2015-16 General Catalog

4111 McHenry
(831) 459-2969
http://www.math.ucsc.edu

Faculty | Course Descriptions

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Program Description

Mathematics is both a fundamental discipline and an essential tool for students of biology, chemistry, computer engineering, computer science, Earth sciences, economics, electrical engineering, information systems management, physics, and psychology. Researchers in all these areas are constantly developing new and cutting-edge ways of applying mathematics to their fields. A strong mathematics background is vital to the advanced study of the physical and biological sciences and plays an integral role in studying the social sciences.

The UCSC mathematics program offers a wide variety of undergraduate mathematics courses:

• Students interested in studying mathematics are strongly encouraged to take algebra, geometry, and trigonometry before entering the university. Students needing mathematics courses for their intended major are encouraged to take the mathematics assessment as early as possible. Failure to begin the calculus series upon arrival at UCSC could delay progress in some majors. Students concerned about their ability to place into courses above Mathematics 2 or Mathematics 3 should consider taking these courses or their equivalent prior to entering UCSC.

• Lower-division courses with numbers in the range 11A-B through 24 (calculus, linear algebra, multivariable calculus, and differential equations) prepare students for further study in mathematics, the physical and biological sciences, or quantitative areas of the social sciences. Science majors take a combination of these courses as part of their undergraduate studies.

• Upper-division courses, with numbers in the range 100-199, are intended for majors in mathematics and closely related disciplines. Some of these courses provide students with a solid foundation in key areas of mathematics such as algebra, analysis, geometry, and number theory, whereas others introduce students to more specialized areas of mathematics. Calculus, linear algebra, multivariable calculus, and proof and problem solving are prerequisite to most of these advanced courses.

Within the major, there are three concentrations leading to the bachelor of arts (B.A.) degree: pure mathematics, computational mathematics, and mathematics education. These programs are designed to give students a strong background for graduate study, for work in industry or government, or for teaching. Each concentration requires nine or ten courses, one of which must be a senior thesis or senior seminar. Please read the pure mathematics, computational mathematics, and mathematics education program descriptions below for specific information about course requirements. A minor in mathematics is also offered.

The mathematics program provides an excellent liberal arts background from which to pursue a variety of career opportunities. UCSC graduates with degrees in mathematics hold teaching posts at all levels, as well as positions in law, government, civil service, insurance, software development, business, banking, actuarial science, forensics, and other professions where skills in logic, numerical analysis, and computing are required. In particular, students of mathematics are trained in the art of problem-solving, an essential skill in all professions.

Program Learning Outcomes

Learning outcomes summarize the most important knowledge, skills, abilities and attitudes that students are expected to develop over the course of their studies. The program learning outcomes clearly communicate the faculty’s expectations to students, provide a framework for faculty evaluation of the curriculum based on empirical data, and help improve and measure the impact of implemented changes.

Mathematics Undergraduate Student Learning Objectives

The mathematics program promotes mathematical skills and knowledge for their intrinsic beauty, effectiveness in developing proficiency in analytical reasoning, and utility in modeling and solving real world problems. To responsibly live within and participate in the transformation of a rapidly changing, complex, and interdependent society, students must develop and unceasingly exercise their analytical abilities. Students who have learned to logically question assertions, recognize patterns, and distinguish the essential and irrelevant aspects of problems can think deeply and precisely, nurture the products of their imagination to fruition in reality, and share their ideas and insights while seeking and benefiting from the knowledge and insights of others.

Students majoring in mathematics attain proficiency in:

Critical thinking. The ability to identify, reflect upon, evaluate, integrate, and apply different types of information and knowledge to form independent judgments. Analytical and logical thinking and the habit of drawing conclusions based on quantitative information.

Problem solving. The ability to assess and interpret complex situations, choose among several potentially appropriate mathematical methods of solution, persist in the face of difficulty, and present full and cogent solutions that include appropriate justification for their reasoning.

Effective communication. The ability to communicate and interact effectively with different audiences, developing their ability to collaborate intellectually and creatively in diverse contexts, and to appreciate ambiguity and nuance, while emphasizing the importance of clarity and precision in communication and reasoning.

