Understanding Fields: How Can They Be Represented and Applied?

[nanoWatt]

I've been wanting to catch back up on my physics and math. I was looking into equations on E&M, which led me to electric fields, which led me through differentiation, and on.

I am looking up on Wikipedia what a mathematical field is. Well, it's an algebraic structure. It's also a ring. A ring has more structure than an abelian group, but less than a field. A field is not just a ring, but a commutative division ring.

And so on. Now we have structures, fields, rings, abelian groups, and properties of commutativity to them.

I feel I'm going in circles. I just want to know what a field is, and if it's the same as a magnetic/electric field. How can I grasp this abstract math? My brain can't seem to grasp it.

I know a set is a group of numbers, like {1,2,3}
I guess this could be a 1x3 matrix too, and in computers it's an array.

How can this object called a "set" be represented as a field, structure, ring, abelian group, etc?

Isn't a field like a 2d or 3d (or n-dimensional) surface of points?

 

[Vid]

You have two different definitions of field here. The algebraic structure called a field is just an abstraction of the real numbers. An electric or magnetic field is a vector field. They are complety unrelated concepts.
http://en.wikipedia.org/wiki/Vector_field

 

[nanoWatt]

Ok, then I guess my next question is, how can I learn something when one thing leads to another, on and on? I'm thinking that Wikipedia isn't a good way to learn a new subject.

Even that link you sent says "In the rigorous mathematical treatment, (tangent) vector fields are defined on manifolds as sections of a manifold's tangent bundle. They are one kind of tensor field on the manifold."

Now I have more terms I don't understand:
1. rigorous mathematical treatment
2. manifolds
3. sections
4. tangent bundle
5. tensor field

So, it really didn't help me understand, although I do understand fields to be collections of vectors. However, wikipedia seems to go on forever.

You have two different definitions of field here. The algebraic structure called a field is just an abstraction of the real numbers. An electric or magnetic field is a vector field. They are complety unrelated concepts.
http://en.wikipedia.org/wiki/Vector_field

 

[Vid]

http://tutorial.math.lamar.edu/Classes/CalcIII/VectorFields.aspx

This is a pretty good site if you need to refresh on any of calculus. I would suggest getting a textbook instead of relying on wikipedia for learning.

 

[nanoWatt]

Thanks. I sent an application to a community college to audit a Calculus I class. I think that would be a good refresher. I also ordered an E&M textbook off Amazon. Since I have a degree in Physics (7 years old), I have covered this before.

Anyway, when I get sufficient knowledge from the Electricity and Magnetism then I'll be at the level where I can go for my masters in Physics.

Good advice though on getting the textbook.