Summer 2019

Prerequisites: Three and one-half years of high school math, including trigonometry and analytic geometry. Students with high school exam credits (such as 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 or N1A

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 or N1A

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: Math 10A or N10A

Description: The sequence Math 10A, Math 10B is intended for majors in the life sciences. Elementary combinatorics and discrete and continuous probability theory. Representation of data, statistical models and testing. Sequences and applications of linear algebra.

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Prerequisites: Three years of high school math, including trigonometry

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: Mathematics 16A or N16A

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: Three years of high school mathematics

Description: Polynomial and rational functions, exponential and logarithmic functions, trigonometry and trigonometric functions. Complex numbers, fundamental theorem of algebra, mathematical induction, binomial theorem, series, and sequences.

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Prerequisites: Mathematics 1B or N1B

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.

Office: Evans 849

Office Hours: MTW 1-2pm

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Course Webpage: bCourses

Prerequisites: Mathematics 1B or N1B

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.

Office: Evans 849

Office Hours: MTW 1-2pm

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Course Webpage: bCourses

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, N1B, 10B, or N10B

Description: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product spaces. Eigenvalues and eigenvectors; orthogonality, symmetric matrices. Linear second-order differential equations; first-order systems with constant coefficients. Fourier series, application to partial differential equations.

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Prerequisites: 1B, N1B, 10B, or N10B

Description: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product spaces. Eigenvalues and eigenvectors; orthogonality, symmetric matrices. Linear second-order differential equations; first-order systems with constant coefficients. Fourier series, application to partial differential equations.

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Prerequisites: 1B, N1B, 10B, or N10B

Description: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product spaces. Eigenvalues and eigenvectors; orthogonality, symmetric matrices. Linear second-order differential equations; first-order systems with constant coefficients. Fourier series, application to partial differential equations.

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Prerequisites: 1B, N1B, 10B, or N10B

Description: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product spaces. Eigenvalues and eigenvectors; orthogonality, symmetric matrices. Linear second-order differential equations; first-order systems with constant coefficients. Fourier series, application to 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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Description: Elements of college algebra. Designed for students who do not meet the prerequisites for 32. Offered through the Student Learning Center.

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Description: Elements of college algebra. Designed for students who do not meet the prerequisites for 32. Offered through the Student Learning Center.

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Description: Elements of college algebra. Designed for students who do not meet the prerequisites for 32. Offered through the Student Learning Center.

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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.

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Prerequisites: 53 and 54

Description: Divisibility, congruences, numerical functions, theory of primes. Topics selected: Diophantine analysis, continued fractions, partitions, quadratic fields, asymptotic distributions, additive problems.

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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:   905 Evans Hall

Office Hours:  To Be Announced

Required Text:   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 ℝ  R by ℂ 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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Prerequisites: 104

Description: (This course is offered in Berlin as part of the "Mathematics in Berlin" Summer program.  Please see http://studyabroad.berkeley.edu/program/summerabroad/berlin for more information.)

Math 185 is a core upper-division Math course focused on a rigorous introduction to complex analysis. It traditionally focuses on analytic functions of a complex variable. Main topics include 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 mappings and various applications to other branches of mathematics as well as physics. Math 185 is often offered during the summer.

Math 185 will be taught using the traditional lecture format and teaching methods, using Visual Complex Functions, An Introduction with Phase Portraits, Birkhauser, by Elias Wegert, as the main textbook, and Complex Analysis, Springer-Verlag by T. W. Gamelin, as a supplementary text.

Notwithstanding the traditional lecture format, this version of Math 185 will be unusual in its unorthodox choice of material and its heavy reliance on visualization. In particular, both the main textbook and the lectures will often involve visual demonstrations using MATLAB and other software of phase portraits of functions, and ‘visual’ applications to various problems of mathematics and physics.

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