Students acquire and enhance these abilities in mathematical contexts, but the acquired habits of rigorous thought and creative problem solving are invaluable in all aspects of life. These skills are acquired through experience in the context of studying specific mathematical topics and exploring problems chosen to challenge students’ abilities, spurring them on to acquire new techniques and abandon familiar but restrictive habits of thought. The overarching objectives can be realized in terms of more focused, appraisable objectives specific to mathematics as follows:

Critical Thinking

Students will:

• understand the basic rules of logic, including the role of axioms or assumptions;

• appreciate the role of mathematical proof in formal deductive reasoning;

• be able to distinguish a coherent argument from a fallacious one, both in mathematical reasoning and in everyday life;

• understand and be able to articulate the differences between inductive and deductive reasoning;

• proficiently construct logical arguments and rigorous proofs; and

• formulate conjectures by abstracting general principles from examples.

Courses: 20AB, 100, 101, 105AB, 110, 111AB, 117, 118, 160, 161.

Problem Solving

Students will be able to:

• formulate and solve abstract mathematical problems;

• recognize real-world problems that are amenable to mathematical analysis, and formulate mathematical models of such problems;

• apply mathematical methodologies to open-ended, real-world problems;

• recognize connections between different branches of mathematics; and

• recognize and appreciate the connections between theory and applications.

Courses: 19AB, 20AB, 22, 23AB, 24, 100, 101, 103AB, 106, 107, 114, 115, 116, 134, 145.

Effective communication

Students will be able to:

• present mathematics clearly and precisely to an audience of peers and faculty;

• appreciate the role of mathematical proof as a means of conveying mathematical knowledge;

• understand the differences between proofs and other less formal arguments;

• make vague ideas precise by formulating them in mathematical language;

• describe mathematical ideas from multiple perspectives; and

• explain fundamental mathematical concepts or analyses of real-world problems to non-mathematicians.

Courses: 100, 101, 105AB, 111AB, 188, 189, 194, 195.

Subject-specific knowledge

Students must demonstrate mastery in the three basic areas of mathematics: algebra, analysis, and topology/geometry on a basic level in lower-division courses and at an advanced level in upper-division courses.

Algebra, number theory, and combinatorics

"We must endeavor to persuade those who are to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study until they see the nature of numbers with the mind only; not like merchants or retail-traders, with a view to buying or selling, but for the sake of their military applications and the benefit of the soul. . . I must add how charming the science is, and in how many ways it conduces to our desired end, if pursued in the spirit of a philosopher, and not of a shopkeeper!"—Plato, The Republic

Abstract algebra involves the study of algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. Linear algebra is a crucial subfield of algebra, both as an introduction to abstract algebraic structures and as a body of advanced results of immense importance in diverse areas of application. Number theorists study properties of the integers, as well as those of mathematical objects constructed from or generalizing the integers. Combinatorics involves finite or countable discrete structures, such as abstract graphs.

Courses: Math 21, 100, 110, 111AB, 114, 115, 116, 117, 118, 120, 134.

Calculus and analysis

Although the nature of is in no way altered when they appear. . . on the right-hand side of the differential equation, nonetheless their role and the character of the equation are thereby altered. . . They are brought into the world unilaterally, shadow figures lacking the body which cast them ...The initiative is thus shifted from the right-hand pole, the algebraic, to the left-hand one, the symbolic.—Karl Marx, On the Differential

Analysis extends and refines calculus; it encompasses differentiation, integration, measure, limits, infinite series, and analytic functions, primarily in the context of real and complex number systems. In much of analysis, the emphasis is not on finding explicit solutions to specific problems, but rather on determining which problems can be solved and what general properties solutions may share. Ordinary and partial differential equations play a central role in analysis, and are widely used in modeling real-world systems.

Courses: Math 19AB, 20AB, 23AB, 24, 100, 103AB, 105AB, 106, 107, 140, 145, 148.

Geometry and topology

"My noble friend, geometry will draw the soul towards truth, and create the spirit of philosophy . . . nothing should be more sternly laid down than that the inhabitants of your fair city should by all means learn geometry. Moreover, the science has significant indirect effects...in all departments of knowledge, as experience proves, anyone who has studied geometry enjoys infinitely quicker understanding than one who has not."—Plato, The Republic

Geometry explores the implementation and far-reaching consequences of systems of measurement; topology addresses questions pertaining to shape and global structure. Non-Euclidean geometry, differential geometry (the extension of calculus to mapping of curves, surfaces, and their generalizations), and algebraic geometry generalize key results and techniques from Euclidean geometry to both familiar and exotic settings. In algebraic and differential topology, techniques from diverse areas of mathematics are used to infer information about the shapes and related properties of spaces.

