# Applied Mathematical Sciences Major

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## About the Major

Mathematics has been a central feature of humanity's intellectual achievements over the past several centuries. Its role in the physical sciences and engineering is well established and continues to aid in their development. However, mathematics is becoming more important in the social sciences and life sciences which are all new application areas for applied mathematical sciences. Thus, the field is undergoing remarkable growth.

### The Program

UC Merced offers an undergraduate major leading to a B.S. degree in the Applied Mathematical Sciences. This educational experience provides the foundations of mathematics and the skills needed to apply mathematics to real-world phenomena in the social sciences, natural sciences and engineering. The curriculum is designed to provide courses in the fundamentals while allowing for building expertise in an application area through the emphasis tracks. There is a core set of courses all mathematical sciences students take. Beyond those courses, students take an emphasis track consisting of courses in other fields. Some examples of emphasis tracks include physics, computational biology, economics, computer science and engineering, and engineering mechanics. New emphasis tracks will be added along side new programs developing at UC Merced. Students may also design their own emphasis track with the approval of the faculty program leads for the Applied Mathematical Sciences major.

## Learning Outcomes

The over-arching goal of the Applied Mathematical Sciences program is to

**Build a community of life-long learners who use the analytical and computational tools of mathematics to solve real-world problems.**

Upon graduating, we expect students completing the Applied Mathematical Sciences major to have become effect problem-solvers, meaning that student will be able to

- Solve mathematical problems using analytical methods.
- Solve mathematical problems using computational methods.
- Recognize the relationships between different areas of mathematics and the connections between mathematics and other disciplines.
- Give clear and organized written and verbal explanations of mathematical ideas to a variety of audiences.
- Model real-world problems mathematically and analyze those models using their mastery of the core concepts.

## Careers

A degree in applied mathematical sciences opens the door to a wide variety of careers. Employers understand that a degree in mathematics means a student has been trained well in analytical reasoning and problem solving. Moreover, applied mathematical sciences majors with skills in scientific computing have the additional leverage of substantial computing experience. The market for applied mathematicians has usually been good, especially for those who can relate their mathematics to real world problems. In particular, applied mathematics majors familiar with concepts in management, biology, engineering, economics or the environmental sciences among others are well suited for many specialized positions. In addition, the breadth and rigor of this program provide an excellent preparation to teach mathematics at the elementary or high school levels.

## Requirements

In addition to adhering to the UC Merced and School of Natural Sciences requirements, the **additional** requirements that must be met to obtain the B.S. degree in the Mathematical Sciences at UC Merced are:

### Requirements for All Emphases

#### Lower Division Courses

- BIO 1: Contemporary Biology [4 units] or
- ESS 1: Introduction to Earth Systems Science [4 units] or
- ESS 5: Introduction to Biological Earth Systems [4 units]
- MATH 21: Calculus I [4 units]
- MATH 22: Calculus II [4 units]
- MATH 23: Vector Calculus [4 units]
- MATH 24: Linear Algebra and Differential Equations [4 units]
- MATH 32: Probability & Statistics [4 units]
- PHYS 8 or 8H: Introductory Physics I [4 units]
- PHYS 9 or 9H: Introductory Physics II [4 units]

#### Upper Division Courses

- MATH 122: Applied Math Methods II [4 units]
- MATH 125: Immediate Differential Equations [4 units]
- MATH 126: Partial Differential Equations [4 units]
- MATH 131: Numerical Analysis I [4 units]
- MATH 132: Numerical Analysis II [4 units]
- MATH 141: Linear Analysis I [4 units]

### Additional Requirements for Emphasis Tracks

The student must complete at least 19 units of approved course work from other programs toward the completion of an emphasis track. At least 12 of these 19 units must be upper division courses. Some examples of emphasis tracks include physics, computational biology, economics, computer science and engineering,and engineering mechanics. More application themes will become available as new programs on campus develop.The student may design their own emphasis track with approval from the faculty program leads for the Applied Mathematical Sciences major.

