Math 140B

• Office: APM 6-101
• Office hour: Monday 2-3, Wednesday 11:30-12:30. You can also speak to me after lecture.
• Email: doprea at math dot ucsd dot edu

• Discussion: APM 7421, Monday, 6:00-6:50PM
• Office: APM 6446
• Office hour: Tuesday 3:30-4:30, Thursday 12-1.
• Email: mbodnar at ucsd dot edu.

Differentiation. Riemann integral. Sequences and series of functions. Special functions. Fourier series. This corresponds to chapters 5-8 in Rudin's book.

Math 140A or permission of instructor. Students will not receive credit for both Math 140 and Math 142. Math 140 is a difficult and time consuming course, so enroll only if your course load allows it.

• Homework 20%, Midterm I 20%, Midterm II 20%, Final Exam 40%.

W. Rudin, Principles of Mathematical Analysis, Third Edition.

Reading the sections of the textbook corresponding to the assigned homework exercises is considered part of the homework assignment. You are responsible for material in the assigned reading whether or not it is discussed in the lecture. It will be expected that you read the assigned material in advance of each lecture.

Homework problems will be assigned on the course homework page. There will be 7 problem sets, typically due on Wednesdays at 4:30PM in the TA's mailbox. The due date of some of the problem sets may change during the quarter depending on the pace at which we cover the relevant topics. The best six problem sets will be used to compute the final grade.

You may work together with your classmates on your homework and/or ask the TA (or myself) for help on assigned homework problems. However, the work you turn in must be your own. No late homework assignments will be accepted.

There will be two midterm exams given on January 28 and February 27. There will be no makeup exams.

Regrading policy: graded exams will be handed back in section. Regrading is not possible after the exam leaves the room.

The final examination will be held on Friday, March 20, 11:30-2:30AM. There is no make up final examination. It is your responsability to ensure that you do not have a schedule conflict during the final examination; you should not enroll in this class if you cannot sit for the final examination at its scheduled time.

• Monday, January 5: First lecture
• Monday, January 19: Martin Luther King Day. No class.
• Wednesday, January 28: Midterm I
• Monday, February 16: Presidents' Day. No class.
• Friday, February 27: Midterm II
• Friday, March 13: Last Lecture
• Friday, March 20: FINAL EXAM, 11:30-2:30.

• Preparation for Midterm 1:
• Practice midterm from Spring 2013 - PDF and Solutions
• Practice midterm from Winter 2014 (from Professor Adrian Ioana) - PDF and Solutions
• Midterm 1 - PDF and Solutions
• Preparation for Midterm 2:
• Practice midterm from Spring 2013 - PDF and Solutions
• Practice midterm from Winter 2014 (from Professor Adrian Ioana) - PDF and Solutions
• Midterm 2 - PDF and Solutions
• Preparation for Final:
• Practice final from Spring 2013 - PDF and Solutions
• Practice Final from Winter 2014 (from Professor Adrian Ioana) - PDF and Solutions
• Final Exam - PDF and Solutions

• Lecture 1: Introduction. Definition of the derivative. Differentiable functions are continuous. Rules for differentiation.
• Lecture 2: Chain rule and proof. Extremal values and derivatives.
• Lecture 3: Rolle, Cauchy and the mean value theorem. Monotonic functions via derivatives. Example of a function with discontinuous derivative.
• Lecture 4: Darboux functions. The derivative of a differentiable function is Darboux. L'Hopital's rule and proof in a particular case.
• Lecture 5: Proof of L'Hopital's rule. Taylor's theorem. Motivation. Taylor polynomial. Statement of the theorem.
• Lecture 6: Proof of Taylor's theorem. Integration. Lower and upper Darboux sums. Definition of the integral. Riemann-Stieltjes integral.
• Lecture 7: Riemann's criterion for integrability. Refinements of partitions and effects on upper and lower sums. Monotonic functions are integrable.
• Lecture 8: Continuous functions are integrable. Functions with finitely many discontinuities are integrable. Composition of continuous and integrable is integrable.
• Lecture 9: Properties of the integral: sums, products, inequalities. Change of variables formula for Riemann-Stieljes integral.
• Lecture 10: Relationship between integration and differentiation. Fundamental theorem of calculus. Integration by parts. Change of variables formula for Riemann integral.
• Lecture 11: Relation between Riemann-Stiltjes integral and Riemann integral. Integration with respect to step functions. The delta "function".
• Lecture 12: Sequences and series of functions. Pointwise limits do not behave well with respect to continuity, derivatives, integration. Uniform convergence and examples. Cauchy criterion.
• Lecture 13: Uniform limit of continuous functions is continuous. Supremum norm. The space of continuous bounded functions in complete.
• Lecture 14: Uniform continuity and integration. Uniform continuity and differentiation.
• Lecture 15: Series and uniform convergence. The Weierstrass M-test. When can series be integrated term by term and differentiated term by term.
• Lecture 16: Example of a function which is continuous but nowhere differentiable. When does a sequence posses an uniformly convergent subsequence. Equicontinuity.
• Lecture 17: Connection between equicontinuity and uniform convergence. Arzela-Ascoli theorem.
• Lecture 18: Proof that every continuous function over a compact interval can be uniformly approximated by polynomials.
• Lecture 19: Algebras of functions. Examples: polynomials in several variables, Fourier series, separation of variables. Algebras which separate points and vanish nowehere. Uniform closure.
• Lecture 20: Algebras which sepate points and vanishing nowehere: constructing functions with prescribed values at two points. Proof of real version of Stone-Weierstrass.
• Lecture 21: Power series. Radius of convergence. Uniform convergence within compact intervals. Power series can be differentiated term by term. Continuity across endpoints/Abel's theorem.
• Lecture 22: Applications of power series: the exponential function. Definition and properties.
• Lecture 23: Applications of power series: sine and cosine. The number pi defined. Properties of the trigonometric functions.
• Lecture 24: Trigonometric polynomials. Fourier series. Fourier coefficients of a function. Example. Statement of Parseval's theorem.
• Lecture 25: L^2 norm and L^2 convergence. The complex exponentials form an orthogonal system for L^2. Least square properties of Fourier partial sums.
• Lecture 26: Approximation of integrable functions by continuous functions in L^2 norm, and eventually by trigonometric polynomials. L^2 convergence of Fourier series. Proof of Parseval.