Courses: Math 23AB, 121AB, 124, 128AB, 129, 130.

This subdivision of mathematics is not sharp: the areas of overlap between the three main areas of mathematics are of great interest and importance. For example, analytic number theory is the study of the integers by means of tools from real and complex analysis, while differential geometry focuses on the interplay between analysis and geometry. The importance of all three areas, the influence of each on the others, and the insight to be gained by considering one area from perspectives commonly associated to another one are all emphasized in the mathematics curriculum—as can be seen in the course matrix, many courses involve material from multiple areas.

The Mathematics Department offers three tracks within the mathematics major:

Pure Mathematics

Students in the Pure Mathematics track often go on to graduate study in mathematics; the pathway emphasizes the importance of a well-rounded, in-depth mathematical education, and includes advanced coursework in algebra, analysis, and geometry.

Computational Mathematics

Students in the Computational Mathematics track explore applications of mathematics in other fields and gain experience in mathematical modeling of real-world phenomena using ordinary and partial differential equations, approximation and optimization techniques, linear programming, or game theory.

Mathematics Education

Students in the Mathematics Education track prepare for a career in K–12 mathematics education; students acquire in-depth knowledge of subjects covered at an introductory level in the classroom, including number theory, classical geometry, and the history of mathematics, and gain experience in teaching mathematics in an accessible and intuitive, but precise, manner.

Curriculum matrix

All of the key objectives are addressed to some extent in all courses. For example, the ability to formulate precise mathematical statements and to reason logically are essential skills that are progressively developed throughout the curriculum. However, some skills are more heavily emphasized and utilized in some courses than in others. Some courses are specifically intended to help students move to a new level of proficiency with a particular portfolio of skills, while others are accessible only to students who have already reached a given level; the latter courses make heavy use of particular skills, and thus enhance and reinforce the student’s mastery of them, but the skills themselves are not the primary focus of such courses. Some connections between the key objectives, main subject-specific areas, and courses are indicated in the following tables of lower- and upper-division mathematics courses.

	Lower-Division Courses	Critical Thinking	Problem Solving	Communication	Algebra	Analysis	Geometry
MATH 2	College Algebra for Calculus		√		√
MATH 3	Precalculus		√		√	●
MATH 4	Mathematics of Choice and Argument	●	√		√
MATH 11AB	Calculus with Applications		√			√
MATH 19AB*	Calculus for Science, Engineering, and Mathematics	●	√			√
MATH 20AB*	Honors Calculus	●	√	●		√
MATH 21	Linear Algebra		√		√
MATH 22	Introduction to Calculus of Several Variables		√			√
MATH 23AB	Vector Calculus	●	√			√	●
MATH 24	Differential Equations		√		●	√
Boldface indicates that a course that is required for at least one of the major pathways.
Boldface* indicates a course that is one of two or three options for satisfying a requirement of one of the major pathways.
√ indicates a course in which the skill or subject is directly addressed, with substantial instruction and assessment emphasis.
● indicates a course in which the skill or subject plays an important role, but is not the primary focus