#### Computational Biology Emphasis Track

- BIO 2: Introduction to Molecular Biology [4 units]

AND

- BIO 175: Biostatistics [4 units] or
- BIO 140: Genetics [4 units]

- BIO 180: Introduction to Scientific Modeling [4 units]
- BIO 181: Survey of Computational Biology [4 units]
- BIO 182: Bioinformatics [4 units]

OR

- BIO 175: Biostatistics [4 units] or
- BIO 180: Introduction to Scientific Modeling [4 units]

- BIO 140: Genetics [4 units]
- BIO 144: Phylogenetics [4 units]
- BIO 183: Population Genetics [4 units]

#### Computer Science Emphasis Track

- CSE 30: Introduction to Computer Science and Engineering I [4 units]
- CSE 31: Introduction to Computer Science and Engineering II [4 units]
- CSE 100: Algorithm Design and Analysis [4 units]

AND

Two of the following courses:

- CSE 111: Database Systems [4 units]
- CSE 140: Computer Architecture [4 units]
- CSE 160: Computer Networks [4 units]
- CSE 170: Computer Graphics [4 units]
- CSE 171: Programming Interactive 3D Graphics and Games [4.0 units]
- CSE 173: Computational Cognitive Neuroscience [4.0 units]
- CSE 175: Introduction to Artificial Intelligence [4.0 units]
- CSE 176: Machine Learning [4.0 units]
- CSE 185: Introduction to Computer Vision [4.0 units]
- ENGR 140: Introduction to Object Oriented Programming [4.0 units]
- ENGR 160: Discrete Mathematics [4.0 units]

#### Economics Emphasis Track

- ECON 1: Introduction to Economics [4 units]
- ECON 10: Statistical Inference [4 units]
- ECON 100: Intermediate Microeconomic Theory [4 units]
- ECON 101: Intermediate Macroeconomic Theory [4 units]
- ECON 130: Econometrics [4 units]

#### Engineering Emphasis Track

- ENGR 57: Dynamics [3 units]
- ME 135: Finite Element Analysis [3 units]

AND

3 of the following courses:

- ENGR 120: Fluid Mechanics [4 units]
- ENGR 130: Thermodynamics [3 units]
- ENGR 135: Heat Transfer [4 units]
- ENGR 151: Strength of Materials [4 units]
- ME 140: Vibration & Control [4 units]

#### Environmental Emphasis Track

Two of the following courses:

- ESS 001: Introduction to Earth Systems Science [4.0 units]
- ESS 002: Sustainability Science [4.0 units]
- ESS 020: Fundamentals of Geology [4.0 units]
- ESS 025: Introduction to Ecosystem Science [4.0 units]
- ENVE 020: Introduction to Environmental Science and Technology [4.0 units]

AND

3 of the following courses:

- ENVE 105: Environmental Data Analysis [3.0 units]
- ENVE 110: Hydrology and Climate [4.0 units] or
- ESS 110: Hydrology and Climate [4.0 units]
- ESS 112: Subsurface Hydrology [4.0 units]
- ENVE 130: Meteorology and Air Pollution [4.0 units]
- ENVE 160: Sustainable Energy [4.0 units]
- ESS 128: Theoretical Ecology [4.0 units]
- ESS 148: Fundamentals of Ecology [4.0 units]

#### Physics Emphasis Track

- PHYS 10: Introductory Physics III [4 units]
- Any Upper Division Physics Courses, except PHYS 116 or PHYS 160 [totaling 14 units]

The Calculus I and Calculus II courses are designed to meet the needs of students needing two or fewer semesters of calculus to meet their educational objectives. View FAQs on the series.

## Courses

### MATH 005: Preparatory Calculus [4]

Preparation for calculus. Elementary functions, trigonometry, polynomials, rational functions, systems of equations and analytical geometry. Course cannot be taken after obtaining credit for MATH 21. Normal Letter Grade only.

