# Applied Mathematics B.S.

## Information and Policies

### Introduction

Applied mathematics is a field of research specializing in the development and application of analytical and numerical tools and techniques toward the solution of complex quantitative problems in science and engineering. A Bachelor of Science (B.S.) in Applied Mathematics prepares graduates for careers in the industry (usually in research and development) or academia (either in teaching and/or in research at university or governmental laboratories or agencies). It can be a terminal degree, or prepare students for graduate school in applied mathematics or related fields.

The applied mathematics major at UC Santa Cruz provides students with a holistic training in mathematical modeling. The core of the degree program includes six courses that introduce modern analytical (and semi-analytical) techniques as well as general aspects of scientific computing. Students gain further depth or breadth by taking three (or more) electives among a large list of upper-division or graduate courses offered by applied mathematics faculty or in related subject areas. Finally, students learn to apply the techniques and tools learned through two capstone courses that focus on modeling “real- life” quantitative problems in science and engineering. In these courses, students also learn additional skills such as mathematical abstraction, critical thinking, and disciplinary communication.

The major has been designed to allow students to easily pursue an additional minor or major in an area of applications of mathematics (such as physics, astrophysics, Earth sciences, computer sciences, computer engineering, electrical engineering), and/or to pursue the 4+1 pathway into the Master of Science (M.S.) degree program in Scientific Computing and Applied Mathematics also offered by the Department of Applied Mathematics.

### Academic Advising for the Program

The Baskin School of Engineering (BSOE) Undergraduate Advising Office is located in the Baskin Engineering Building, Room 225. It can be contacted by email at bsoeadvising@ucsc.edu or at the __Undergraduate Advising website__.

Transfer students to the program should consult the __Transfer Students section __of the Baskin Engineering Undergraduate Affairs page.

### Getting Started in the Major

It is recommended that high school students intending to apply to this major have completed four years of mathematics (through advanced algebra and trigonometry) and three years of science in high school. Comparable college mathematics and science courses completed at another institution also serve to properly prepare students for these majors

### Program Learning Outcomes

Recipients of a B.S. degree in applied mathematics at UC Santa Cruz are expected to have the following skills and experiences:

- To be able to take a real-life science or engineering problem, and create a mathematical model for it, under the supervision of a professor.
- To be competent with a number of analytical methods for the solution of linear algebra problems, as well as ordinary and partial differential equations.
- To be competent with constructing numerical algorithms for the solution of linear algebra problems and ordinary differential equations.
- To be competent in at least two scientific computing languages such as: Fortran, C, Python, R, Matlab, etc. and to be familiar with other computational elements such as Unix-type operating systems, the use of compilers, professional scientific computing libraries, efficient IO algorithms, data visualization tools, etc.
- To be able to analyze critically the results from the model obtained, and identify when the model is inappropriate.
- To be able to communicate clearly and coherently with professionals (orally and/or in writing), in order to: (1) understand what is needed of the mathematical model prior to the investigation; and (2) report on the results of the model after the investigation

### Major Qualification Policy and Declaration Process

#### Major Qualification

In order to be admitted into the applied mathematics major, students must be listed as a proposed major within the School of Engineering or within the Division of Physical and Biological Sciences. Please refer to __Prepare to Declare a BSOE Major__.

Transfers to the program should consult the __Transfer Students section __of the Baskin Engineering Undergraduate Affairs page.

In addition to being listed as a proposed School of Engineering or Physical and Biological Sciences major, admission to the applied mathematics major is based on passing the following foundational courses:

##### Foundation Courses

###### One of the following series:

Either these courses | ||

MATH 19A | Calculus for Science, Engineering, and Mathematics | 5 |

MATH 19B | Calculus for Science, Engineering, and Mathematics | 5 |

or these courses | ||

MATH 20A | Honors Calculus | 5 |

MATH 20B | Honors Calculus | 5 |

###### Plus these courses:

AM 30 | Multivariate Calculus for Engineers | 5 |

CSE 16 | Applied Discrete Mathematics | 5 |

###### Plus one of the following:

AM 10 | Mathematical Methods for Engineers I | 5 |

MATH 21 | Linear Algebra | 5 |

###### Plus one of the following:

AM 20 | Mathematical Methods for Engineers II | 5 |

MATH 24 | Ordinary Differential Equations | 5 |

Declaration of the major can happen no sooner than the student's second quarter, and no later than the campus deadline.

At most, one unsuccessful attempt (grade C-, D+, D, D-, F, or NP) for a foundation course is permitted.

