 UC Merced's Applied Mathematics courses emphasize learning fundamental mathematics as well as its application to other fields of study such as biology, economics and physics. To that end, we emphasize learning mathematics conceptually and concretely by exploring topics analytically, graphically, computationally and verbally. Below we describe each of our undergraduate course offerings.

## Lower Division Courses

Lower division mathematics courses (e.g. pre-calculus and calculus) are required for many of the majors at UC Merced because mathematics is the foundation for many different areas of study. To help our students to learn mathematics as best as they can, our math instructors provide students a variety of experiences to learn mathematics through lectures, discussion sections, homework and various workshops. In addition, there is free tutoring available on campus through the Math Center.

Math 5: Pre-calculus

Preparation for calculus. Analyzing data by means of functions (linear, quadradic, polynomial, logarithmic, exponential and trigonometric) and graphs with an emphasis on mathematical modeling of real-world applications.

Math 11: Calculus I

An introduction to differential and integral calculus of functions of one variable, including exponential, logarithmic and trigonometric functions. Emphasis is put on conceptual understanding and applying mathematical concepts to real-world problems such as approximation and optimization.

Math 12: Calculus II

Continuation of MATH 11. An introduction to integral calculus of functions of one variable and differential equations. Emphasis is put on conceptual understanding and applying mathematical concepts to analyze problems in biology, economics, business, and other fields.

Math 15: Introduction to Scientific Data Analysis

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.

Corequisite: Math 5.

Math 18: Statistics for Scientific Data Analysis

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.

Prerequisite: Math 5 and Math 15.

Math 21: Calculus I for Physical Sciences and Engineering

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 natural sciences and engineering.

Prerequisite(s): Passing score on calculus readiness placement test, or score of 3 or higher on AP Calculus AB exam or grade of C- or higher in Math 5.

Math 22: Calculus II for Physical Sciences and Engineering

Continuation of MATH 21. Analytical and numerical techniques of integration with applications, infinite sequences and series, first order ordinary differential equations.

Prerequisite(s): Score of 4 or higher on AP Calculus AB exam OR Score of 3 or higher on AP Calculus BC exam OR grade of C- or better in MATH 21 or ICP 1

Math 23: Vector Calculus

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

Math 24: Linear Algebra and Differential Equations

Introduces ordinary differential equations, systems of linear equations, matrices, determinants, vector spaces, linear transformations and linear systems of differential equations.

Prerequisite: Math 22

Math 32: Probability and Statistics

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 1

## Upper Division Courses

#### Math 122: Complex Variables and Applications

Introduction to complex variables, analytic functions, contour integration and theory of residues. Mappings of the complex plane. Introduction to mathematical analysis.

Prerequisite(s): Math 23 and Math 24

#### Math 125: Intermediate Differential Equations

This course introduces advanced solution techniques for ordinary differential equations (ODE) and elementary solution techniques for partial differential equations (PDE). Specific topics include higherorder 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 and Math 24

#### Math 126: Partial Differential Equations

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.

Prerequisites: Math 125

#### Math 130: Numerical Analysis

Introduction to numerical methods with emphasis on the analysis and implementation of numerical methods. Topics covered: computer arithmetic, solution of nonlinear equations in one variable, interpolation and polynomial approximation, elements of approximation theory, numerical differentiation and integration, and introduction to initial-value problems for ordinary differential equations.

Prerequisite(s): Mechanical Engineering 21 (ME 21) or (CSE 020 and CSE 021 or equivalent exam), and Math 24. Open only to major(s): Applied Mathematical Sciences. Course may not be taken for credit after obtaining credit for Math 131.

#### Math 131: Introduction to Numerical Analysis

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

#### Math 132: Numerical Solution of Differential Equations

A continuation of Math 131. Initial-value problems for ordinary differential equations, iterative 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 131

#### Math 141: Linear Analysis

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.

Prerequisite: Math 23, Math 24

#### Math 146: Numerical Linear Algebra

Matrix factorization and iterative methods for solving systems of linear equations. Topics include floating point arithmetic, eigenvalue problems, conditioning and stability, LU factorization, QR factorization, and SVD with applications in science and engineering.

Prerequisite: Math 141, Math 130 (or Math 131) any of which may be taken concurrently) and (ME 021 or (CSE 020 and CSE 021 or equivalent exam).

#### Math 180: Modern Applied Statistics

Introduction to modern applied statistics emphasizing computational methods to deal with high-dimensional data. Multivariate linear and nonlinear regression, model selection, overfitting, cross-validation, bootstrapping and quantification of uncertainty in model parameters and predictions, principal component analysis, and classification.

Prerequisite: Math 24 and Math 32.

#### Math 181: Stochastic Processes

Introduction to stochastic processes with emphasis on problem-solving using both analytical and computational techniques. Markov chains in discrete and continuous time, martingales, branching processes, renewal processes, and Brownian motion.

Prerequisite: Math 24 and Math 32.