[Syllabus]

### MATH 011: Calculus I [4]

Introduction to differential and integral calculus of functions of one variable, including exponential, logarithmic and trigonometric functions, emphasizing conceptual understanding and applying mathematical concepts to real-world problems (approximation, optimization). Course may not be taken for credit after obtaining credit for MATH 021. Course does not lead to MATH 23, 24. [Syllabus]

### MATH 012: Calculus II [4]

Continuation of MATH 011. Introduction to integral calculus of functions of one variable and differential equations, emphasizing conceptual understanding and applying mathematical concepts to real-world problem. Course may not be taken for credit after obtaining credit for MATH 022. Course does not lead to Math 23, 24. Prerequisites: MATH 011 or MATH 021. [Syllabus]

### MATH 015: Introduction to Scientific Data Analysis [2]

Fundamental analytical and computational skills to find, assemble and evaluate information and to teach the basics of data analysis and modeling using spreadsheets, statistical tool, scripting languages and high-level mathematical languages. This course is not for students from School of Engineering. [Syllabus]

### MATH 018: Statistics for Scientific Data Analysis [4]

Analytical and computational methods for statistical analysis of data. Descriptive statistics, graphical representations of data, correlation, regression, causation, experiment design, introductory probability, random variables, sampling distributions, inference and significance. Course can not be taken for credit after obtaining credit for Math 32. Prerequisite: MATH 5 and MATH 15. Normal Letter Grade only. [Syllabus]

### MATH 021: Calculus I for Physical Sciences & Engineering [4]

An introduction to differential and integral calculus of functions of one variable. Elementary functions such as the exponential and the natural logarithm, rates of change and the derivative with applications to physical sciences and engineering. Course may not be taken for credit after obtaining credit for MATH 011. Prerequisite: MATH 005 or equivalent score on Math Placement Exam. Normal Letter Grade only. [Syllabus]

### MATH 022: Calculus II for Physical Sciences & Engineering [4]

Continuation of MATH 021. Analytical and numerical techniques of integration with applications, infinite sequences and series, first order ordinary differential equations. Course may not be taken for credit after obtaining credit for Math 012. Prerequisite: MATH 21. Normal Letter Grade only. [Syllabus]

### MATH 023: Vector Calculus [4]

Calculus of several variables. Parametric equations and polar coordinates, algebra and geometry of vectors and matrices, partial derivatives, multiple integrals and introduction to theorems of Green, Gauss and Stokes. Prerequisite: MATH 22. Normal Letter Grade only. [Syllabus]

### MATH 024: Linear Algebra and Differential Equations [4]

Introduces ordinary differential equations, systems of linear equations, matrices, determinants, vector spaces, linear transformations and linear systems of differential equations. Prerequisite: MATH 22. Normal Letter Grade only. [Syllabus]

### MATH 030: Mathematical Biology [4] (discontinued)

A version of MATH 22 for students majoring in the life sciences. Analytical and numerical techniques of integration, first-order ordinary differential equations and methods in discrete math with applications to questions from biology and medicine. Prerequisite: MATH 21 or ICP 1A.**Course discontinued, please see Math 12.**

[Syllabus]

### MATH 032: Probability and Statistics [4]

Concepts of probability and statistics. Conditional probability, independence, random variables, distribution functions, descriptive statistics, transformations, sampling errors, confidence intervals, least squares and maximum likelihood. Exploratory data analysis and interactive computing. Prerequisite: MATH 21 or ICP 1A. [Syllabus]

### MATH 90X: Freshman Seminar [1]

Examination of a topic in mathematics.

### MATH 091: General Topics in Applied Mathematics [1]

Introduction to a variety of concepts useful in applied mathematics. Topics covered included floating point arithmetic, methods of proofs, random walks, stereographic projections, transforms, etc. Students are exposed to advanced mathematical topics in preparation for their ongoing studies. Prerequisite: MATH 023 and MATH 024. Either of which may be taken concurrently.

[Syllabus]

### MATH 95: Lower Division Undergraduate Research [1 – 5]

Supervised research. Permission of instructor required.

### MATH 98: Lower Division Directed Group Study [1 – 5]

Permission of instructor required. Pass/No Pass grading only.

