Geometrical methods of mathematical physics

In summary: However, if you tried to find a vector field that had no zero at all, you would eventually hit a point where the tangent vector and the vector field both have the same magnitude, and at that point, the vector field would be zero.
  • #1
pmb_phy
2,952
1
I'm reading Schutz's text Geometrical methods of mathematical physics right now as a part of a diredcted study (Two of my former math professors and I wish to learn the subject together so we formed a group for self-study). I know the material in a general sense but not at the precision of this wonderful text. But there is a comment that I don't understand on page 39 which, regarding the tangent-bundle on a 2-sphere, states that
...there is no Cinf vector field on S2 which is nowhere zero. This is a consequence of the famous but difficult fixed-point theorem of the sphere, that every 1-1 map (difffeomorphism) of S2 onto itself leaves one point fixed, provided that the map is a member of a continuous family of maps containing the identity map. A nowhere-zero vector field would generate such a map with no fixed point, ...
Okay. I may be burned out on this stuff by now but frankly I don't see how they reach the conclusion that there is no Cinf vector field on S2 which is nowhere zero.

Help? :confused:

Pete :smile:
 
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  • #2
That's known as the "hairy ball" theorem. TIf you imagine a ball (or someone's head) completely covered with hair, there must be a "whorl" or a place where there is no hair. Basically, if you start with a continuous, non-zero, vector field at a point and follow along both directions of great circles around the ball, at some point, the directions will conflict.

If you are asking "how they reach the conclusion that there is no Cinf vector field on S2 which is nowhere zero" in that quote, they don't, they simply assert it. If memory serves, that is a difficult theorem to prove.
 
  • #3
Likewise consider the atmosphere of the earth; the theorem states that at any given moment, somewhere on earth, the wind is calm.

The reason is basically that if you had a vector field with no zero, you could use it to construct a shrinking of the sphere, in itself, to a point, and this is impossible; the 2-sphere is o-connected (i.e you can get from any point on it to any other by a continuous path). And it is 1-connected (i.e you can shrink any closed curve, in the sphere, to a point), but it is not 2-connected.
 
  • #4
A nowhere-zero vector field would generate such a map with no fixed point, ...
It's saying that you find some way to use your vector field to define a family of continuous maps from the sphere to itself. You could imagine the vector field as describing something sophisticated like a flow... or something simpler like "this point moves that-a-way".

Because the sphere is compact, that guarantees you can find a parameter sufficiently small so that the only way a point could be mapped to itself is if the corresponding tangent vector at that point is zero.
 

1. What are geometrical methods of mathematical physics?

Geometrical methods of mathematical physics refer to the use of geometric concepts and techniques to study and understand physical phenomena. This includes the application of differential and integral calculus, vector analysis, and other mathematical tools to describe and analyze physical systems.

2. What are some examples of geometrical methods used in mathematical physics?

Some common examples of geometrical methods used in mathematical physics include the use of vector calculus to describe the motion of particles in space, differential geometry to study the curvature of space-time in general relativity, and group theory to understand symmetries in physical systems.

3. How are geometrical methods different from analytical methods in mathematical physics?

Geometrical methods in mathematical physics focus on the use of visual and geometric intuition to understand physical phenomena, while analytical methods rely on algebraic equations and mathematical models. Geometrical methods can often provide a more intuitive understanding of physical systems, while analytical methods are often more precise and rigorous.

4. What are the benefits of using geometrical methods in mathematical physics?

Geometrical methods can help to simplify complex physical problems and provide a visual representation of physical phenomena, making them easier to understand and analyze. They can also reveal underlying symmetries and patterns in physical systems, leading to new insights and discoveries.

5. How are geometrical methods used in real-world applications of mathematical physics?

Geometrical methods have a wide range of applications in the real world, such as in the study of fluid dynamics, electromagnetism, and quantum mechanics. They are used to model and predict the behavior of physical systems, as well as in the design and optimization of technologies such as aircrafts, computer graphics, and medical imaging devices.

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