#### Appeal Process

Denials of admission to the major may be appealed by submitting a letter to the School of Engineering Undergraduate Advising office, addressed to the program undergraduate director within 15 days from the date the notification was mailed. The appeal letter must describe why the prior performance is not an accurate reflection of the student's potential. Within 15 days of receipt of the appeal, the Undergraduate Advising office will notify the student and their college of the decision.

#### How to Declare a Major

There are four steps to declaring a BSOE major. For a detailed guide to this process, please consult Baskin Engineering's Declare Your Major website.

Students should start the declaration of major process by completing Step One on the BSOE "Declare Your Major" website as soon as they complete the major qualification courses or reach their declaration deadline quarter, whichever comes first.

Students petitioning when the campus declaration deadline is imminent (i.e., in their sixth quarter, for students admitted as frosh), will either be approved, denied, or provided with conditions (e.g., completion of some courses with certain grades) that will be resolved within at most one more enrolled quarter, even if they have not completed major qualification courses.

### Transfer Information and Policy

#### Transfer Admission Screening Policy

To be considered for admission to the applied mathematics major, incoming transfer students should complete the following courses:

##### All of the following courses:

AM 10 | Mathematical Methods for Engineers I | 5 |

AM 20 | Mathematical Methods for Engineers II | 5 |

AM 30 | Multivariate Calculus for Engineers | 5 |

CSE 16 | Applied Discrete Mathematics | 5 |

##### Plus one of the following options:

Either these courses | ||

MATH 19A | Calculus for Science, Engineering, and Mathematics | 5 |

MATH 19B | Calculus for Science, Engineering, and Mathematics | 5 |

or these courses | ||

MATH 20A | Honors Calculus | 5 |

MATH 20B | Honors Calculus | 5 |

Transfer students should have completed as many general education requirements as possible if they wish to graduate in two years.

#### Getting Started at UCSC as a Transfer Student

Transfer students should declare their major in their first quarter at UC Santa Cruz. Instructions for declaring a major in the Baskin School of Engineering are on the department's major declaration page.

### Letter Grade Policy

The Baskin School of Engineering requires letter grades for all courses in an engineering major.

### Course Substitution Policy

Undergraduate engineering students who wish to substitute a major course with a course from UC Santa Cruz must first consult the School of Engineering Undergraduate Advising Office. The advising office requires a Petition for Course Substitution be approved before credit for an alternate course can be applied to any School of Engineering major requirement. Petition forms are available at the undergraduate advising office and on the SoE advising website.

Petitions and procedures for approval must be obtained from and submitted to the Undergraduate Advising Office.

### Double Major Policy

Students who are planning a double-major with physics can replace the AM 100, AM 112 and STAT 131 (or CSE 107) courses with the PHYS 116A, PHYS 116B, and PHYS 116C series. Special approval by both undergraduate directors will be required to do so.

### Honors

Students must obtain a GPA of 3.8 or higher in the courses in the major to be considered for the distinction of “Highest Honors in the Major.” Students must obtain a GPA of 3.5 or higher in the courses in the major to be considered for the distinction of “Honors in the Major.” The School of Engineering reserves the right to withhold honors based on other criteria, such as an incident of academic dishonesty.

### School of Engineering Policies

Please refer to the School of Engineering's Admission to School of Engineering Majors section of the catalog for additional policies that apply to all School of Engineering programs. These policies include admission to the major and the need for students to obtain preapproval before taking courses elsewhere.

## Requirements and Planners

### Course Requirements

Course requirements are divided into foundational lower-division courses and advanced upper-division courses:

#### Lower-Division Courses

##### Choose one of the following series:

Either these courses | ||

MATH 19A | Calculus for Science, Engineering, and Mathematics | 5 |

MATH 19B | Calculus for Science, Engineering, and Mathematics | 5 |

or these courses | ||

MATH 20A | Honors Calculus | 5 |

MATH 20B | Honors Calculus | 5 |

Credit for one or both MATH 19A and MATH 19B may be granted with adequate performance on the CEEB calculus AB or BC Advanced Placement examinations.

##### Plus one of the following courses:

AM 10 | Mathematical Methods for Engineers I | 5 |

MATH 21 | Linear Algebra | 5 |

AM 10 is preferred.

##### Plus one of the following courses:

AM 20 | Mathematical Methods for Engineers II | 5 |

MATH 24 | Ordinary Differential Equations | 5 |

AM 20 is preferred.

