Mathematics
201718 General Catalog
4111 McHenry
(831) 4592969
http://www.math.ucsc.edu
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 ways of applying mathematics to their fields. A strong mathematics background is vital to the advanced study of many disciplines including the physical and biological sciences, engineering and 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 3 should consider taking Mathematics 2 or its equivalent prior to entering UCSC.

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

Upperdivision courses, with numbers in the range 100199, 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, vector 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 problemsolving, 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 including 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, collaborate intellectually and creatively in diverse contexts, and 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 realworld problems that are amenable to mathematical analysis, and formulate mathematical models of such problems;

apply mathematical methodologies to openended, realworld problems;

recognize connections between different branches of mathematics; and

recognize and appreciate the connections between theory and applications.
Courses: 19AB, 20AB, 21, 22, 23AB, 24, 100, 101, 103AB, 106, 107, 114, 115, 116, 134, 145, 148.
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 realworld problems to nonmathematicians.
Courses: 100, 101, 105AB, 111AB, 188, 189, 194, 195.
Subjectspecific knowledge
Students must demonstrate mastery in the three basic areas of mathematics: algebra, analysis, and topology/geometry on a basic level in lowerdivision courses and at an advanced level in upperdivision 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 retailtraders, 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 righthand 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 righthand pole, the algebraic, to the lefthand 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 realworld 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 farreaching consequences of systems of measurement; topology addresses questions pertaining to shape and global structure. NonEuclidean 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 wellrounded, indepth 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 realworld 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 indepth 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 subjectspecific areas, and courses are indicated in the following tables of lower and upperdivision mathematics courses.
LowerDivision 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 
UpperDivision 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 105AB  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 NonEuclidean 
√ 
√ 
√ 
√ 

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
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 Mathematics Department website 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 with requirements, course curriculum, and departmental policy.
Enrollment Requirements
Students who plan to take a mathematics course at UCSC must first demonstrate sufficient preparation for that course by completing the mathematics placement process, 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.
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 assesses 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. Students intending to take one or more mathematics courses at UCSC should begin the placement process as early as possible to fully benefit from the process.
Students completing placement by assessing using ALEKS PPL should familiarize themselves with the assessment instructions and guidelines, course eligibility cutoffs, and score posting schedule.
All students should review the requirements for their major or intended major and possibly consult with the department sponsoring their major (or expected major) before deciding which mathematics to take.
College Board Advanced Placement Calculus Examinations
Students completing placement requirements by using their scores from the College Board Advanced Placement Calculus Exam should refer to the table below for assistance in deciding which course to enroll.
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, Vector Calculus.
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, Vector Calculus

Mathematics 23B, Vector 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 mathematics adviser 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 lowerdivision and upperdivision 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 upperdivision 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 NonEuclidean, 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 20AB or 19AB, 21, 23A 
2nd year 
Mathematics 23B, 24, 100, 103A, 110 or 128A 
3rd year 
Mathematics 105AB, 111AB, 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, 19AB, 21 
2nd year 
Mathematics 23AB, 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 148/L, Numerical Analysis or Mathematics 145, Introductory Chaos Theory, or Applied Mathematics and Statistics 114, Introduction to Dynamical Systems, Applied Mathematics and Statistics 147, Computational Methods and Applications;
 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 107, 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 upperdivision 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 
19AB, 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 (K12) 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 NonEuclidean;

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 singlesubject 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 teachingcredential 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 19AB, 21, 23A 
2nd year 
Mathematics 23B, 100; 103A, Applied Mathematics and Statistics 5 
3rd year 
Mathematics 101, 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, 19AB, 21 
2nd year 
Mathematics 23AB, 100; Applied Mathematics and Statistics 5 
Disciplinary Communication (DC) Requirement
Students of every major must satisfy that major’s upperdivision 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 19AB, 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 secondlevel 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 firstlevel pass on at least one of the three written preliminary examinations and a secondlevel 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 firstlevel pass;

satisfy the foreignlanguage 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 place into the 400 mathematics tier are strongly encouraged to take the 19AB sequence, which is required for most upperdivision mathematics courses. Laboratory sections are mandatory. 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 mandatory.
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 mandatory.
Mathematics 20A and 20B, Honors Calculus, are recommended for prospective math majors or any well prepared student interested in exploring a rigorous development of single variable calculus with conceptual understanding and problem solving. Laboratory sections are mandatory.
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 100 and 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 23AB and not 22. Laboratory sections are mandatory.
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 mandatory.
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 upperdivision courses and partially fulfills the Disciplinary Communication (DC) requirement.
Mathematics 200+, Graduatelevel 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.
Revised: 09/01/17