Summer 2018

Prerequisites: Three and one-half years of high school math, including trigonometry and analytic geometry, plus a satisfactory grade in one of the following: CEEB MAT test, an AP test, the UC/CSU math diagnostic test, or 32. Consult the mathematics department for details. Students with AP credit should consider choosing a course more advanced than 1A

Description: This sequence is intended for majors in engineering and the physical sciences. An introduction to differential and integral calculus of functions of one variable, with applications and an introduction to transcendental functions.

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Prerequisites: 1A

Description: Continuation of 1A. Techniques of integration; applications of integration. Infinite sequences and series. First-order ordinary differential equations. Second-order ordinary differential equations; oscillation and damping; series solutions of ordinary differential equations.

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Prerequisites: 1A

Description: Continuation of 1A. Techniques of integration; applications of integration. Infinite sequences and series. First-order ordinary differential equations. Second-order ordinary differential equations; oscillation and damping; series solutions of ordinary differential equations.

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Prerequisites: Continuation of 10A

Description: Elementary combinatorics and discrete probability theory. Introduction to graphs, matrix algebra, linear equations, difference equations, and differential equations.

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Prerequisites: Three years of high school math, including trigonometry, plus a satisfactory grade in one of the following: CEEB MAT test, an AP test, the UC/CSU math diagnostic exam, or 32. Consult the mathematics department for details

Description: This sequence is intended for majors in the life and social sciences. Calculus of one variable; derivatives, definite integrals and applications, maxima and minima, and applications of the exponential and logarithmic functions.

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Prerequisites: 16A

Description: Continuation of 16A. Application of integration of economics and life sciences. Differential equations. Functions of many variables. Partial derivatives, constrained and unconstrained optimization.

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Prerequisites: Mathematics 1B

Description: Parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and Stokes.

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Prerequisites: Mathematics 1B

Description: Parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and Stokes.

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Prerequisites: Mathematics 1B

Description: Parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and Stokes.

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Prerequisites: Mathematics 1B or equivalent

Description: Parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and Stokes.

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Prerequisites: 1B

Description: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product as spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous ordinary differential equations; first-order differential equations with constant coefficients. Fourier series and partial differential equations.

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Prerequisites: 1B

Description: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product as spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous ordinary differential equations; first-order differential equations with constant coefficients. Fourier series and partial differential equations.

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Prerequisites: 1B

Description: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product as spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous ordinary differential equations; first-order differential equations with constant coefficients. Fourier series and partial differential equations.

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Prerequisites: 1B

Description: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product as spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous ordinary differential equations; first-order differential equations with constant coefficients. Fourier series and partial differential equations.

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Prerequisites: Mathematical maturity appropriate to a sophomore math class. 1A-1B recommended

Description: Logic, mathematical induction sets, relations, and functions. Introduction to graphs, elementary number theory, combinatorics, algebraic structures, and discrete probability theory.

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Prerequisites: Mathematical maturity appropriate to a sophomore math class. 1A-1B recommended

Description: Logic, mathematical induction sets, relations, and functions. Introduction to graphs, elementary number theory, combinatorics, algebraic structures, and discrete probability theory.

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Prerequisites: 53 and 54

Description: The real number system. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.

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Prerequisites: 53 and 54

Description: The real number system. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.

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Prerequisites: 54 or a course with equivalent linear algebra content

Description: Matrices, vector spaces, linear transformations, inner products, determinants. Eigenvectors. QR factorization. Quadratic forms and Rayleigh's principle. Jordan canonical form, applications. Linear functionals.

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Prerequisites: 54 or a course with equivalent linear algebra content

Description: Matrices, vector spaces, linear transformations, inner products, determinants. Eigenvectors. QR factorization. Quadratic forms and Rayleigh's principle. Jordan canonical form, applications. Linear functionals.

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Prerequisites: 54 or a course with equivalent linear algebra content

Description: Sets and relations. The integers, congruences, and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.

Office: 783 Evans

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Required Text: John B. Fraleigh, "A first course in abstract algebra" 7th edition

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Prerequisites: 54 or a course with equivalent linear algebra content

Description: Sets and relations. The integers, congruences, and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.

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Prerequisites: 53 and 54

Description: Waves and diffusion, initial value problems for hyperbolic and parabolic equations, boundary value problems for elliptic equations, Green's functions, maximum principles, a priori bounds, Fourier transform.

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Prerequisites: 53 and 54

Description: Programming for numerical calculations, round-off error, approximation and interpolation, numerical quadrature, and solution of ordinary differential equations. Practice on the computer.

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Prerequisites: 104

Description: Analytic functions of a complex variable. Cauchy's integral theorem, power series, Laurent series, singularities of analytic functions, the residue theorem with application to definite integrals. Some additional topics such as conformal mapping.

Office: 903 Evans Hall

Office Hours: MTWTh 1pm-2pm

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Recommended Reading: Complex analysis, 4th edition Springer, Serge Lang.

Comments: The material in this class is both deep and very useful. We will study functions defined on the complex plane and taking values in the complex plane. This innocent looking replacement of $\mathbb R$ by $\mathbb C$ brings in many consequences. We will be learning about tools that are crucial both in mathematics itself as well as in many applications, like mechanics, hydrodynamics, the design of airfoils, etc.

Since all these ideas have to be mastered in eight weeks students should be ready to work hard from the very beginning. There is no chance to catch up with the material later on. The class is structured in such a way that everything that we do will play a role later on ( for us this may mean in the same or following weeks after new concepts are introduced).

A list of topics includes:

• Complex numbers
• Differentiability
• Power series
• Cauchy's theorem
• Winding numbers
• Laurent series
• Residues
• Evaluation of definite integrals
• Conformal mappings
• Harmonic functions

In the first hour or each 2 hours day we will do some theory, and in the second hour we will discuss problems that have been assigned previously. Class participation in this second hour will be an important part of the grade. Details about the grading policy follow:

Grading: Homework will be assigned in class and collected every Monday in class. Students will get credit for attempting complete solutions. This work will count for 30% of the class grade.

In class discussion of the homework will count for an extra 20% of the grade. Students should be prepared to discuss on the blackboard what they have done as homework the previous week.

We will have a midterm on the Thursday of the fourth week. It will take one hour and will count for 20% of the grade.

We will have a final exam on the last day of classes, it will take two hours and will count for 30% of the grade.

No one should be surprised if the problems in the midterm and/or final are very similar to problems that have been discussed (maybe assigned as homework) during the course of the eight weeks. A good way to prepare for these tests is to take the homework very seriously.

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