##### Plus one of the following options:

AM 30 | Multivariate Calculus for Engineers | 5 |

or these courses | ||

MATH 23A | Vector Calculus | 5 |

MATH 23B | Vector Calculus | 5 |

AM 30 is preferred.

##### Plus the following course:

CSE 16 | Applied Discrete Mathematics | 5 |

##### Plus one of the following programming courses

CSE 20 | Beginning Programming in Python | 5 |

CSE 13S | Computer Systems and C Programming | 7 |

ECE 13 | Computer Systems and C Programming | 7 |

The programming requirement can be satisfied if students have a prior programming course, AP credit, or clearing the “Test-out” bar.

Students who did not take either AM 10 or AM 20 as part of the lower-division core requirement are strongly encouraged to study the Matlab self-paced tutorial prior to taking AM 114 and AM 147.

#### Lower-Division Electives

Students are required to take two lower-division electives from the following list, in preparation for the upper-division electives they are later required to take. Students are encouraged to plan ahead carefully in consultation with undergraduate advising in making their selection.

##### EART, ECE, PHYS lower-division electives

Students interested in EART, ECE, PHYS upper-division electives should consider taking some of the following PHYS lower-division electives:

PHYS 5A | Introduction to Physics I | 5 |

PHYS 5B | Introduction to Physics II | 5 |

PHYS 5C | Introduction to Physics III | 5 |

PHYS 6A | Introductory Physics I | 5 |

PHYS 6B | Introductory Physics II | 5 |

PHYS 6C | Introductory Physics III | 5 |

Only one course out of PHYS 5A and PHYS 6A, one course out of PHYS 5B and PHYS 6B, and one course out of PHYS 5C and PHYS 6C can be taken. Several of these courses have corequisite lab courses.

##### CSE or ECE lower-division electives

Students interested in CSE or ECE upper-division electives should consider taking some of the following CSE or ECE lower-division electives (in addition to possible PHYS electives):

CSE 30 | Programming Abstractions: Python | 7 |

ECE 9 | Statics and Mechanics of Materials | 5 |

##### ECON lower-division electives

Students interested in ECON upper-division electives should consider taking some of the following ECON or STAT lower-division electives:

ECON 1 | Introductory Microeconomics: Resource Allocation and Market Structure | 5 |

ECON 2 | Introductory Macroeconomics: Aggregate Economic Activity | 5 |

STAT 5 | Statistics | 5 |

##### Lower-division electives for mathematical biology

Students interested in mathematical biology (in particular AM 115) should consider taking some of the following BIOL or BIOE lower-division electives:

BIOL 20A | Cell and Molecular Biology | 5 |

BIOE 20C | Ecology and Evolution | 5 |

#### Upper-Division Courses

##### Complete the following core courses:

AM 100 | Mathematical Methods for Engineers | 5 |

AM 129 | Foundations of Scientific Computing for Scientists and Engineers | 5 |

AM 147 | Computational Methods and Applications | 5 |

##### Plus one of the following courses:

AM 112 | Introduction to Partial Differential Equations | 5 |

AM 212A | Applied Partial Differential Equations | 5 |

##### Plus one of the following courses:

AM 114 | Introduction to Dynamical Systems | 5 |

AM 214 | Applied Dynamical Systems | 5 |

##### Plus one of the following courses:

STAT 131 | Introduction to Probability Theory | 5 |

CSE 107 | Probability and Statistics for Engineers | 5 |

Students who intend to pursue an M.S. degree in scientific computing and applied mathematics later are strongly encouraged to take the AM 212A and AM 214 options instead of AM 112 and AM 114.

#### Upper-Division Electives

Students are required to take three upper-division elective courses from the following list of possible electives. Note that many of these electives have lower- division prerequisites. Students should plan carefully which ones to take to ensure they are prepared for their selected upper-division electives. Also note that enrollment in the graduate courses is by permission of the instructor, who will verify adequate preparation.

##### Possible AM Electives

Any 5-credit upper-division AM course that is not already a core course. Any 5-credit graduate AM course **with the exception*** *of AM 200, AM 211, AM 212A and AM 214.

##### Possible CSE Electives

Note that many require lower-division CSE courses. Lecture-lab combinations count as one course.

