ForMyThunder said:Right now, I'm reading through Lee's Intro to Smooth Manifolds and I was wondering if there is a website somewhere that has problems different from the book. Or if there is another book out there that covers about the same material as Lee's that would be good to.
Thanks in advance.
A smooth manifold is a mathematical space that can be locally approximated by a Euclidean space, such as a plane or a curve. It is a generalization of the concept of a manifold, which is a space that is locally similar to a Euclidean space but may have more complicated global structure.
Smooth manifolds are important in mathematics because they provide a framework for studying and understanding more complicated geometric objects, such as curves and surfaces. They also have applications in physics, where they are used to describe the space-time continuum in general relativity.
A smooth function on a manifold is a function that is defined on the manifold and is differentiable to all orders. This means that the function has continuous derivatives of all orders, making it a very well-behaved function on the manifold.
Some common problems on smooth manifolds include determining the curvature of the manifold, finding geodesics (the shortest paths between points) on the manifold, and studying the topology of the manifold (i.e. the properties that are preserved under continuous deformations).
Smooth manifolds are a foundational concept in differential geometry, which is the study of geometric objects using calculus and analysis techniques. Smooth manifolds provide the framework for understanding and analyzing more complicated geometric structures, making them an important tool in differential geometry.