PDE or differentiable manifolds?

In summary: Differentiable manifolds sounds like a really interesting subject, would you recommend it to somebody who is doing a math masters ?? (Assuming that they are not familiar with these concepts)Differentiable manifolds sounds like a really interesting subject, would you recommend it to somebody who is doing a math masters ??Depends on what somebody is looking for, if somebody just wants to know about the subject and doesn't want to get into too much details then I would say go for it, if somebody is looking for a deeper understanding then I would say differential geometry or linear algebra are better courses.
  • #1
mathmari
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Hello! :eek:

I am doing my master in the field Mathematics in Computer Science. I am having a dilemma whether to take the subject Partial differential equations- Theory of weak solutions or the subject differentiable manifolds.

Could you give me some information about these subjects?? (Wondering)

What knowledge is required?? (Wondering)
 
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  • #2
Hi mathmari,

I don't really know what you could do in PDE's but I had a course called "differentiable manifolds" also, at that course I was supposed to have two linear algebra courses, two differential calculus course, two integral calculus courses, introduction to ODE's, introduction to PDE's, topology, curves and surfaces geometry and a course called "Global theory of surfaces".

In fact I don't remember too much about that course but I remember it was really interesting. I was taught about "inmersions and submersions" (direct translation from Spanish ), bundles, integral curves, local flows, lies derivatives , Frobenius theorem (do not confound with Rouche-Frobenius) and this kind of things.

As always it will depend on who, what and how your professor explain but in my case I remember it was a hard course. (Which is obviously good :p)
 
  • #3
Fallen Angel said:
I don't really know what you could do in PDE's but I had a course called "differentiable manifolds" also, at that course I was supposed to have two linear algebra courses, two differential calculus course, two integral calculus courses, introduction to ODE's, introduction to PDE's, topology, curves and surfaces geometry and a course called "Global theory of surfaces".

I haven taken subjects like multivariable analysis, differential geometry, ordinary differential equations, linear algebra, introduction to ODE/PDE, etc.

But I haven't taken subjects like topology...

Do you think that these subjects are a good backround for the subject "Differentiable Manifolds" ?? (Wondering)
 
  • #4
Hi mathmari,

I just told you what I was supposed to know, of course differential geometry is essential but topology ... I just remember to have used a couple of simple facts at the begginning and maybe some isolated theorem in advance, I mean, if you know the basics you don't need a course on topology, but probably will need to read about some result you didn't know (or maybe not).
 

1. What is a PDE?

A PDE, or partial differential equation, is a type of mathematical equation that involves multiple independent variables and their partial derivatives. It is commonly used to describe physical phenomena in various fields such as physics, engineering, and finance.

2. What is the difference between ordinary and partial differential equations?

The main difference between ordinary differential equations (ODEs) and partial differential equations (PDEs) is the number of independent variables involved. ODEs only involve one independent variable, while PDEs involve multiple independent variables and their partial derivatives.

3. What are some applications of PDEs in science?

PDEs have numerous applications in various scientific fields. They are often used to describe physical processes such as heat transfer, fluid dynamics, and quantum mechanics. They are also used in image and signal processing, finance, and population dynamics.

4. What is a differentiable manifold?

A differentiable manifold is a mathematical concept that generalizes the idea of a smooth surface in higher dimensions. It is a topological space that is locally similar to Euclidean space, and can be described using a set of coordinate charts and transition maps.

5. How are PDEs and differentiable manifolds related?

PDEs can be used to describe the behavior of differentiable manifolds. For example, PDEs can be used to describe the flow of a fluid on a manifold, or the propagation of a wave on a curved surface. Additionally, the study of PDEs on manifolds has led to the development of geometric analysis, which uses differential geometry to study PDEs and their solutions.

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