	Upper-Division Courses	Critical Thinking	Problem Solving	Communication	Algebra	Analysis	Geometry
MATH 100	Introduction to Proof and Problem Solving	√	●	√	●	●
MATH 101	Mathematical Problem Solving	√	√	√	●	●	●
MATH 103A*B	Complex Analysis	√		√		√
MATH 105ABC	Real Analysis	√	●	√		√
MATH 106*	Systems of Ordinary Differential Equations	●	√	●	●	√	●
MATH 107*	Partial Differential Equations		√			√
MATH 110	Introduction to Number Theory	√	●	√	√
MATH 111AB	Algebra	√	●	√	√
MATH 114	Introduction to Financial Mathematics	●	√	●	√	●
MATH 115	Graph Theory	●	√	●	√
MATH 116	Combinatorics	●	√	●	√
MATH 117	Advanced Linear Algebra	√	√	●	√
MATH 118	Advanced Number Theory	√	√	●	√	●
MATH 120	Coding Theory	●	√	●	√
MATH 121A*B	Differential Geometry	√	●	●		√	√
MATH 124	Introduction to Topology	√	●	●	√	√
MATH 128A	Classical Geometry: Euclidean and Non-Euclidean	√	√	√			√
MATH 128B	Classical Geometry: Projective	√	√	√			√
MATH 129	Algebraic Geometry	√	●	●	√		√
MATH 130	Celestial Mechanics	●	√	●		√	√
MATH 134	Cryptography	√	√	●	√
MATH 140	Industrial Mechanics	●	√	√		√
MATH 145*	Introductory Chaos Theory	●	√	√		√	√
MATH 148	Numerical Analysis	●	√	●	√	√
MATH 160	Mathematical Logic I	√	●	√
MATH 161	Mathematical Logic II	√	●	√
MATH 181	History of Mathematics	√	●	√	●	√	●
MATH 188	Supervised Teaching		√	√
MATH 189	ACE Program Service Learning		●	√	●	●
MATH 194*	Senior Seminar	√	●	√
MATH 195*	Senior Thesis	√	●	√
MATH 199	Tutorial	●	●	●	●	●	●
Boldface indicates that a course that is required for at least one of the major pathways.
Boldface* indicates a course that is one of two or three options for satisfying a requirement of one of the major pathways.
√ indicates a course in which the skill or subject is directly addressed, with substantial instruction and assessment emphasis.
● indicates a course in which the skill or subject plays an important role, but is not the primary focus

Academic Advising

Academic advising is available at the Undergraduate Affairs Office . The undergraduate adviser provides information about requirements, prerequisites, policies and procedures, learning support, scholarships, and special opportunities for undergraduate research. In addition, the adviser assists with the drafting of study plans, as well as certifying degrees and minors. Students are urged to stay informed and involved with their major, as well as to seek advice should problems arise.

The Undergraduate Affairs web site is a critical resource for students. Here you will find a link to the undergraduate program; the materials at that link constitute the undergraduate handbook. Students should visit this first to seek answers to their questions, because it hosts a wealth of information. Each student in the major is encouraged to regularly review the materials posted to stay current about requirements, course curriculum, and departmental policy.

Requirements

Students who plan to take a mathematics course at UCSC must demonstrate sufficient preparation by their score on either the mathematics placement examination (MPE), the College Entrance Examination Board Advanced Placement (AP) calculus examination, the International Baccalaureate Higher Level Mathematics Examination, or by passing the appropriate prerequisite course.

Students who have passed course 2 may enroll in course 3. Students who have passed course 3 may enroll in course 11A or 19A. Students who have passed a precalculus course at a college or university may enroll in course 11A or 19A, but they must first verify eligibility of the course and course completion with the mathematics adviser (mathadvising@ucsc.edu).

Transfer students interested in a mathematics or science major should take courses equivalent to course 19A, 19B, and 21 before enrolling at UCSC. Courses equivalent to 23A, 23B, and 24 are also recommended before transferring to UCSC.

UCSC Mathematics Placement

The mathematics placement process is designed to assess student readiness for their first UCSC mathematics class. Students whose areas of study require precalculus or calculus courses are strongly advised to complete placement and any required courses early in their academic careers. Information on registering and taking the placement examination is provided at: http://undergrad.pbsci.ucsc.edu/enrollment/math/math-placement/index.html. Course eligibility cut-offs are listed at: http://undergrad.pbsci.ucsc.edu/enrollment/math/math-placement/mp-verify-your-placement.html.

All students are advised to consult with the department sponsoring their major before deciding which mathematics pathway to take. Many majors in science and engineering require MATH 19A/B.

College Board Advanced Placement Calculus Examinations

Students who have received 4 credits for the College Entrance Examination Board Advanced Placement (AP) calculus examination should normally enroll in course 19B, and those with 8 credits should normally enroll in course 23A. However, students who received a score of 3 on the calculus AB or BC AP examination, should enroll in course 19A or 19B, respectively, to improve their knowledge of calculus before continuing their studies. Students who wish to challenge themselves, and who received a score of 4 or 5 on the AB or a score of 3, 4, or 5 on the BC examination may choose course 20A, Honors Calculus. Non-mathematics majors should consult their major departments before enrolling in a mathematics course.

If your AP AB score is	May enroll in this course
3	Mathematics 11A or 19A
4 or 5	Mathematics 20A or 11B or 19B

If your AP BC score is	May enroll in this course
3	Mathematics 11B or 19B or 20A
4 or 5	Mathematics 20A or 22 or 23A

International Baccalaureate Higher Level Examination in Mathematics

Students who have received a score of 5, 6, or 7 on the International Baccalaureate (IB) Higher Level Examination in Mathematics may enroll in course 20A, Honors Calculus; 22, Calculus of Several Variables; or 23A, Multivariable Calculus. Non-mathematics majors should consult their major departments before enrolling in a mathematics course.

