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## What is an Applied Mathematics Degree?

Mathematics helps companies perform better in our data-driven marketplace. That is the foundation of applied mathematics. A career in the field is about more than just crunching numbers.

Applied mathematicians and computational scientists use theories and techniques, such as mathematical modeling and computational methods, to formulate and solve practical problems in business, government, engineering, and the physical, life, and social sciences. They may find themselves exploring questions like these: How can an airline use smarter scheduling to reduce costs of aircraft parking? Can biofuel production be optimized to protect the world’s environment and economy? How can one design a detailed plan for a clinical trial? Can we measure sentiment change as a result of social media shares, likes, and comments?

In short, the field of applied mathematics brings math to life. It allows us to better understand the world around us. And to that end, its students learn to manipulate abstract objects and ideas, translate real-life data into mathematical language, and understand fundamental concepts of real and complex analysis.

## Program Options

**Bachelor’s Degree in Applied Mathematics – Four Year Duration**

At the bachelor’s level, the applied mathematics major offers a program in mathematical techniques and ways of thinking. It focuses on mathematical and computational methods applicable in the sciences, engineering, and industry.

Here is a snapshot of an undergraduate curriculum in applied mathematics:

- Computing in MATLAB – an introduction to MATLB, a multi-paradigm programming language and numeric computing environment which allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages
- Applications in MATLAB – an introduction to some particular applications using MATLAB; students will reinforce their programming skills as they are exposed to problems where computation is required
- Calculus I – an introductory course in the theory and applications of calculus; topics include functions, limits and continuity, differentiation of algebraic, trigonometric, exponential, and logarithmic functions, curve sketching, optimization, and L’Hôpital’s rule (a technique to evaluate limits of indeterminate forms)
- General Physics I – mechanics, gravitation, energy conservation, fluids, and waves; discussion of biological applications
- Calculus II – a second course in the theory and applications of calculus; topics include the definite integral and the Fundamental Theorem of Calculus, applications of the definite integral, techniques of integration, improper integrals, and sequences and series including power and Taylor series
- General Physics II – electricity and magnetism; optics; atomic, molecular, and nuclear physics; discussion of biological applications
- Calculus III – a third course in the theory and applications of calculus; topics include vectors in two or three dimensions, lines and planes in space, parametric equations, vector functions and their derivatives, functions of several variables, partial derivatives, multiple integrals, line integrals, and Green’s theorem, Divergence theorem, and Stokes’ theorem
- Basic Concepts of Math – an introduction to mathematical abstraction and the language of mathematical proof; topics include logic, sets, integers, induction and modular arithmetic, functions, and cardinality
- Differential Equations with Linear Algebra – a course in ordinary differential equations that focuses on the use of linear algebra
- Probability Theory – counting techniques, axiomatic definition of probability, conditional probability, independence of events, Bayes’ theorem, random variables, discrete and continuous probability distributions, expected values, moments and moment generating functions, joint probability distributions, functions of random variables, covariance, and correlation

- Modern Algebra – a thorough introduction to the theory of groups and rings
- Numerical Analysis I – computer arithmetic, pitfalls of computation, iterative methods for the solution of a single nonlinear equation, interpolation, least squares, numerical differentiation, numerical integration, and solutions of linear systems by direct and iterative methods
- Real and Complex Analysis – real and complex number systems, completeness, sequences and series and their limits, continuity of real and complex functions, derivative, analytic functions, and power series
- Numerical Analysis II – solution of systems of nonlinear equations, solution of initial value problems, matrix norms and the analysis of iterative solutions, numerical solution of boundary value problems and partial differential equations, and introduction to the finite element method
- Theoretical Linear Algebra – vector spaces, linear transformations, determinants, eigenvalues and eigenvectors, canonical forms, inner product spaces, and bilinear forms
- Applied Mathematics – construction and study of mathematical models for physical, economic, and social processes
- Partial Differential Equations – the solution and properties of first and second order equations, heat and wave equation, elliptic boundary value problems and Green’s function, hyperbolic problems and the theory of characteristics, finite difference methods, the equations of mathematical physics
- Senior Problem Solving – mathematical discovery through problem solving; students will be expected to develop two or three areas of mathematics by solving assigned problems

**Master’s Degree in Applied Mathematics – Two Year Duration**

The applied mathematics master’s degree prepares students for careers in science, engineering, and business, where advanced methods in differential equations, nonlinear optimization, statistics, and computational mathematics play a significant role in technology development and innovation. At this level students choose a specialty, and coursework is customizable and goes far beyond foundational math. It is quite common for schools to offer both thesis and non-thesis options.

**Sample Core Courses**

- Functional Analysis
- Probability
- Numerical Analysis

**Sample Concentrations and Applicable Courses**

**Differential Equations**

- Intermediate Differential Equations
- Intermediate Partial Differential Equations
- Numerical Solutions of Partial Differential Equations
- Complex Analysis

**Optimization of Stochastic Systems** (for a system to be stochastic, one or more parts of the system has randomness associated with it; a stochastic system does not always produce the same output for a given input)

- Statistical Methods
- Stochastic Processes
- Nonlinear Optimization
- Stochastic Optimization

**Data Science**

- Numerical Linear Algebra for Big Data
- Mathematical Statistics
- Time Series Analysis
- Dynamic Programming and Reinforcement Learning

**Doctoral Degree in Applied Mathematics – Five Year Duration**

Doctoral programs in advanced mathematics are composed of coursework, independent study, and opportunities to conduct original, creative research. Graduates typically go to careers in academia and industrial research.

