Instructor Dragos Oprea
Lectures: MWF, 11pm-11:50pm, AP&M 2-301.
Course
Assistants		 William Wood
  • Discussion: Tuesday 4-4:50AM, APM B-412
  • Office: APM 5-412
  • Office Hours: Tuesday, Wednesday, Friday 1-2.
  • Email: wfwood at ucsd dot edu.
Course
Content		 Differentiation. Riemann integral. Sequences and series of functions. Special functions. Fourier series. This corresponds to chapters 5-8 in Rudin's book.
Prerequisities: Math 140A or permission of instructor. Students will not receive credit for both Math 140 and Math 142.
Grade
Breakdown 		The grade is computed as the following weighed average:
  • Homework 20%, Midterm I 20%, Midterm II 20%, Final Exam 40%.
Textbook:W. Rudin, Principles of Mathematical Analysis, Third Edition.
ReadingsReading 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 Homework problems will be assigned on the course homework page. There will be 7 problem sets, due certain Wednesdays at 4:30PM in the TA's mailbox. 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.
Midterm
Exams		There will be two midterm exams given on April 24 and May 22. There will be no makeup exams.
Final
Exam		 The final examination will be held on Friday, June 14, 11:30-2:30. 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.
Announcements and Dates
  • Monday, April 1: First lecture
  • Wednesday, April 24: Midterm I
  • Wednesday, May 22: Midterm II
  • Monday, May 27: Memorial Day. No class
  • Friday, June 7: Last Lecture
  • Friday, June 14: FINAL EXAM, 11:30AM-2:30PM pm
Exams
Lecture Summaries
  • Lecture 1: Definition of the derivative. Differentiable functions are continuous. Rules for differentiation.
  • Lecture 2: Rules for differentiation. Local minima and local maxima. Critical points. The generalized mean value theorems.
  • Lecture 3: More on mean value theorem. Monotonic functions and the derivative. Intermediate value property. Darboux functions.
  • Lecture 4: The derivative is a Darboux function. Discontinuities of Darboux functions. L'Hopital rule. Applications of L'Hopital's rule.
  • Lecture 5: Taylor's theorem. Smooth functions, functions of class C^k, real analytic functions. Vector valued functions. Mean value theorem for vector valued functions.
  • Lecture 6: Integrability. Definition of Riemann integral and Riemann-Stiltjes integral. Partitions. Upper and lower sums, upper and lower integrals.
  • Lecture 7: Riemann's criterion for integrability. Refinements of partitions and effects on upper and lower sums. Integrability and continuity. Continuous functions are integrable.
  • Lecture 8: More on integrability and continuity: discontinuous functions at finitely many points are integrable. Composition of a continuous function with an integrable function is integrable. Properties of the integral.
  • Lecture 9: Properties of the integral: sums and products. Change of variables. Integration with respect to step functions. The delta "function".
  • Lecture 10: Series are particular cases of Riemann-Stiljes integral. Relation between Riemann-Stiltjes integral and Riemann integral.
  • Lecture 11: Relationship between integration and differentiation. Fundamnetal theorem of calculus. Integration by parts. Vector valued functions.
  • Lecture 12: Sequences and series of functions. Pointwise convergence. Pointwise limits do not behave well with respect to continuity, derivatives, integration.
  • Lecture 13: Uniform convergence of sequences and series. Examples. Cauchy criterion. Weierstrass M-test.
  • Lecture 14: Uniform limit of continuous functions is continuous. The space of continuous bounded functions in complete.
  • Lecture 15: Uniform convergence and integration. Term by term integration of series of functions.
  • Lecture 16: Uniform convergence and differentiation. Example of a function which is continuous but nowhere differentiable.
  • Lecture 17: Compactness in spaces of functions. Equicontinuous families. Statement of Arzela-Ascoli.
  • Lecture 18: Connection between equicontinuity and uniform convergence. Proof of Arzela-Ascoli.
  • Lecture 19: Proof that every continuous function over a compact interval can be uniformly approximated by polynomials.
  • Lecture 20: Apporximating functions of several variables by polynomials, Fourier series, separation of variables. Algebras of functions, algebras which separate points, self-adjoint algebras.
  • Lecture 21: Stone-Weierstrass theorem and its proof. Real and complex version.
  • Lecture 22: Power series and analytic functions. Power series can be differentiated term by term. Abel's theorem.
  • Lecture 23: Exponential function. Definition and properties.
  • Lecture 24: Trigonometric functions in terms of complex exponential. Definition and properties.
  • Lecture 25: Trigonometric polynomials. Approximation of continuous periodic functions by trigonometric polynomials. Fourier coefficients.
  • Lecture 26: Generalized Fourier coefficients. Least square properties of Fourier partial sums. Bessel's inequality. L^2 norm and L^2 Hermitian products.
  • Lecture 27: Approximation of integrable functions by trigonometric polynomials in L^2 norm. Fourier series converge to the function in L^2 sense. Parseval's theorem.