Declaration of the Mathematics Major

Admission to the mathematics major (all concentrations) is contingent on students successfully passing the following introductory courses or their equivalents:

• Mathematics 19A, Calculus for Science, Engineering, and Mathematics

• Mathematics 19B, Calculus for Science, Engineering, and Mathematics

• Mathematics 21, Linear Algebra

• Mathematics 23A, Multivariable Calculus

• Mathematics 23B, Multivariable Calculus

• Mathematics 100, Introduction to Proof and Problem Solving

Students may only declare once they have passed all introductory courses or their equivalent courses with a grade of C or better. Students who receive two grades of NP, C-, D+, D, D-, or F in the introductory courses are not eligible to declare in the major. Students who are not eligible to declare may submit an appeal to the department's undergraduate vice chair. The advising office will subsequently notify the student, and their college, of the decision, no later than 15 business days after the submission of the appeal.

It should be emphasized that the nature of mathematics changes dramatically between lower-division and upper-division courses. Students often find that the material becomes far more abstract and theoretical. In addition, the role of computation in assignments diminishes and a greater weight is placed on deductive reasoning and the integral role of mathematical proofs. The Mathematics Department recommends that students interested in a mathematics major enroll in Mathematics 100 as early as prerequisites allow in order to decide whether they are interested in upper-division mathematics courses. It is strongly recommended that only students who earn grades of B- or better in Mathematics 100 consider applying to the major in mathematics. Students with a grade less than B in Mathematics 100 are urged to take Mathematics 101.

Major Requirements

Pure Mathematics

This concentration is intended for students who desire a comprehensive understanding of mathematics, including those considering graduate studies in the natural sciences. Students are required to complete at least 11 courses (with laboratories, if appropriate).

Eight of these courses must be:

• Mathematics 24, Ordinary Differential Equations;

• Mathematics 100, Introduction to Proof and Problem Solving;

• Mathematics 103A, Complex Analysis;

• Mathematics 105A, Real Analysis;

• Mathematics 111A, Algebra;

• Mathematics 117, Advanced Linear Algebra;

• one of Mathematics 121A, Differential Geometry, Mathematics 124, Introduction to Topology, or Mathematics 128A, Classical Geometry: Euclidean and Non-Euclidean, or Mathematics 129, Algebraic Geometry;

• and either Mathematics 194, Senior Seminar, or Mathematics 195, Senior Thesis.

The remaining three courses are selected by the student from among any mathematics course numbered above 100 (excluding Mathematics 188 and Mathematics 189) and Applied Mathematics and Statistics (AMS) 100 or above. Only one of the three courses can be from the AMS series.

A typical program for a pure mathematics major might include the following:

1st year	Mathematics 20A-B or 19A-B, 21, 23A
2nd year	Mathematics 23B, 24, 100, 103A, 110 or 128A
3rd year	Mathematics 105A-B, 111A-B, 106
4th year	Mathematics 107, 117, 121A, 194 or 195

The first two years of a typical program for a pure mathematics major who begins mathematics studies with precalculus might include the following:

1st year	Mathematics 3, 19A-B, 21
2nd year	Mathematics 23A-B, 24, 100, 103A or 128A

Computational Mathematics

This concentration is intended to prepare students for technical careers in industry or government while providing a solid mathematical background. Students are required to complete a minimum of eight mathematics courses (with laboratories, if appropriate) as follows:

• Mathematics 24, Ordinary Differential Equations;

• Mathematics 100, Introduction to Proof and Problem Solving;

• Mathematics 103A, Complex Analysis, or Mathematics 105A, Real Analysis;

• Mathematics 106, Systems of Ordinary Differential Equations, or Mathematics 107, Partial Differential Equations;

• Mathematics 110, Introduction to Number Theory;

• Mathematics 111A, Algebra, or Mathematics 117, Advanced Linear Algebra;

• Mathematics 106, Systems of Ordinary Differential Equations, or Mathematics 145, Introductory Chaos Theory, or Applied Mathematics and Statistics 114, Introduction to Dynamical Systems;

• and either Mathematics 194, Senior Seminar, or Mathematics 195, Senior Thesis.

