From the New College of Florida 2011-2012 General Catalog.
The (minimal) course work for a slash degree in Applied Mathematics includes the following:
• Calculus 1, Calculus 2, Calculus 3
• Linear Algebra
• Ordinary Differential Equations
• Mathematical Modeling
• Numerical Analysis
In addition, a course in Programming is highly recommended.
The (minimal) course work for a major in Applied Mathematics includes the following:
The requirements for a slash degree in Applied Mathematics.
• Partial Differential Equations
• Probability/Statistics
• Advanced Linear Algebra
• A course in programming.
In addition, a course in Complex Analysis is 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.
Course Descriptions
Intro to Scientific Programming
Introduction to Programming with Matlab and C++. Fundamental concepts and skills of programming in a high-level language. Flow of control: selection, iteration, subprograms. Data structures: strings, arrays, records, lists, tables. Algorithms using selection and iteration (decision making, finding maxima/minima,searching, sorting, simulation, etc.) Good program design, structure and style are emphasized. Testing and debugging. The first part of the course is going to concentrate on Matlab. Then we move to C++ and
continue with C++ in the next term.
Prerequisites: Permission 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
method. Implement these methods in a computer language MATLAB).
Prerequisites: Permission of instructor.
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.
Calculus with Theory I
This course is the first in a two semester sequence designed as a rigorous introduction to the calculus. This class targets students that want a deep understanding of the theoretical under-pinnings of calculus and the ability to reprove the classical theorems of calculus. This course will cover considerably more detail than a regular calculus course and includes an introduction to writing proofs. The first semester will cover differential calculus with an in-depth look at limits, continuity, and differentiability as well as applications such as optimization and linear approximation. We will complete the course by rigorously developing the Riemann integral and proving the fundamental theorem of calculus.
Prerequisites: Permission of Instructor.
Calculus with Theory II
This is a continuation of Calculus with Theory 1. This course will continue with techniques of integration, logarithms and exponential functions, infinite sequences and sums and power series. If time remains the course will touch on Fourier series.
Prerequisites: Calculus With Theory 1.
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.
Computational Ordinary Differential Equations
This course will focus on differential equations and computational methods using Matlab/Maple. It is intended for Mathematics and Science students who are going to apply these techniques in their coursework. Reflecting the shift in emphasis from traditional methods to new computer-based methods, we will focus on the mathematical modeling of real-world phenomena as the goal and constant motivation for the study of differential equations. Topics covered include some material on complex numbers and matrix algebra, first and second order linear and non-linear systems with applications, introductory
numerical methods, and Laplace transform techniques.
Prerequisites: Calculus I and II.
Probability*
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.
Discrete Dynamical Modeling
An important problem in science is to predict the behavior of systems that change in time. Such systems are called dynamical systems. This course introduces students to a set of mathematical methods used to model dynamical systems. It focuses on discrete dynamical models in which time is viewed as a sequence of steps. Students will learn how to translate real world problem into mathematical equations and they also learn how to use mathematical and computational methods to analyze the problem and make prediction. Mathematical concepts on steady states, cycles and chaos will be introduced. Concrete examples will be drawn from biology when possible.
Prerequisite: Calculus
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).
Mathematical Biology
This course introduces the study of nonlinear interactions in biology and medicine. We consider physical problems which are well modeled by systems of coupled ordinary differential equations and develop techniques to obtain qualitative information about such systems. Mathematical concepts on nonlinear dynamics and chaos, qualitative and quantitative mathematical techniques as local and global stability theory, bifurcation analysis, phase plane analysis, and numerical simulation will be introduced. Concrete and detailed examples will be drawn from molecular, cellular and population biology and mammalian physiology.
Prerequisite: Calculus, Differential Equations. Programming experience preferred.
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.
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.
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.
Computational Fluid Mechanics
This course is an interdisciplinary introduction to Computational Fluid Mechanics. The course focuses on physical and mathematical foundations of computational fluid mechanics with emphasis on applications. We will consider solution methods for model equations and the Euler and the Navier-Stokes equations; the finite volume formulation of the equations; classification of partial differential equations and solution techniques; truncation errors, stability, conservation, and monotonicity. The main programming language is Matlab.
Prerequisites: Calculus III, Ordinary Differential Equations.
Computational Partial Differential Equations
This course will focus on applied partial differential equations and their computational methods. It is intended for math and science students who apply these techniques in their work. Topics we will consider include, but are not limited to heat, wave, and Laplace equation, harmonic functions, Fourier
series expansions, separation of variables, spherical and cylindrical Bessel functions, and Legendre polynomials. For each topic we will study numerical and computer algebra approaches with Matlab and Maple.
Prerequisites: Calculus III, Ordinary Differential Equations.
Mathematics Seminar
Math Seminar has been a traditional forum for students interested in mathematics. The purpose of this seminar is to cover many interesting or advanced topics in mathematics that cannot be titled under one subject. Students enrolled in this seminar are expected to present several lectures prepared under supervision of the math faculty.
Prerequisites: None
Click here for the General Catalog.
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