CSE 101 | Introduction to Data Structures and Algorithms | 5 |

CSE 102 | Introduction to Analysis of Algorithms | 5 |

CSE 106 | Applied Graph Theory and Algorithms | 5 |

CSE 140 | Artificial Intelligence | 5 |

CSE 142 | Machine Learning | 5 |

CSE 144 | Applied Machine Learning | 5 |

CSE 160 | Introduction to Computer Graphics | 5 |

CSE 160L | Introduction to Computer Graphics Laboratory | 2 |

CSE 161 | Introduction to Data Visualization | 5 |

CSE 161L | Data Visualization Laboratory | 2 |

CSE 162 | Advanced Computer Graphics and Animation | 5 |

CSE 162L | Advanced Computer Graphics and Animation Laboratory | 2 |

Note that most of these courses require CSE 101 as prerequisite and that enrollment restrictions vary and might apply to any of the CSE courses listed here on short notice. CSE 101, CSE 102, CSE 140, CSE 142, CSE 144 are courses for which enrollment restrictions may apply, and that may only be appropriate for double-majors (or major-minor combinations). CSE 162 & CSE 162L is a course that has more than one upper-division prerequisite beyond those that are already part of the core requirements. This course may only be appropriate for double-majors (or major-minor combinations).

##### Possible EART Electives

Note that many require lower-division PHYS or CHEM courses:

EART 124 | Modeling Earth's Climate | 5 |

EART 160 | Planetary Science | 5 |

EART 162 | Planetary Interiors | 5 |

###### Either of the following courses:

EART 125 | Statistics and Data Analysis in the Geosciences | 5 |

EART 225 | Statistics and Data Analysis in the Geosciences | 5 |

###### Either of the following courses:

EART 172
/OCEA 172
| Geophysical Fluid Dynamics | 5 |

EART 272
/OCEA 272
| Geophysical Fluid Dynamics | 5 |

##### Possible ECE Electives

Lecture-lab combinations count as one course.

ECE 101 | Introduction to Electronic Circuits | 5 |

ECE 101L | Introduction to Electronic Circuits Laboratory | 2 |

ECE 103 | Signals and Systems | 5 |

ECE 115 | Introduction to Solid Mechanics | 5 |

ECE 135 | Electromagnetic Fields and Waves | 5 |

ECE 135L | Electromagnetic Fields and Waves Laboratory | 2 |

ECE 141 | Feedback Control Systems | 5 |

ECE 153 | Digital Signal Processing | 5 |

Note that most of these courses require ECE 101 as prerequisite. ECE 141 and ECE 153 are courses that have more than one upper-division prerequisite beyond those that are already part of the core requirements. These courses may only be appropriate for double-majors (or major-minor combinations).

##### Possible ECON Electives

Note that many require ECON lower-division courses:

ECON 100M | Intermediate Microeconomics, Math Intensive | 5 |

ECON 100N | Intermediate Macroeconomics, Math Intensive | 5 |

ECON 113 | Introduction to Econometrics | 5 |

ECON 166A
/CSE 166A
| Game Theory and Applications I | 5 |

ECON 166A is a course that has more than one upper-division prerequisite beyond those that are already part of the core requirements. This course may only be appropriate for double-majors (or major-minor combinations).

##### Possible MATH Electives

MATH 105A | Real Analysis | 5 |

MATH 110 | Introduction to Number Theory | 5 |

MATH 111A | Algebra | 5 |

MATH 111T | Algebra | 5 |

MATH 115 | Graph Theory | 5 |

MATH 116 | Combinatorics | 5 |

MATH 117 | Advanced Linear Algebra | 5 |

MATH 118 | Advanced Number Theory | 5 |

MATH 120 | Coding Theory | 5 |

MATH 134 | Cryptography | 5 |

MATH 160 | Mathematical Logic I | 5 |

Note that many MATH electives require MATH 100 as a prerequisite.

##### Possible OCEA Electives

Note that some require lower-division PHYS electives, or upper-division ESCI electives:

OCEA 260
/EART 260
| Introductory Data Analysis in the Ocean and Earth Sciences | 5 |

OCEA 286 | Introduction to Ocean Modeling | 5 |

###### Either of the following courses:

OCEA 100 | Physical Oceanography | 5 |

OCEA 200 | Physical Oceanography | 5 |

###### Either of the following courses:

OCEA 111 | Climate Dynamics | 5 |

OCEA 211 | Climate Dynamics | 5 |

##### Possible PHYS Electives

Note that many require lower-division PHYS courses:

PHYS 105 | Mechanics | 5 |

PHYS 110A | Electricity, Magnetism, and Optics | 5 |

PHYS 110B | Electricity, Magnetism, and Optics | 5 |

PHYS 139A | Quantum Mechanics I | 5 |

PHYS 139B | Quantum Mechanics II | 5 |

PHYS 150 | Quantum Computing | 5 |

PHYS 171
/ASTR 171
| General Relativity, Black Holes, and Cosmology | 5 |

The prerequisites of PHYS 116A, PHYS 116B, and PHYS 116C is waived for students who have taken AM 100 and AM 112 and STAT 131 (or CSE 107).

