Mathematical engineering (or engineering mathematics) is a branch of applied mathematics, concerning mathematical methods and techniques that are typically used in engineering and industry. Along with fields like engineering physics and engineering geology, both of which may belong in the wider category engineering science, engineering mathematics is an interdisciplinary subject motivated by engineers' needs both for practical, theoretical and other considerations outside their specialization, and to deal with constraints to be effective in their work.
Historically, engineering mathematics consisted mostly of applied analysis, most notably: differential equations; real and complex analysis (including vector and tensor analysis); approximation theory (broadly construed, to include asymptotic, variational, and perturbative methods, representations, numerical analysis); Fourier analysis; potential theory; as well as linear algebra and applied probability, outside of analysis. These areas of mathematics were intimately tied to the development of Newtonian physics, and the mathematical physics of that period. This history also left a legacy: until the early 20th century subjects such as classical mechanics were often taught in applied mathematics departments at American universities, and fluid mechanics may still be taught in (applied) mathematics as well as engineering departments. [1]
The success of modern numerical computer methods and software has led to the emergence of computational mathematics, computational science, and computational engineering (the last two are sometimes lumped together and abbreviated as CS&E), which occasionally use high-performance computing for the simulation of phenomena and the solution of problems in the sciences and engineering. These are often considered interdisciplinary fields, but are also of interest to engineering mathematics. [2]
Specialized branches include engineering optimization and engineering statistics.
Engineering mathematics in tertiary education typically consists of mathematical methods and models courses. [3]
Discrete mathematics is the study of mathematical structures that can be considered "discrete" rather than "continuous". Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no exact definition of the term "discrete mathematics".
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.
Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, and maintain mechanical systems. It is one of the oldest and broadest of the engineering branches.
The following outline is provided as an overview of and topical guide to physics:
Computational physics is the study and implementation of numerical analysis to solve problems in physics. Historically, computational physics was the first application of modern computers in science, and is now a subset of computational science. It is sometimes regarded as a subdiscipline of theoretical physics, but others consider it an intermediate branch between theoretical and experimental physics — an area of study which supplements both theory and experiment.
An academic discipline or field of study is a branch of knowledge, taught and researched as part of higher education. A scholar's discipline is commonly defined by the university faculties and learned societies to which they belong and the academic journals in which they publish research.
Theoretical computer science is a subfield of computer science and mathematics that focuses on the abstract and mathematical foundations of computation, such as the theory of computation, formal language theory, the lambda calculus and type theory.
Computer science and engineering (CSE) is an academic program at many universities which comprises approaches of computer science and computer engineering. There is no clear division in computing between science and engineering, just like in the field of materials science and engineering. However, some classes are historically more related to computer science, and other to computer engineering. CSE is also a term often used in Europe to translate the name of technical or engineering informatics academic programs. It is offered in both undergraduate as well postgraduate with specializations.
Lists of mathematics topics cover a variety of topics related to mathematics. Some of these lists link to hundreds of articles; some link only to a few. The template to the right includes links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a manner better suited for browsing. Lists cover aspects of basic and advanced mathematics, methodology, mathematical statements, integrals, general concepts, mathematical objects, and reference tables. They also cover equations named after people, societies, mathematicians, journals, and meta-lists.
Computational science, also known as scientific computing, technical computing or scientific computation (SC), is a division of science that uses advanced computing capabilities to understand and solve complex physical problems. This includes
Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics. Before the emergence of computational science as a "third way" besides theoretical and experimental sciences, computational mechanics was widely considered to be a sub-discipline of applied mechanics. It is now considered to be a sub-discipline within computational science.
Computational Engineering is an emerging discipline that deals with the development and application of computational models for engineering, known as Computational Engineering Models or CEM. Computational engineering uses computers to solve engineering design problems important to a variety of industries. At this time, various different approaches are summarized under the term Computational Engineering, including using computational geometry and virtual design for engineering tasks, often coupled with a simulation-driven approach In Computational Engineering, algorithms solve mathematical and logical models that describe engineering challenges, sometimes coupled with some aspect of AI, specifically Reinforcement Learning.
Computational mathematics is the study of the interaction between mathematics and calculations done by a computer.
The branches of science, also referred to as sciences, scientific fields or scientific disciplines, are commonly divided into three major groups:
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models.
Mathematics is a broad subject that is commonly divided in many areas that may be defined by their objects of study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the study of natural numbers.
The TUM School of Computation, Information and Technology (CIT) is a school of the Technical University of Munich, established in 2022 by the merger of three former departments. As of 2022, it is structured into the Department of Mathematics, the Department of Computer Engineering, the Department of Computer Science, and the Department of Electrical Engineering.