### MATH 99: Lower Division Individual Study [1 - 5]

Permission of instructor required. Pass/No Pass grading only

### MATH 101: Analysis

### MATH 121: Applied Math Methods I: Introduction to Partial Differential Equations [4]

Introduction to Fourier series. Physical derivation of canonical partial differential equations of mathematical physics (heat, wave and Laplace’s equation). Separation of variables, Fourier integrals and general eigenfunction expansions. Prerequisite: MATH 23 and MATH 24. Normal Letter Grade only. [Syllabus]

### MATH 122: Complex Variables and Applications [4]

Introduction to complex variables, analytic functions, contour integration and theory of residues. Mappings of the complex plane. Introduction to mathematical analysis. Prerequisite: MATH 23 and MATH 24. Normal Letter Grade only. [Syllabus]

### MATH 125: Intermediate Differential Equations [4]

This course introduces advanced solution techniques for ordinary differential equations (ODE) and elementary solution techniques for partial differential equations (PDE). Specific topics include higher-order linear ODE, power series methods, boundary value problems, Fourier series, Sturm-Liouville theory, Laplace transforms, Fourier transforms, and applications to one-dimensional PDE. Prerequisites: MATH 23, MATH 24. [Syllabus]

### MATH 126: Partial Differential Equations [4]

This course introduces students to the theory of boundary value and initial value problems for partial differential equations with emphasis on linear equations. Topics covered include Laplace's equation, heat equation, wave equation, application of Sturm-Liouville's theory, Green's functions, Bessel functions, Laplace transform, method of characteristics. [Syllabus]

### MATH 131: Numerical Analysis I [4]

Introduction to numerical methods with emphasis on algorithm construction, analysis and implementation. Programming, round-off error, solutions of equations in one variable, interpolation and polynomial approximation, approximation theory, direct solvers for linear systems, numerical differentiation and integration, initial-value problems for ordinary differential equations. Prerequisite: MATH 24. Normal Letter Grade only. [Syllabus]

### MATH 132: Numerical Analysis II [4]

A continuation of MATH 131. Initial-value problems for ordinary differential equations, interactive techniques for solving linear systems, numerical solutions of nonlinear systems of equations, boundary-value problems for ordinary differential equations, numerical solutions to partial differential equations. Prerequisite: MATH 121 and MATH 131 Normal Letter Grade only. [Syllabus]

### MATH 140: Mathematical Methods for Optimization [3]

Linear programming and a selection of topics from among the following: matrix games, integer programming, semidefinite programming, nonlinear programming, convex analysis and geometry, polyhedral geometry, the calculus of variations and control theory. Prerequisite: MATH 23 or MATH 25. Normal Letter Grade only.

### MATH 141: Linear Analysis I [4]

Applied linear analysis of finite dimensional vector spaces. Review of matrix algebra, vector spaces, orthogonality, least-squares approximations, eigenvalue problems, positive definite matrices, singular value decomposition with applications in science and engineering.

[Syllabus]

### MATH 142: Linear Analysis II [4]

Applied linear analysis of infinite dimensional vector spaces. Inner product spaces, operators,adjoint operators, Fredholm alternative, spectral theory, Sturm-Liouville operators, distributionsand Green's functions with applications in science and engineering. Prerequisite: MATH 141. Normal Letter Grade only. [Syllabus]

### MATH 150: Mathematical Modeling [4]

Introduction to the basics of mathematical modeling emphasizing model construction, analysis and application. Using examples from a variety of fields such as physics, biology, chemistry and economics, students will learn how to develop and use mathematical models of real-world systems. Prerequisites: Math 131, Math 132, Math 141. [Syllabus]

### MATH 195: Upper Division Undergraduate Research [1 – 5]

Supervised research. Permission of instructor required.

### MATH 198: Upper Division Directed Group Study [1 – 5]

Permission of instrucor required. Pass/No Pass grading only. [Syllabus]

## Assessment Plan

### Timeline and Goals

Through a five-year assessment schedule, data gathered will determine the degree to which students achieve the desired program learning outcomes as a result of completing the major or minor in applied mathematical sciences at UC Merced. The over-arching goal of the Applied Mathematical Sciences program is to:

Build a community of life-long learners that use the analytical and computational tools of mathematics to solve real-world problems.

