What Are the Math Major Requirements at the University of California, Berkeley?

The University of California Berkeley is a public research university in Berkeley, California. It's the flagship university of the University of California System.

UC Berkeley offers a variety of degree programs, including a Bachelor of Arts in mathematics through its College of Letters & Sciences. Keep reading as we break down the math major requirements at Berkeley.

What Are the Math Major Requirements at UC Berkeley?

The Bachelor of Arts in mathematics degree at UC Berkeley has several math requirements, including lower-division courses, upper-division courses, and general education courses.

Lower Division Course Requirements

There are seven mathematics courses that are math major requirements at Berkeley. These courses cover calculus, linear algebra, and discrete mathematics.

Calculus and Multivariable Calculus

Students are required to take two calculus and calculus courses. Calculus is the mathematical study of continual change.

The first calculus course introduces differential and integral calculus of functions of one variable and presents applications and an introduction to transcendental functions.

The second course covers things like techniques and applications of integration, infinite sequences and series, ordinary differential equations, and oscillation and damping.

Once students have completed the calculus courses, they can move on to the multivariable calculus course.

This course focuses on parametric equations and polar coordinates, vectors in 2 and 3-dimensional Euclidean spaces, partial derivatives, multiple integrals, and theorems of Green, Gauss, and Stokes.

Linear Algebra and Differential Equations

Students who have completed their calculus courses are also able to move on to the courses in linear algebra. There are two options: Linear Algebra or Linear Algebra and Differential Equations.

The Linear Algebra course covers a variety of topics, including:

• algebra and geometry of vectors and matrices
• systems of linear equations and Gaussian elimination
• eigenvalues and eigenvectors
• Gram-Schmidt and least squares
• symmetric matrices and quadratic forms
• singular value decomposition and other factorizations

The Linear Algebra and Differential Equations course covers many of the same things but also covers differential equations, which are equations with one or more unknown functions.

Discrete Mathematics

The final lower-division requirement for math majors at Berkeley is a course in discrete mathematics. The Discrete Mathematics course studies logic, mathematical induction sets, relations, and functions. It also introduces:

• graphs
• elementary number theory
• combinatorics
• algebraic structures
• discrete probability theory

Upper Division Course Requirements

Once students have completed their lower division requirements, they'll be able to take the upper division requirements. There are four core upper-division courses required as well as electives.

The first core course is Introduction to Analysis. Students learn concepts like:

• the real number system
• sequences, limits, and continuous functions
• metric space
• uniform convergence
• interchange of limit operations
• infinite series
• mean value theorem and applications
• Reimann integral

Next is Abstract Linear Algebra, which will build on what was learned in the lower-division linear algebra course. It focuses on topics like:

• matrices, vector spaces, linear transformations, inner products, and determinants
• Eigenvectors
• QR factorization
• Quadratic forms and Rayleigh's principle
• Jordan canonical form
• linear functionals

The third required core mathematics major course is Introduction to Abstract Algebra. It covers a wide range of concepts, including:

• sets and relations
• integers, congruences, and the Fundamental Theorem of Arithmetic
• groups and their factor groups
• commutative rings, ideals, and quotient fields
• polynomials
• Euclidean algorithm and unique factorizations
• Fundamental Theorem of Algebra
• fields and field extensions

The final course needed for the math major is Introduction to Complex Analysis, which focuses on analytic functions of a complex variable, Cauchy's integral theorem, power series, Laurent series, singularities of analytic functions, and the residue theorem with application to definite integrals.

Upper Division Elective Courses

The math major also requires students to complete two semi-electives, or electives chosen from a set list. Students must choose two courses from three focus areas: computing, geometry, and logic and functions.

Computing

There are two course options in the computing category. Students can choose Programming for Mathematical Applications or Numerical Analysis.

Programming for Mathematical Applications offers an introduction to computer programming with a focus on the solution of mathematical and scientific problems.

Numerical Analysis, meanwhile, covers programming for numerical calculations, round-off error, approximation and interpolation, numerical quadrature, and solution of ordinary differential equations.

Geometry

There are five course options available in the geometry area.

Groups and Geometries covers isometries of Euclidean space, the Platonic solids and their symmetries, crystallographic groups, projective geometry, and hyperbolic geometry

In Metric Differential Geometry, students learn Frenet formulas, isoperimetric inequality, local theory of surfaces in Euclidean space, fundamental forms, Gaussian and mean curvature, isometries, geodesics, parallelism, and the Gauss-Bonnet-Von Dyck Theorem.

The Elementary Differential Topology course goes over manifolds in n-dimensional Euclidean space and smooth maps, Sard's Theorem, classification of compact one-manifolds, transversality, and intersection modulo 2.

Elementary Algebraic Topology covers the topology of one and two-dimensional spaces, including manifolds and triangulation, classification of surfaces, Euler characteristics, and fundamental groups.

Elementary Algebraic Geometry provides an introduction to basic commutative algebra, algebraic geometry, and computational techniques.

Logic and Functions

Only two courses are available in the area of logic and functions: Mathematical Logic and Introduction to the Theory of Sets.

Mathematical Logic teaches students sentential and quantificational logic, formal grammar, semantical interpretation, formal deduction, and their interrelation, as well as applications to formalized mathematical theories.

Theory of Sets covers multiple topics, including set-theoretical paradoxes, sets, relations, functions, order, and well-order, proof by transfinite induction, and Cardinal and ordinal numbers and their arithmetic.

General Education Requirements

General education courses are courses that all bachelor's students must take to earn their degree. UC Berkeley has only four requirements for general courses.

The first is an entry-level writing requirement, which can be completed through an English composition course or by passing the university's writing assessment.

Next, students must take an American history course. There are 15 courses students can choose from, covering areas like African American history, American foreign policy, American religious history, and immigration in U.S. History.

Students must also take a course in American Institutions. There are two options here: Immigrants and Immigration as U.S. History and Introduction to American Politics.

Students who take the Immigrants and Immigration as U.S. History will fulfill both requirements.

Finally, all students are required to take the American Cultures course, which centers on the study of race, ethnicity, and culture in the United States.

The College of Letters & Science also has a few requirements, including a quantitative reasoning course, a foreign language course, and a course in reading and comprehension.

Complete a Math Major at UC Berkeley

The math major requirements at Berkeley include courses in calculus, linear algebra, discrete mathematics, analysis, and complex algebra. For more information about the math major at UC Berkeley, contact Learn.org.