**Sample Courses**

- Applied Analysis
- Computational Mathematics
- Probability
- Discrete Applied Mathematics
- Mathematical Statistics

**Sample Research Areas and Topics**

- Mathematical Biology and Bioinformatics – detection of functional signals in biological sequences and protein networks
- Image and Vision Sciences – imaging science, computer vision, information theory, machine learning
- Mathematical Finance and Mathematical Economics – pricing of financial derivatives, portfolio selection, risk measure, credit risk, trading strategy, game theory (the branch of applied mathematics that provides tools for analyzing situations in which parties, called players, make decisions that are interdependent; this interdependence causes each player to consider the other player’s possible decisions in formulating strategy; game theory is sometimes referred to as the science of strategy)
- Numerical Analysis and Computing – numerical linear algebra, computational fluid dynamics, computational materials science
- Mathematical Modeling – for the control of forest fires
- Optimization – optimal harvesting of natural resources
- Statistics and Big Data Analysis – applications to biology, medicine, finance, and the energy industry
- Control of Mechanical Systems – by moving coordinates and locomotion in fluids

## Degrees Similar to Applied Mathematics

**Accounting**

Degree programs in accounting prepare students for the work of gathering, recording, analyzing, interpreting, evaluating, and communicating financial information. This includes examining accounting records, reconciling accounts, preparing financial reports, and completing tax returns. The typical curriculum includes classes in mathematics, business management, business communication, business research, finance, and economics.

**Actuarial Science**

This degree program provides students with in-depth training in mathematics, statistics, and probability. It teaches the use of models in analyzing and solving financial problems and includes coursework in economics, finance, accounting, and computer science.

**Civil Engineering**

This degree field is focused on the processes of design and planning of civil infrastructure like roads, tunnels, bridges, dams, railroads, and airports. In their work, civil engineers are concerned with such things as how much weight a structure can support and the environmental issues presented by construction. The emphasis of civil engineering degree programs is math, statistics, engineering systems and mechanics, building codes, and statistical analysis.

**Computer Science**

The field of computer science is focused on computer systems and how humans interact with them. Courses cover mathematics for computer science, artificial intelligence, data structures and algorithms, and introduction to program design.

**Engineering Physics**

Students of engineering physics learn how to use physics to solve practical problems. For this reason, the field is sometimes referred to as the bridge between physics and engineering. Coursework includes computational physics, materials science, thermodynamics, and nanotechnology.

**Industrial Engineering**

Industrial engineering majors learn how to improve the way that industries and organizations, such as hospitals and factories, operate. They draw on their knowledge in math, science, business, and psychology to consider factors like materials, equipment, and people.

**Management Information Systems**

This degree field is focused on information systems and how they are used by businesses and organizations to improve their operations. Classes cover computer databases, networks, computer security, and related project management.

**Mathematics**

Degree programs in mathematics typically teach both the theory and abstract of pure mathematics and its practical application to the world, known as applied mathematics. In other words, math majors study algebra, geometry, calculus, and statistics; but most pair this mathematics concentration with classes that reveal how math concepts are used in business management, computer science, economics, finance, music, philosophy, physics, and sports science.

**Simulation Programming**

Simulation programmers develop computer *simulations* that allow us to predict, see, think about, test, and manipulate real-world products, services, systems, processes, conditions, situations, and issues, without taking the risk and incurring the costs of doing so *in* the real world. Math, engineering, and computer science are the overlapping disciplines that simulation relies on.

Degree programs in the field are made up of courses in these technical and scientific areas, but they are also focused on teaching the skills of abstracting, theorizing, hypothesizing, and intellectualizing. In other words, simulation programming students learn everything they need to conceptualize the world into models that are designed to reach solutions to many of the world’s challenges and problems.

**Statistics**

The degree field of statistics is focused on the study of probability theory and sampling theory. Students use techniques like sample survey theory and variance analysis (the quantitative investigation of the difference between actual and planned behavior) to examine the relationships between groups and measurements. In simple terms, statistics is about collecting data, organizing it, analyzing it, and interpreting it in practical ways that guide decision making in both business sectors and politics.

## Skills You’ll Learn

Thinking mathematically involves four fundamental processes:

- Specializing – looking at examples and special cases
- Generalizing – looking for patterns and relationships
- Conjecturing – predicting relationships and results
- Convincing – finding and communicating reasons why something is true

It is not difficult to see how these processes can be applied to almost any field of work and how they leave applied mathematics graduates with the following set of transferable skills:

- Analytical Thinking
- Critical Thinking
- Problem Solving
- Quantitative Reasoning
- Ability to manipulate precise and intricate ideas
- Ability to construct logical arguments and expose illogical arguments
- Communication
- Time Management
- Teamwork
- Independence

## What Can You Do with an Applied Mathematics Degree?

Application of mathematical modeling and computational methods is a key process in many fields.

These are some sectors in which it is becoming an increasingly important tool:

- Systems Biology
- Data Mining and Data Privacy
- Materials Science
- Computer Animation and Digital Imaging
- Finance and Economics
- Ecology / Epidemiology / Environment
- Climatology

Here are examples of organizations that hire applied mathematics graduates:

- Academic institutions and research institutes
- Aerospace and transportation equipment manufacturers or service providers
- Analytics and forecasting organizations
- Chemical / pharmaceutical manufacturers
- Communications services providers
- Computer information and software firms (established or start-ups)
- Consumer products companies
- Energy systems firms
- Electronics and computer manufacturers
- Engineering research organizations
- Financial service and investment management firms
- Government labs, research offices, and agencies
- Insurance companies
- Medical device companies
- Producers of petroleum and petroleum products

Titles held by those working in applied mathematics include:

- Research Scientist
- Data Scientist
- Operations Researcher / Analyst
- Quantitative Analyst
- Investment Fund Manager
- Financial Analyst
- Actuary
- Research and Development Engineer
- Modeling Engineer
- Mathematician / Statistician
- Mathematics / Science Teacher
- Mathematics Professor