PDEs, Manifolds & Frobenius: Intro Course Insights

In summary, PDEs are mathematical equations used to model physical phenomena, while manifolds are geometric objects used to study spaces and their properties. PDEs and manifolds are related as PDEs can be used to study the behavior of manifolds over time. The Frobenius theorem is a fundamental result in differential geometry that states a smooth distribution of vector fields on a manifold can be integrated into a foliation. These concepts have various applications in fields such as physics, engineering, and economics, and can help in solving real-world problems.
  • #1
redrzewski
117
0
I'm looking to delve into PDEs. I'm reading thru Lee's Smooth Manifolds, and he has a chapter on integral manifolds, and how they relate to PDE solutions via Frobenius' theorem. I find the hint of geometrical aspects very appealing.

Evans' PDE book (that I was planning on picking up) doesn't seem to mention these geometrical aspect of PDEs - no manifolds, frobenius, differential forms in the index. For instance, going by the index, Taylor does.

https://www.amazon.com/dp/0387946543/?tag=pfamazon01-20

Yet many of the reviews I've read, and browsing course homepages, tend to favor Evans' treatment for intro PDEs.

Can anyone give me insight into this? Are the geometrical aspects of PDEs via manifolds not that compelling/useful? Or is it too much additional complexity for an intro course? Or what?

thanks
 
Last edited by a moderator:
Physics news on Phys.org
  • #2

I can definitely understand your interest in exploring the geometrical aspects of PDEs. Manifolds and differential forms can provide a powerful and elegant framework for studying PDEs, and Frobenius' theorem is a key tool in this approach. However, I can also understand your confusion about why these aspects are not mentioned in Evans' PDE book.

Firstly, I want to assure you that the geometrical approach to PDEs is indeed useful and compelling. In fact, many advanced research areas in PDEs heavily rely on this framework. However, it may not be the most suitable approach for an introductory course.

Evans' book is widely favored for introductory PDE courses because it strikes a balance between rigor and accessibility. It provides a solid foundation in the theory of PDEs, with a focus on the analytical and computational aspects. This makes it a great resource for students who are just starting out in PDEs and need a strong understanding of the fundamentals.

On the other hand, books like Lee's Smooth Manifolds and Taylor's Introduction to PDEs may be more suitable for students who have a strong background in geometry and topology, and are looking for a deeper understanding of the geometrical aspects of PDEs. These books tend to be more advanced and may not be the best choice for an introductory course.

In summary, while the geometrical approach to PDEs is certainly valuable, it may not be the most suitable for an introductory course. I would recommend starting with Evans' book to build a strong foundation in PDE theory, and then exploring the geometrical aspects in more depth as you progress in your studies.

I hope this helps clarify your doubts. Best of luck in your exploration of PDEs!
 

Related to PDEs, Manifolds & Frobenius: Intro Course Insights

1. What are PDEs?

PDEs, or partial differential equations, are mathematical equations that involve multiple variables and their partial derivatives. They are used to model and describe physical phenomena in various fields such as physics, engineering, and economics.

2. What are manifolds?

Manifolds are geometric objects that can be thought of as curved spaces. In mathematics, they are used to study the properties and structures of spaces, such as surfaces, curves, and higher-dimensional spaces.

3. How are PDEs and manifolds related?

PDEs can be used to study and describe the behavior of manifolds. In particular, they are used to describe the evolution of geometrical objects on a manifold over time.

4. What is the Frobenius theorem?

The Frobenius theorem is a fundamental result in differential geometry that states that a smooth distribution of vector fields on a manifold can be integrated into a foliation, which is a partition of the manifold into smooth submanifolds. It is an important tool in the study of manifolds and their properties.

5. How can I apply the concepts of PDEs, manifolds, and the Frobenius theorem in my research?

The concepts of PDEs, manifolds, and the Frobenius theorem have applications in various fields such as physics, engineering, and economics. They can be used to model and analyze complex systems, study the behavior of physical phenomena, and solve real-world problems. For example, PDEs are used in fluid dynamics to describe the motion of fluids, and manifolds are used in computer vision to analyze and classify images. The Frobenius theorem can help in understanding the structure and dynamics of complex systems and can aid in developing efficient algorithms and methods for solving problems.

Similar threads

Deciding Between DG & PDE Graduate Math Classes
    • Differential Geometry
Replies
14
Views
3K
  • Poll
Geometry A Comprehensive Introduction to Differential Geometry series by Spivak
    • Science and Math Textbooks
Replies
15
Views
19K

Hot Threads

Recent Insights

Back
Top