"The topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes."
The study of numbers, quantities, and shapes, and their relationships.
Calculus: The study of rates of change and accumulation, used extensively in chemical engineering for modeling processes and optimization.
Linear algebra: The study of linear equations and matrices, used in chemical engineering for solving systems of equations and analyzing systems of linear equations.
Differential equations: The study of equations involving derivatives, used in chemical engineering for modeling dynamic behavior and systems of chemical reactions.
Probability and statistics: The study of random events and data analysis, used in chemical engineering for process control, quality control, and risk assessment.
Thermodynamics: The study of energy and its transformations, used in chemical engineering for designing and analyzing systems involving heat, work, and mass transfer.
Fluid mechanics: The study of the mechanics of fluids, used in chemical engineering for designing and analyzing systems involving fluid flow and transport.
Materials science: The study of the properties and behavior of materials, used in chemical engineering for designing and selecting materials for various applications.
Process control: The study of how to control industrial processes, used in chemical engineering for optimizing processes and ensuring consistency and safety.
Transport phenomena: The study of mass, heat, and momentum transfer, used in chemical engineering for designing and analyzing processes involving fluid- or gas-phase reactions and separations.
Optimization: The study of how to maximize or minimize a specific objective function, used in chemical engineering for designing and optimizing processes to meet specific objectives.
Algebraic equations: Which are used to represent chemical reactions and their stoichiometry.
Differential equations: Which describe changes in concentration, temperature, and pressure within a reactor, or any process system.
Linear algebra: Widely used in modeling and solving linear systems of equations found in chemical engineering problems, including mass balance and energy balance equations.
Vector calculus: Used to describe fluid flow and transport phenomena in chemical engineering systems.
Probability & Statistics: Used to describe the uncertainty and variability of observed events, values, and measurements. It is also used to build mathematical models of experimental data and to test the validity of the models.
Optimization: Used to maximize/minimize an objective function within a given set of constraints. It's extensively utilized in chemical engineering applications, such as chemical production and process control.
Partial Differential Equations: Used to describe systems involving multi-dimensional space and time, primarily in the field of transport phenomena, including heat transfer, mass transfer, and fluid dynamics.
Numerical Analysis: Used to solve a wide range of mathematical problems and equations that are difficult to solve analytically.
Control systems: Used to control the behavior of a chemical process, such as temperature, pressure, and flow rate.
Graph theory: Used to model and analyze complex systems, including chemical processes and networks.
Game theory: Used for analyzing the decision-making processes among multiple players or even teams, in terms of selecting the best process or technology for a given project.
Computational science: Coupled with numerical methods, used to create simulations of chemical processes that are either too complex or impossible to observe experimentally.
"Number theory, algebra, geometry, and analysis."
"No general consensus among mathematicians about a common definition."
"The discovery of properties of abstract objects and the use of pure reason to prove them."
"A proof consists of a succession of applications of deductive rules to already established results."
"Independent from any scientific experimentation."
"The natural sciences, engineering, medicine, finance, computer science, and the social sciences."
"Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications."
"Other areas are developed independently from any application (and are therefore called pure mathematics), but often later find practical applications."
"The problem of integer factorization, for example, which goes back to Euclid in 300 BC, had no practical application before its use in the RSA cryptosystem, now widely used for the security of computer networks."
"Historically, the concept of a proof and its associated mathematical rigor first appeared in Greek mathematics, most notably in Euclid's Elements."
"Mathematics was essentially divided into geometry and arithmetic (the manipulation of natural numbers and fractions)."
"The 16th and 17th centuries."
"The interaction between mathematical innovations and scientific discoveries has led to a rapid lockstep increase in the development of both."
"The foundational crisis of mathematics at the end of the 19th century."
"It heralded a dramatic increase in the number of mathematical areas and their fields of application."
"More than 60 first-level areas of mathematics."
"Mathematics is extensively used for modeling phenomena."
"Certain properties called axioms."
"Some basic properties that are considered true starting points of the theory under consideration."