This five-year assessment schedule will allow us to develop and assess program learning outcomes in a strategic manner, detailed in *Process* section (below).

### Programmatic Learning Outcomes

Upon graduating, we expect students completing the Applied Mathematical Sciences major to have become effect *problem-solvers*, meaning that student will be able to:

- Solve mathematical problems using analytical methods.
- Solve mathematical problems using computational methods.
- Recognize the relationships between different areas of mathematics and the connections between mathematics and other disciplines.
- Give clear and organized written and verbal explanations of mathematical ideas to a variety of audiences.
- Model real-world problems mathematically and analyze those models using their mastery of the core concepts.

The Applied Mathematics faculty plan to communicate the program goals and learning outcomes in the following forums by the start of the 2009 – 2010 academic year.

- The Applied Math website;
- A revision of the catalog copy;
- Documents included with orientation materials for lecturers and teaching assistants;
- Included on slides given for student recruitment and orientation meetings.

### Evidence

The procedure for assessing achievement of the program learning outcomes will utilize four methods — two direct and two indirect. These are:

**Embedded Midterm and Final Exam Questions**— These questions will be embedded in key lower division and upper division mathematics courses to assess students at various stages of learning — introductory, developing, and mastery levels. Each exam question will be assessed using the rubric agreed upon by the applied mathematics faculty appropriate for each level.**Capstone Course**— By fall 2012, we will have developed the Capstone course in applied mathematics. This course will serve several functions. One such function is to provide a unique experience for the major students and present them with a series of activities designed to enhance their knowledge and appreciation of applied mathematics while preparing them for entry into a graduate program or placement in industry.& Other functions include avenues to assess the program through projects, presentations, and research.**Senior Exit Survey**— Using the graduating senior survey administered through Alumni and Career Services, data gathered will provide information about the program, minors obtained, job placement and future plans, and reasons why the student selected the major. Furthermore, we hope to identify which parts of the program work well and areas which need improvement.**Alumni Survey**— A survey will be distributed to one, five, and ten year alumni to gather information on whether applied mathematics graduates pursue graduate degrees, obtain meaningful jobs, and perception of preparedness for careers. In Spring 2009, we will administer a prototype exit survey to our first graduating class. Through analyzing the data we collect, we will continue to revise and develop our alumni survey further. Since we have a manageably small number of students graduating, we plan to keep in touch with our alumni informally and monitor their progress.

### Process

For a plan to assess these program learning outcomes, the applied math faculty have decided to follow a five-year schedule. Each year will assess the corresponding program learning outcomes in the order that they appear. In other words, our five year schedule is as follows:

**2009-2010 Academic Year**— Develop and implement measures to assess PLO 1: Embedded exam questions in Math 22, Math 24, and Math 131 will be identified and assessment rubrics will be created, senior exit survey data gathered, alumni surveys administered.**2010-2011 Academic Year**— Develop and implement measures to assess PLO 2: Embedded exam questions in Math 22, Math 131, and Math 132, senior and alumni surveys.**2011-2012 Academic Year**— Develop and implement measures to assess PLO 3: Embedded exam questions in Math 32 and Math 131, senior and alumni surveys.**2012-2013 Academic Year**— Develop and implement measures to assess PLO 4: Capstone course, embedded exam questions in Math 24, senior and alumni surveys.**2013-2014 Academic Year**— Develop and implement measures to assess PLO 5: Capstone course, senior and alumni surveys.

### Participants

The Mathematics Coordinator will compile senior exit and alumni survey responses and collect embedded exam questions of applied math majors for faculty review. All applied math faculty will be included in the interpretation of evidence through a series of three faculty meetings. Data collected will be reviewed and analyzed using agreed upon assessment rubrics with the goal to create a report. This report will be forwarded to Associate Dean and Dean of Natural Sciences. Using this report, the faculty can re-evaluate course structures and teaching practices, modify assessment measures, and examine student skill development for continuous quality of improvement.