Why is differential geometry not required in undergraduate physics curriculum?

In summary: Differential geometry can be quite important, and is often covered in graduate school. However, it is not typically covered in undergraduate school. A sound knowledge of the basics of analysis, vector spaces, and topology is necessary to study differential geometry. For instance, a differential manifold is a topological space with an atlas, which is a collection of open subset U_a, for a in A (possibly a finite index , possibly infinite) such that each U_a is homeomorphic to a neighbourhood of the origin R^n and such that it is well 'behaved on intesections' ot things like (U_a)n(U_b) {i don't think it helps to talk too much about
  • #1
leright
1,318
19
differential geometry...

I am constantly being told that this is important stuff for undergradautes to know about and it is taught by the end of the secon year...however, I am not going to be seeing ANY of this stuff at all during undergraduate school. And there isn't even a class offered in differential geometry. However, I was told my the mcs chair that he would set up a directed study in differential geometry since many other students are very interested in it.

Is this differential geometry THAT imporant in a ugrad physics curriculum, and if so, why doesn't my school require it?
 
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  • #2
Anyone have a comment?
 
  • #3
It CAN be very important- much of relativity is based on differential geometry just as much of quantum physics is based on group theory. It is, however, fairly advanced- I would consider analysis and abstract geometry prerequisites. Typically small colleges do not have regular differential geometry courses. Generally, you can, if necessary make it up in graduate school. I'm not crazy about most "individual" or "directed" study classes since the interaction with other students is an important part of a class but if there are several students who would be taking it- go for it!
 
  • #4
HallsofIvy said:
It CAN be very important- much of relativity is based on differential geometry just as much of quantum physics is based on group theory. It is, however, fairly advanced- I would consider analysis and abstract geometry prerequisites. Typically small colleges do not have regular differential geometry courses. Generally, you can, if necessary make it up in graduate school. I'm not crazy about most "individual" or "directed" study classes since the interaction with other students is an important part of a class but if there are several students who would be taking it- go for it!

would calc 1-3, diffEQ, linear algebra, prob/stats, adv. calc and adv. eng. math (similar to mathematical physics) be sufficient to study differential geometry?
 
  • #5
course names are not universal; as you indicate you university isn't teaching what you expected in other places, so why should these be any different?

in order to stfy differential geometry you should have a sound knowledge of the basicas of analysis (limits, differentiation of vector functions, integration), vector spaces (say to the point where you understand that the dual space of a finite dimensional vector space is isomorphic [but unnaturally] to its dual space), and topology with perticular regard to metric spaces.

for instance, a differential manifold is a topological space with an atlas, that is a collection of open subset U_a, for a in A (possibly a finite index , possibly infinite) such that each U_a is homeomorphic to a nieghbourhood of the origin R^n and such that it is well 'behaved on intesections' ot things like (U_a)n(U_b) {i don't think it helps to talk too much about this. we then can transfer the analytic properties of these nbds in R^n to the U_a by pullnig back via these maps.

i would be surprised if anywhere in the US taught this before the 3/4th year of undergraduacy if at all. In the Uk I'd have it as a third year course, possibly a second year one depending on how much detail they did it in. eslewhere in europe it could well be in the second year.
 
  • #6
matt grime said:
i would be surprised if anywhere in the US taught this before the 3/4th year of undergraduacy if at all. In the Uk I'd have it as a third year course, possibly a second year one depending on how much detail they did it in. eslewhere in europe it could well be in the second year.
Yea I just looked and at my school(in the US) it's a graduate course. I guess won't be taking it, or maybe I'll take it my last semester. I have a friend who did his undergraduate studies in Europe and he says it's a lot more work. Is this the case in general?
 

1. What is differential geometry?

Differential geometry is a branch of mathematics that studies the geometric properties of curves, surfaces, and higher-dimensional objects using calculus and linear algebra. It also investigates the relationships between these objects and the underlying geometric structures such as curvature and distance.

2. What are some real-world applications of differential geometry?

Differential geometry has many practical applications in fields such as physics, engineering, computer graphics, and robotics. It is used to model and analyze the behavior of physical systems, design efficient structures, and create realistic computer-generated images.

3. How is differential geometry different from traditional geometry?

Traditional geometry focuses on the properties and relationships of objects in a flat, Euclidean space. In contrast, differential geometry considers curved spaces and uses techniques from calculus to study the behavior of objects in these spaces.

4. What are some key concepts in differential geometry?

Some key concepts in differential geometry include curves, surfaces, manifolds, curvature, geodesics, and tensors. These concepts are used to describe and analyze the geometric properties of objects in different contexts.

5. How is differential geometry related to other branches of mathematics?

Differential geometry has connections to many other branches of mathematics, including topology, differential equations, and algebraic geometry. It also has applications in fields such as physics, computer science, and economics, making it an interdisciplinary subject.

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