What exactly is differential geometry?

In summary: But the point is, you could use differential and integral calculus to find the best path.So in summary, differential geometry is used in problems where a straight line calculation wouldn't work, or in scenarios where we want to find the shortest path.
  • #1
U.Renko
57
1
Hello.
I'm new here and I'm not sure if I should post this topic here or in general math, or anywhere else. Feel free to move the topic elsewhere if needed.

Just a bit of explanation first.
I'm not from USA (so forgive any grammar errors) and I don't understand completely your academic system with "minor-major" "freshman-senior-junior" and the such. Therefore I find it a little hard to explain "how far I am", but I guess you could call me a college freshman in a physics school (if that is how you call it).
Just to give an idea we're currently studying linear algebra, starting with multivariable calculus and finishing introductory mechanics.

Anyways, next semester my college will be offering a course on differential geometry and I want to know:

- How useful it is for physics major?
- what skills are necessary?
- should I try it or wait some more?

Well, that`s basically it.

Thanks in advance.
 
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  • #3
U.Renko said:
Hello.
I'm new here and I'm not sure if I should post this topic here or in general math, or anywhere else. Feel free to move the topic elsewhere if needed.

Just a bit of explanation first.
I'm not from USA (so forgive any grammar errors) and I don't understand completely your academic system with "minor-major" "freshman-senior-junior" and the such. Therefore I find it a little hard to explain "how far I am", but I guess you could call me a college freshman in a physics school (if that is how you call it).
Just to give an idea we're currently studying linear algebra, starting with multivariable calculus and finishing introductory mechanics.

Anyways, next semester my college will be offering a course on differential geometry and I want to know:

- How useful it is for physics major?
- what skills are necessary?
- should I try it or wait some more?

Well, that`s basically it.

Thanks in advance.

Hello U.Renko and welcome to the forums.

Lets start off with something very basic, yet important that will help you understand this better: that is, the idea of distance.

In Cartesian geometry, it is very easy to measure the distance between two points: you use the pythagorean theorem in n dimensions.

But what happens when lines between points are not straight? You can't use the pythagorean theorem. What if you are trying to measure the difference between two points on a sphere like you do when you are measuring the distance between two cities on the opposite side of the world to each other? The distance is not a straight line distance, since you have to travel on the boundary of the sphere: you can't just go through the middle of the earth!

So how do you find the distance? Well if you do first year calculus, you learn about the concept of arc-length which allows you to find the length of a curve, and this is how you can use differential and integral calculus to find various properties of a geometry and also things like length with respect to a specific geometry, like for example a 2D sphere (like the earth) or maybe even a 3D sphere (you can't really picture that because it requires another dimension).

Now interestingly enough, relativity theory uses geometry that is not euclidean to describe space and time. Finding differences between different points is not as easy as doing a straight line calculation: it is a lot harder. But this is the theory that is used to describe these physical quantities, so physicists use this model, because so far it has been pretty good at being accurate.

With regard to other things, differential geometry can be used in optimization.

In many problems, we might want to minimize or maximize something that is dependent on a lot of constraints (mathematical rules). These things that we want to minimize or maximize might be represented with mathematical objects that are highly non-linear (think highly curved).

Now the problem may end up being a distance problem. We might have this really complicated mathematical object, it could even be in double digit dimensions. The problem will boil down to finding the shortest path from one point to the other given this highly complex object.

If we had a flat geometry, the answer is really simple: it's just the straight line path. But if its complicated and curvy, it's not so simple anymore!

I'll give you a concrete example.

Let's say you have someone that asks you what is the best route to work, and they give you a geometry that encodes the level of "resistance" when you travel along that path. You are given the geometry, and the two points, and you have to find the path that is the shortest since the shortest path will be the one that has the least amount of resistance.

You can model the above scenario in differential geometry. You could make it relatively simple or really complex by taking into account less or more variables, but the idea is still the same: minimize distance between points and find the path that gives us this solution. Also note that in some geometries, you can have more than one solution!

Now there is a lot more to differential geometry than this, but that should give you an idea of how it can be used.
 
  • #4
It looks interesting and useful indeed.
according to the other threads its not a vey basic subject, so it would probably be better to wait a little bit?

Anyways, thanks a lot for the explanations
 
  • #5
U.Renko said:
It looks interesting and useful indeed.
according to the other threads its not a vey basic subject, so it would probably be better to wait a little bit?

Anyways, thanks a lot for the explanations

You can still get some idea by looking at arc-length and its derivation to get an idea of finding the length of a path on some object. If you are curious you could derive the arc-length of various curves like distance between one point and another that both lie on a circle or an ellipse, and maybe if you are keen, to do it on a sphere. You don't have to find the shortest path, just use arc-length to find a path that you know what the answer will be "like a circular path", and that will help you see how calculus is used in this case.

It will get more complicated than this no doubt, but that doesn't mean that you can't get a good idea of what it is about.
 

1. What is the definition of differential geometry?

Differential geometry is a branch of mathematics that studies the properties of curves, surfaces, and other geometric objects using tools from calculus and linear algebra. It focuses on the study of smooth or differentiable manifolds, which are spaces that locally resemble Euclidean space but may have more complicated global structures.

2. What are the main applications of differential geometry?

Differential geometry has many applications in physics, engineering, and other branches of mathematics. Some examples include the study of general relativity, which uses the tools of differential geometry to describe the curvature of spacetime, and computer graphics, which uses differential geometry to model and render 3D shapes and surfaces.

3. How does differential geometry differ from other branches of geometry?

Differential geometry differs from other branches of geometry, such as Euclidean geometry or algebraic geometry, in that it focuses on the study of smooth or differentiable objects. This means that the objects being studied have well-defined tangent spaces and can be described using calculus techniques.

4. What are some key concepts in differential geometry?

Some key concepts in differential geometry include curvature, which measures how much a surface deviates from being flat, and geodesics, which are the paths of shortest distance on a curved surface. Other important concepts include metrics, which define distances and angles on a manifold, and connections, which describe how to differentiate vector fields on a manifold.

5. What are some famous theorems in differential geometry?

Some famous theorems in differential geometry include the Gauss-Bonnet theorem, which relates the curvature of a surface to its topology, and the fundamental theorem of surfaces, which states that every smooth surface can be locally parameterized by a map from a 2D plane to 3D space. Other notable results include the Nash embedding theorem, which shows that any Riemannian manifold can be isometrically embedded in a higher-dimensional Euclidean space, and the Poincaré conjecture, which was famously proved by Grigori Perelman in 2002.

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