How Can a Grade 12 Student Progress to Advanced Geometry?

[mathsTKK]

Hi, everyone, I am new to Physic Forum, hope that member in physics forum can help me to overcome my difficulties in math^^
Thank you.

Here comes the problem:
I have a great interest in Einstein's Theory if Relativity, but I understand that theory requires deep understanding on Geometry.

I am now a Grade 12 student, so can someone tell me what is the pathway to advanced geometry? I want to learn full set of knowledge about geometry by myself, but I do not know how to start and where to start? Can someone help me to list out the topics in geometry step y step to advanced geometry such as manifolds?

I would appreciate if someone can help me out^^

Thank you very much to all!^^

 

[chiro]

Hi, everyone, I am new to Physic Forum, hope that member in physics forum can help me to overcome my difficulties in math^^
Thank you.

Here comes the problem:
I have a great interest in Einstein's Theory if Relativity, but I understand that theory requires deep understanding on Geometry.

I am now a Grade 12 student, so can someone tell me what is the pathway to advanced geometry? I want to learn full set of knowledge about geometry by myself, but I do not know how to start and where to start? Can someone help me to list out the topics in geometry step y step to advanced geometry such as manifolds?

I would appreciate if someone can help me out^^

Thank you very much to all!^^

The first you will have to do is to learn calculus, up to Calculus III.

Also you will to learn concepts in Euclidean Geometry. Things like projections, planes, metrics (distance functions), and angle.

When you're dealing with geometry the two most important things you deal with is distance and angle. You generally start off with a metric that describes distance between two points in the geometry. You've already learned the Euclidean metric as well as inner product that is used to define the angle. You should also learn that you can relate a metric to an inner product in any geometry as long as certain axioms are specified.

There are other things in geometry, but the distance and the angle are key things.

Another way to think of geometry is to think of a euclidean geometry in N dimensions, and then taking that geometric object (line, surface, volume, hypervolume etc) and basically "mould it like clay". So for a line, its like say a rope where you are bending it and twisting it and deforming it at will. The surface is like you are given a sheet of paper and you are doing the same thing with it folding it, twisting it, maybe even joining ends together. The volume is like having a block of clay and then you take that piece of clay and you stretch it and mould it like you did with the piece of paper.

One thing to note about the above is that when you are doing the "transforming" the geometric object has to keep its topological properties, that means that objects with holes, keep their holes with transformations. If your are in doubt about this a book on topology will clarify these things.

So I guess in conclusion, learn the basics first and then think about geometry in the topological terms I've said above. There is more but hopefully this will give you a nudge in the right direction.

 

[lavinia]

Hi, everyone, I am new to Physic Forum, hope that member in physics forum can help me to overcome my difficulties in math^^
Thank you.

Here comes the problem:
I have a great interest in Einstein's Theory if Relativity, but I understand that theory requires deep understanding on Geometry.

I am now a Grade 12 student, so can someone tell me what is the pathway to advanced geometry? I want to learn full set of knowledge about geometry by myself, but I do not know how to start and where to start? Can someone help me to list out the topics in geometry step y step to advanced geometry such as manifolds?

I would appreciate if someone can help me out^^

Thank you very much to all!^^

The Theory of Relativity relies on Differential Geometry in 4 dimensions. The prerequisite is multivariate calculus and highly useful after that would be a course in the classical differential geometry of 2 dimensional surfaces in three dimensional space. Then calculus on manifolds and finally four dimensional differential differential geometry.

Thorne and Wheeler have introductory books that show how geometry and gravity are intertwined with very little calculus at all. You could read these excellent books right now. The one I am reading is called Black Holes and they have another called Space Time Geometry which covers Special Relativity.

 

[WannabeNewton]

I have found that the best path is to study texts on tensor analysis on manifolds and modern differential geometry BEFORE going into GR books. The reason being that once you understand how pseudo - Riemannian manifolds work and how tensor fields work on such manifolds (along with some knowledge of topology) then the connections you make between the geometry and the physics become a lot easier to conceptualize.

 

[Fredrik]

There are many similar threads in the academic guidance section (possibly in the science book forum). I suggest you start by finding a few of those.

 

[mathwonk]

one such is: what is mathematics? by courant and robbins.

 

[mathsTKK]

It seems that it may e a long journey to go, but I am happy that finally I have a basic guidance to advanced geometry and GR. Should I have any other question and problems, I will be happy and glad to have all of you as a guidance^^

Once again, thanks to all of you!