Editing Linear algebra (section)

=== Fluid mechanics, fluid dynamics, and thermal energy systems ===
<ref>{{Cite web|title= MIT OpenCourseWare. Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations - Mechanical Engineering |url= https://ocw.mit.edu/courses/2-035-special-topics-in-mathematics-with-applications-linear-algebra-and-the-calculus-of-variations-spring-2007/}}</ref><ref>{{Cite web|title= FAMU-FSU College of Engineering. ME Undergraduate Curriculum |url= https://engineering.ucdenver.edu/electrical-engineering/research/energy-and-power-systems#:~:text=Power%20systems%20analysis%20deals%20with,the%20analysis%20of%20power%20systems}}</ref><ref>{{Cite web|title= University of Colorado Denver. Energy and Power Systems |url= https://eng.famu.fsu.edu/me/undergraduate-curriculum#:~:text=MAS%203105%20Linear%20Algebra%20%283%29,and%20eigenvectors%2C%20linear%20transformations%2C%20applications)}}</ref>

Linear algebra, a branch of mathematics dealing with [[vector spaces]] and [[linear mapping]]s between these spaces, plays a critical role in various engineering disciplines, including [[fluid mechanics]], [[fluid dynamics]], and [[thermal energy]] systems. Its application in these fields is multifaceted and indispensable for solving complex problems.

In [[fluid mechanics]], linear algebra is integral to understanding and solving problems related to the behavior of fluids. It assists in the modeling and simulation of fluid flow, providing essential tools for the analysis of [[fluid dynamics]] problems. For instance, linear algebraic techniques are used to solve systems of [[differential equations]] that describe fluid motion. These equations, often complex and [[non-linear]], can be linearized using linear algebra methods, allowing for simpler solutions and analyses.

In the field of fluid dynamics, linear algebra finds its application in [[computational fluid dynamics]] (CFD), a branch that uses [[numerical analysis]] and [[data structure]]s to solve and analyze problems involving fluid flows. CFD relies heavily on linear algebra for the computation of fluid flow and [[heat transfer]] in various applications. For example, the [[Navier–Stokes equations]], fundamental in [[fluid dynamics]], are often solved using techniques derived from linear algebra. This includes the use of [[Matrix (mathematics)|matrices]] and [[Vector (mathematics and physics)|vectors]] to represent and manipulate fluid flow fields.

Furthermore, linear algebra plays a crucial role in [[thermal energy]] systems, particularly in [[power systems]] analysis. It is used to model and optimize the generation, [[Electric power transmission|transmission]], and [[Electric power distribution|distribution]] of electric power. Linear algebraic concepts such as matrix operations and [[eigenvalue]] problems are employed to enhance the efficiency, reliability, and economic performance of [[power systems]]. The application of linear algebra in this context is vital for the design and operation of modern [[power systems]], including [[renewable energy]] sources and [[smart grid]]s.

Overall, the application of linear algebra in [[fluid mechanics]], [[fluid dynamics]], and [[thermal energy]] systems is an example of the profound interconnection between [[mathematics]] and [[engineering]]. It provides engineers with the necessary tools to model, analyze, and solve complex problems in these domains, leading to advancements in technology and industry.