Differential Geometry - useful in engineering?

[donnylee]

Dear fellow Mathematicians and Physicists,

As the fall term closes and spring term starting next year, I am deciding on which math class to take. Simply put, I'm a math major (possibly engineering if there are enough slots in my schedule), more applied, seeking to apply math concepts to science and engineering. I have taken a substantial amount of classes in control systems and fluid dynamics to know that area of engineering well enough.

I have already covered the fundamental math classes - Analysis, Algebra, Topology, elementary ODE and PDE. Then when looking at the classes which caught my eye, one of them stood out: DIFFERENTIAL GEOMETRY. I talked to the professor and he said given my background, I should find the material accessible. In short, we will be studying Riemann Geometry and develop the theory using rigorous real analysis.

So far, differential geometry gives me ideas of arc length, tangent, surface integral - all the multivariable calculus stuff. Yet, I am well aware this graduate level class is NOT about that. A quick look at the synopsis, I see stuff like 2-forms, implicit function theorem, manifolds, imbedding, (general case) of Stoke's theorem.

Thus my question: Is this class, Differential Geometry (graduate level), be useful in an applied field like physics or engineering? I heard the phrase that General Relativity is written in the language of Differential Geometry so I feel somewhat glad that there is some applicability in this subject. Well, certainly I don't mind learning the theory if it is solely in the realm of Pure math, but it'll be good if there are some real world applications, at least from my point of view.

Oh, and what the heck is the difference between Differential Geometry and Riemann Geometry?

Thanks.

 

[xepma]

My experience is that differential geometry is one of the most fun and comprehensible subject in mathematics that I learned in my master's (theoretical physics). For a theoretical phycisist it should be part of your basic math knowledge. For a math major I could recommend it as well, but maybe not everyone will agree with me.

You're right that General Relativity is just applied differential geometry (any book on GR starts with an intro on differential geometry). Differential geometry is fun to do, and can be very useful when it comes into play.

I doubt however that you will have a direct application in engineering. You will notice that everything that you have learned and will learn about physical systems 'fits' in the language of differential geometry, but most of the time this is actually quite trivial. The main reason is that differential geometry really only comes into play if the 'manifold' (= the space you are working on) is a non-trivial geometrical structure (e.g. the surface of a sphere). But since most of the time you'll be working in flat, infinitely large spaces you won't need all the fancy machinery. In this case differential geometry just reduces to ordinary linear algebra. Differential geometry is, in some sense, all the linear algebra and calculus learned in the undergraduate courses, but now applied to non-trivial geometrical structures. Still a lot of fun though!

P.S. Riemann Geometry is just a discipline within differential geometry. It deals with smooth manifolds which have a Riemann metric. The metric, in general, defines the inner product between vectors. A Riemann metric makes sure all possible inner products are >= 0. But GR, for instance, uses Lorentzian manifolds instead of Riemann -- this all falls under the regime of differential geometry.

 

[zpconn]

I believe it's fairly common to take a first course in more classical differential geometry covering curves and surfaces directly (almost as a direct extension of a multivariable analysis course) and then to take a course in calculus on manifolds, often covering Riemannian geometry (which consists mostly of the study of Riemannian manifolds). To see the difference between the two, you may look at these two books by Do Carmo (a renowned author on differential geometry):

https://www.amazon.com/dp/0132125897/?tag=pfamazon01-20

and

https://www.amazon.com/dp/0817634908/?tag=pfamazon01-20

The latter is quite a bit more difficult and abstract than the former. Analysis on manifolds also relies heavily on the machinery of linear algebra (tensor products, dual spaces, etc.), whereas the former still relies on linear algebra but is accessible to one who has had only a first undergraduate course in that subject.

 

[Andy Resnick]

Differential geometry is hugely useful in physics and engineering- fluid surfaces, stress-strain analysis, continuum mechanics... I'm really happy I learned some of it, I use the formalism regularly in my own work.

 

[atyy]

http://www.math.rutgers.edu/~sontag/mct.html
Take a look at Chapter 4, and references [199] and [311] that he mentions in section 4.7

Or chapter 8 of http://books.google.com/books?id=Tny497jne7kC&dq=sastry+nonlinear+systems&source=gbs_navlinks_s