PHYS 139A, PHYS 139B, PHYS 171 are courses that have more than one upper-division prerequisite beyond those that are already part of the core requirements. These courses may only be appropriate for double majors (or major-minor combinations).

##### Possible STAT Electives

STAT 132 | Classical and Bayesian Inference | 5 |

All students, but especially those doing a double major or a major-minor combination, may also petition to count courses that are not already on the list as electives, subject to approval

#### Disciplinary Communication (DC) Requirement

The DC requirement in the Applied Mathematics B.S. is satisfied by completing the capstone course AM 170A (see below).

#### Comprehensive Requirement

Students satisfy the senior comprehensive requirement by receiving a passing grade in the two Mathematical Modeling courses:

AM 170A | Mathematical Modeling 1 | 5 |

AM 170B | Mathematical Modeling 2 | 5 |

### Planners

The following are four sample academic plans that students can use to plan their sequence of courses in the major. The first two plans are suggested guidelines for students who begin their studies in their frosh year. Such students, if they plan carefully, will have several openings free to take other breadth courses they find interesting, or pursue an additional minor or major. The other two plans are for students transferring to UC Santa Cruz as juniors.

**Plan One: **This planner is for a student entering UCSC in their frosh year who is prepared to go directly into MATH 19A or MATH 20A.

Fall | Winter | Spring | |
---|---|---|---|

1st Year |
MATH 19A | CSE 20 | AM 20 |

AM 10 | Lower div elective | CSE 16 | |

MATH 19B | |||

2nd Year |
AM 30 | STAT 131 | |

Lower-div elective | |||

3rd Year |
AM 100 | AM 170A | Upper-div elective |

AM 114 | AM 112 | ||

Upper-div elective | |||

4th Year |
AM 129 | AM 147 | AM 170B |

Upper-div elective |

The SI general education requirement is satisfied by passing the capstone course AM 170B. The MF general education requirement is satisfied by passing any of the lower-division mathematical foundations requirements.

**Plan Two: **This planner is for a student entering UCSC their frosh year who needs to take preparatory courses prior to MATH 19A

__to ensure a successful outcome in this course.__

Fall | Winter | Spring | |
---|---|---|---|

1st Year |
MATH 2 | MATH 3 | MATH 19A |

Lower div elective | Lower div elective | ||

2nd Year |
AM 10 | AM 20 | AM 30 |

MATH 19B | CSE 20 | CSE 16 | |

3rd Year |
AM 100 | AM 112 | |

AM 114 | STAT 131 | Upper-div elective | |

Upper-div elective | |||

4th Year |
AM 129 | AM 147 | AM 170B |

Upper-div elective | AM 170A |

The SI general education requirement is satisfied by passing the capstone course AM 170B. The MF general education requirement is satisfied by passing any of the lower-division mathematical foundations requirements.

**Plan Three: **This is a sample planner for a transfer student. It assumes the student has taken the majority of their general education requirements prior to joining UCSC.

Fall | Winter | Spring | |
---|---|---|---|

Junior Year |
CSE 20 | AM 112 | Upper-div elective |

AM 100 | STAT 131 | ||

Lower-div elective | Lower-div elective | ||

Senior Year |
AM 114 | AM 170A | AM 170B |

AM 129 | AM 147 | ||

Upper-div elective | Upper-div elective |

**Plan Four: **This is a sample planner for a transfer student interested in the 4+1 program (e.g. thesis track; replace Independent Study with M.S. elective for coursework track). Note that students **must **take the graduate version of PDEs and Dynamical Systems courses (212A and 214 respectively) to count toward the M.S. core requirements. One additional elective must be taken during the M.S. year to ensure that there are at least 35 credits taken as a graduate student.

Fall | Winter | Spring | |
---|---|---|---|

Junior Year |
CSE 20 | AM 212A | Upper-div elective |

AM 100 | STAT 131 | ||

Lower-div elective | Lower-div elective | ||

Senior Year |
AM 129 | AM 147 | AM 170B |

AM 214 | AM 170A | Upper-div elective | |

Upper-div elective | |||

M.S. Year |
M.S. elective | AM 213A | AM 213B |

M.S. elective | AM 250 | ||

Indep Study | Indep Study |