In addition, students must complete two courses selected from the following:

• Applied Mathematics and Statistics, 100 or above

• Biomolecular Engineering 110

• Computer Engineering 107, 108, 153, 177

• Computer Science 101, 102, 104A, 109, 112, 130, 132, 142

• Earth and Planetary Sciences 172

• Economics 113

• Electrical Engineering 103, 130, 135, 151, 154

• Physics 115

Some of these courses have prerequisites within their departments. Students are encouraged to plan their computational electives early, so that all prerequisites can be satisfied in a timely manner. Other upper-division courses with heavy emphasis on computational mathematics may occasionally be accepted with permission of the Mathematics Department.

Mathematics majors who wish to enroll in Computer Science 101 should contact the instructor to request a permission code.

A typical program for a computational mathematics major might include the following:

1st year	19A-B, 23A, CMPS 12A/L and 12B/M
2nd year	21, 23B, 24, 100, 110, CMPE 16
3rd year	103A, 105A, 145/L or AMS 147, CMPS 101
4th year	106, 111A, CMPS 109, 194

Mathematics Education

This concentration is intended to prepare students for teaching kindergarten through high school (K-12) mathematics. Students are required to complete the following 10 courses:

• Applied Mathematics and Statistics 5, Statistics

• Mathematics 100, Introduction to Proof and Problem Solving;

• either Mathematics 103A, Complex Analysis, or 105A, Real Analysis;

• Mathematics 110, Introduction to Number Theory;

• Mathematics 111A, Algebra;

• Mathematics 128A, Classical Geometry: Euclidean and Non-Euclidean;

• Applied Mathematics and Statistics 131, Introduction to Probability Theory;

• Mathematics 181, History of Math;

• Either Mathematics 188, Supervised Teaching Experience; or Education 50B, CalTeach 1: Mathematics, plus Education 100B, CalTeach 2: Mathematics

• and either Mathematics 194, Senior Seminar, or Mathematics 195, Senior Thesis.

UCSC students can pursue a degree in mathematics while preparing to teach at the secondary level. In California, students seeking a single-subject credential (for secondary teaching) in mathematics are required to take the CSET, a series of examinations that must be passed in order to enter a teaching-credential program (formerly The National Teachers Examination). Students who complete the mathematics education track, plus three additional specified courses, qualify for the California Single Subject Program, exempting themselves from the CSET. Both the Mathematics Department undergraduate adviser and the Education Department advising office have more information about the additional required courses for the Subject Matter Program.

A typical program for a mathematics education major might include the following:

1st year	Mathematics 19A-B, 21, 23A
2nd year	Mathematics 23B, 100; Applied Mathematics and Statistics 5
3rd year	Mathematics 101, 103A, 110, 181; Applied Mathematics and Statistics 131
4th year	Mathematics 111A, 128A, 188, 194

The first two years of a typical program for a mathematics education major who begins mathematics studies with precalculus might include the following:

1st year	Mathematics 3, 19A-B, 21
2nd year	Mathematics 23A-B, 100; Applied Mathematics and Statistics 5

Disciplinary Communication (DC) Requirement

Students of every major must satisfy that major’s upper-division Disciplinary Communication (DC) requirement. The DC requirement in mathematics is satisfied by Mathematics 100, Introduction to Proof and Problem Solving, and either Mathematics 194, Senior Seminar, or Mathematics 195, Senior Thesis.

Honors

Honors in the Mathematics Department are awarded to graduating students whose academic performance in the major demonstrates excellence at a GPA of 3.5 or above. Highest Honors are determined by a cumulative review of student performance in mathematics courses. They are awarded to students who excel in challenging courses and in their capstone projects.

Minor Requirements

The minor is intended for students who are interested in mathematics and want a strong mathematical foundation for studying in areas that rely heavily on analytical skills. Students are required to complete at least eight courses as follows:

• Mathematics 21, Linear Algebra;

• Mathematics 23A, Vector Calculus; and

• Mathematics 23B, Vector Calculus;

• Mathematics 100

• and any four mathematics courses numbered above 100.

No senior seminar or thesis is required.

