textbook was selected
for 3 reasons: 1. it is free for NTHU students; 2. materials in there are
standard and well-organized; 3. it is a very readable book with many good
exercises.
References:
[1]. C. Canuto,
A. Tabacco: Mathematical Analysis I, Springer, 2008.
[2]. W.Rudin:
Principles of Mathematical Analysis, 3rd edition, 1976
[3]. J.Marsden
and M.Hoffman: Elementary Classical Analysis, 2nd
edition, 1993.
Reference [1] contains
materials for a typical year-long course on single-variable calculus. It is
also available in pdf format at the NTHU library. You
may refresh your memories about first year calculus with this book or whichever
textbook you used.
Refernces [2], [3] are standard textbooks for advanced
calculus, often inteded for math major students. They
are resources for those mathematically inclined students who are not contented
by our textbook.
Grading:
Quizzes: 20% -- Quizzes will
be based on homework assignments. Contact your teaching assistants for details.
Midterm Exams: 50% -- April
7 (Thursday), May 19 (Thursday) from 8:00-9:50am
Final exam: 30% -- June 16
(Thursday) from 8:00-11:00am
Attendance:
Students are expected to attend every scheduled
class. It is the student's responsibility to keep informed of any
announcements, syllabus adjustments or policy changes made during scheduled
classes.
Syllabus:
We shall follow the textbook but add, rearrange, and
skip a few topics as we move along. Here are topics to be covered in this
semester:
Chapter 1. Numerical Series
1.1. Sequences and convergence
1.2. Series and convergence
1.3. The algebra of series
Chapter 2. Series of Functions and Power Series
2.1. Sequences of functions
2.2. Properties of uniformly convergent sequences
2.3. Series of functions
2.4. Power series
2.5. Analytic functions
Chapter 3. Fourier Series
3.1. Trigonometric polynomials and series
3.2. Fourier coefficients and Fourier series
3.3. Convergence of Fourier series
3.4. A few applications
Chapter 4. Functions between Euclidean Spaces
4.1. Algebraic structure of Euclidean spaces
4.2. Topology of Euclidean spaces
4.3. Limits and continuity
4.4. Continuity and compactness
Chapter 5. Differential Calculus for Scalar Functions
5.1. First derivatives
5.2. Mean value theorem
5.3. Second and higher derivatives
5.4. Taylor expansions