Why Is Real Analysis Critical in Science and Engineering?

[-IamLittle-]

i like limit, continuity,differentiation in real analysis, they are interesting, but i don't know what is their importance?
And about lebesgue integration, i don't think it is interesting, and it seems it is useless

 

[uman]

It depends on what you mean by "important". If you mean "useful in other branches of mathematics", then absolutely. If you mean "worth learning because math is an interesting subject for its own sake", then it depends on whether you appreciate mathematics as an art.

If you mean "useful in one's daily life", then probably not, except for a tiny minority of professions.

 

[-IamLittle-]

i mean useful in other branches of mathematics

 

[uman]

Yes; for example, real analysis is of paramount importance in differential geometry, ODEs, PDEs, probability, statistics, and other subfields...

 

[rezeew]

Real analysis is a fundamental branch of mathematics that plays a crucial role in various fields of science and engineering. It is the study of the properties and behavior of real numbers, functions, and sequences. The concepts of limit, continuity, and differentiation are the building blocks of real analysis and are essential in understanding the behavior of functions and their applications.

Limit allows us to examine the behavior of a function as its inputs approach a particular value. This is important in understanding the behavior of natural phenomena and making predictions based on mathematical models.

Continuity is a fundamental concept in real analysis that helps us understand the smoothness of functions and their behavior over a given interval. This is essential in many areas of science, such as physics, where smooth and continuous functions are used to describe the behavior of physical systems.

Differentiation is a powerful tool in real analysis that allows us to study the rate of change of a function and its behavior at a specific point. This is crucial in fields such as economics, where the concept of marginal analysis is used to make decisions based on the rate of change of a function.

Lebesgue integration, while not as intuitive or widely used as other concepts in real analysis, is still an important tool in mathematics. It allows us to extend the concept of integration to a wider class of functions, making it a powerful tool in areas such as probability theory and functional analysis.

In conclusion, real analysis is a fundamental and important branch of mathematics that provides the foundation for many other areas of science and engineering. Its concepts and techniques are crucial in understanding and describing the behavior of natural phenomena and making predictions based on mathematical models. While some concepts may seem less interesting or useful at first glance, they still play a significant role in mathematics and its applications.