A typical mathematics minor program for a physics major might be:

1st year	Mathematics 19A-B, 23A
2nd year	Mathematics 21, 23B, 100
3rd year	Mathematics 103A, 121A or 124
4th year	Mathematics 117, 145 or Applied Mathematics and Statistics 114

Combined Majors

Economics and Mathematics

The combined major in economics and mathematics is designed to meet the needs of undergraduate students who plan to pursue doctoral study in economics or business, or who wish to pursue a career as an actuary or other professional requiring a sophisticated understanding of economics and mathematics. The major combines the main undergraduate content of both economics and mathematics within a programmatic structure that joins the two disciplines. It provides a coursework combination required to prepare for a modern economics Ph.D. program, or for technically demanding professional careers. A full description can be found in the economics section of this catalog. The combined major, requiring fewer courses than a double major, is administered through the Economics Department.

Graduate Program

The Mathematics Department offers programs leading to the master of arts (M.A.) and doctor of philosophy (Ph.D.) degrees. Contact the Division of Graduate Studies for further information on the M.A. and Ph.D. programs, as well as on university application procedures.

M.A. Degree Requirements

Students are required to complete two of Mathematics 200, 201, 202, 203; two of Mathematics 204, 205, 206; one of Mathematics 208, 209, 210; and complete five additional courses in mathematics or a related subject by approval. In addition, students must do one of the following:

• obtain a second-level pass on one of three written preliminary examinations;

• write a master’s thesis.

Ph.D. Degree Requirements

All of the following are required:

• obtain a first-level pass on at least one of the three written preliminary examinations and a second-level pass on at least one other. Students must complete the full course sequence in the track associated with the preliminary examination in which they did not achieve a first-level pass;

• satisfy the foreign-language requirement;

• pass the oral qualifying examination;

• complete three quarters as a teaching assistant;

• complete six graduate courses in mathematics other than Mathematics 200, 201, 202, 204, 205, 206, 208, 209, and 210. No more than three courses may be independent study or thesis research courses;

• write a Ph.D. thesis.

Students admitted to the Ph.D. program may receive an M.A. degree en route to the Ph.D.

Course Information

Mathematics 2, College Algebra for Calculus, is designed for students who do not meet the requirements for admission to Mathematics 3, Precalculus, and who need comprehensive and careful preparation for calculus. Mathematics 2 emphasizes algebra, graphs, and functions.

Mathematics 3, Precalculus, is recommended for students who need some preparation in algebra and trigonometry prior to taking calculus. This course covers functions and their inverse, exponentials, logarithms, and trigonometry.

Mathematics 11A and 11B, Calculus with Applications, are intended for biology and Earth sciences majors. However, students in these majors who score 400 or more points on the mathematics placement examination are strongly encouraged to take the 19A-B sequence, which is required for most upper-division mathematics courses. Laboratory sections are strongly advised.

Mathematics 19A and 19B, Calculus for Science, Engineering, and Mathematics, are intended for chemistry, computer engineering, computer science, electrical engineering, information systems management, mathematics, and physics majors. Laboratory sections are strongly advised.

Mathematics 20A and 20B, Honors Calculus, are intended for students who would enjoy delving particularly deeply into the foundational and theoretical issues of calculus. Laboratory sections are strongly advised.

Mathematics 21, Linear Algebra, covers vector spaces, matrices, determinants, systems of linear equations, and eigenvalues. It is intended for students in the physical and biological and social sciences and is prerequisite to Mathematics 111A.

Mathematics 22, Introduction to Calculus of Several Variables, is intended for science students whose schedules do not permit a full and comprehensive two quarters of multivariable calculus. Students who intend to pursue further studies in mathematics must take Mathematics 23A-B and not 22. Laboratory sections are strongly advised.

Mathematics 23A and 23B, Vector Calculus, are intended for mathematics majors and minors and students in computer engineering, computer science, electrical engineering, information systems management, and physics majors which require more rigorous mathematical training. Laboratory sections are strongly advised.

Mathematics 100, Introduction to Proof and Problem Solving, is an introduction to the methodology of advanced mathematics, emphasizing proof techniques. Basic areas such as set theory and logic are introduced, together with extensive applications within mathematics. This course serves as a prerequisite for nearly all upper-division courses and partially fulfills the Disciplinary Communication (DC) requirement.

Mathematics 200+, Graduate-level courses. All graduate courses are open to undergraduates who have taken the recommended prerequisites; students should consult with the course instructor. Advanced undergraduates are strongly advised to take or audit graduate courses that interest them.

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Revised: 09/01/15