Applied Mathematics Curriculum

 Many New College students pursue an Applied Mathematics AOC all on its own, while others combine the major with studies in BiologyPhysics and other concentrations in what we call a “slash” degree. Your faculty advisor can assist you in determining which path is best for your special interests and goals.

The (minimal) course work for a slash degree in Applied Mathematics includes the following:

• Calculus 1
• Calculus 2
• Calculus 3
• Linear Algebra
• Mathematical Modeling
• Numerical Analysis
• Ordinary Differential Equations

A course in Programming is also recommended.

In addition to the coursework listed above, the (minimal) course work for a stand-alone major in Applied Mathematics includes the following:

• A course in programming
• Advanced Linear Algebra
• Partial Differential Equations
• Probability/Statistics

A course in Complex Analysis is also highly recommended.

Other requirements for the major include:

• A two-semester introductory sequence (or two semesters of more advanced material) in either Biology, Chemistry or Physics
• Three semesters of Math Seminar
• A senior thesis involving Applied Mathematics

For detailed requirements, check out our General Catalog.

Here’s a list of recent course offerings in Applied Mathematics:

Calculus I
Calculus is a means for calculating the rate of change of a quantity which varies with time and the total accumulation of the quantity whose rate of change varies with time. Although calculus is only about three centuries old, calculus ideas are the basis for most modern applications of mathematics, especially those underlying our technology. The development of the calculus is one of the great intellectual achievements of Western civilization. A balance will be struck between presenting calculus as a collection of techniques for computation, and as a handful of difficult but very powerful concepts. Wherever possible, we will motivate the ideas as ways of answering questions about real world problems. Prerequisites: Complete the math placement exam.

Calculus II
This course takes up where Calculus I leaves off. The topics covered include integration techniques, sequences, series, Taylor series, complex numbers, areas and volumes. This course is recommended for students pursuing interests in the physical sciences, applied mathematics and economics. Prerequisite: Calculus I and instructor's permission.

Calculus III

This class is a continuation of Calculus I and II. We will cover the calculus in n-dimensional Euclidean space. The topics covered during the course of the semester include the fundamental constructions of the calculus of multivariable functions (vector fields, gradients, line integrals, surface integrals etc) and the associated fundamental results (Green's Theorems, Gauss' Theorem, Stokes' Theorem, etc). The course will focus on application and computation and will include an introduction to differential equations. Prerequisite: Calculus II. 

Linear Algebra

This course is an introduction to the theory of vector spaces and linear transformations and to their representation by means of matrices. The topics that will be covered are: matrices and linear systems of equations, algebra of matrices, determinants, vector spaces, linear transformations, eigenvalues and eigenvectors, matrix diagonalization, and inner product spaces. Prerequisites: Calculus or the consent of instructor. 

Mathematical Modeling I
Mathematical modeling plays a central role in understanding of complex systems that are changing in time. Such systems are called dynamical systems. This course is designed to introduce students to the elements of dynamical systems. Both continuous and discrete systems will be covered. In the course of the term, students will come to understand how mathematical models are formulated, and how their short and long term behaviors can be uncovered through a combination of analysis and computer simulation. Qualitative, quantitative and graphical techniques will be used to analyze and understand mathematical models and to compare theoretical predictions with available data. Mathematical concepts of steady states, cycles and chaos will be introduced. Examples will be given from physics, biology, chemistry and economics. Prerequisites: Calculus and differential equations (or the approval of instructor).

Introduction to Numerical Methods
This is a survey course of the basic numerical methods which are used to solve practical scientific problems. Important concepts such as accuracy, stability, and efficiency and convergence are discussed. The course provides an introduction to MATLAB, an interactive program for numerical linear algebra. Objectives of the course: Develop numerical methods for approximately solving problems from continuous mathematics on the computer. Examine the accuracy, stability, and failure modes of these methods. Implement these methods in a computer language MATLAB. Prerequisites: Calculus and Differential Equations.

Ordinary Differential Equations
Familiarity with the material covered in a first course in differential equations is essential for those interested in advanced work in pure and applied mathematics. Topics covered during the semester include; first order equations, second order linear equations, series solutions, Laplace transform, systems of first order linear equations, qualitative properties of nonlinear equations, boundary value problems and Sturm-Liouville theory. Prerequisites: Calculus II.

Introduction to Programming in Python
This course is an interdisciplinary introduction to Programming in Python. It satisfies LAC curriculum requirements. The course introduces students to the most important programming concepts such as algorithms, sequences, selections, loops, functions, methods, numeric and string types, file processing, collections, classes and object-oriented programming, and recursion. This course serves as an informal prerequisite for many science classes which require programming. Students enrolled in this course MUST also attend the mandatory workshop. Prerequisites: The course is at the introductory – freshman level. It has no prerequisites and no prior programming experience is assumed. However, it requires a large time commitment and the ability to work with computers for extended periods of time.

Advanced Linear Algebra
Linear algebra is a critical mathematical tool in all of the sciences. Therefore, an in-depth knowledge of linear algebra is useful not only to mathematicians, but also to any scientist using mathematics. Topics to be covered include a review of basic linear algebra, the Moore-Penrose Pseudoinverse, singular value decompositions, generalizations of matrix equations, projections and inner products, least squares problems, Jordan canonical form, linear differential equations and the matrix exponential, and difference equations. Prerequisite: Linear Algebra or permission of the instructor.

Partial Differential Equations
This course is designed to prepare students for advanced work in geometry and mathematical physics by developing the knowledge of partial differential equations common to both topics. Topics covered during the semester include: Laplace equations, wave equations, heat equations, Hamilton-Jacobi equations, Fourier theory, and the theory of distributions. Prerequisites: Calculus III and Ordinary Differential Equations.

The course will consist of two parts. In the first part, we will begin by studying discrete spaces and simple games of chance. We will introduce and study the basic notions of probability including random variables, distribution, expectation, and variance. We will study continuous distributions as they relate to approximations of various discrete objects. In the second part of the course we will use our knowledge of simple games of chance to construct discrete models of simple physical systems. The models and the ideas behind their construction have found applications in many different areas (Physics, Chemistry, Biology, Economics, etc.). Time permitting; we will study several such examples in detail. Prerequisite: Calculus.

Complex Analysis
Complex numbers were introduced in the study of the roots of polynomial equations and have found applications in nearly every branch of modern mathematics. This course will develop the notion of a function of a complex variable and the corresponding calculus. The theorems and applications to be discussed are some of the most beautiful results of modern mathematics. Topics for the course include analytic functions, complex integration and the Cauchy integral formula, series representations, residues, the Dirichlet problem, and conformal mappings. Prerequisites: Real Analysis I or permission of instructor.

For a complete list of courses, click here.


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