- #1
saminator910
- 96
- 1
One can do calculus on a differentiable manifold, what does that mean? Does it mean you can use differential forms on the manifold, or that you can find tangent vectors, What is certified as "calculus on a manifold".
Tenshou said:do you know about the calculus of index or index calculus?
Tenshou said:It is just notation, I am no expert on neither manifolds or index calculus, but you are interested in them I suggest learning topology! Topology deals with manifolds, like ##\delta## complexes and betti numbers (which I recently started to read up on), also I would assume doing calculus on a manifold would be like doing arithmetic geometry(or something analogous to) it is just a way of understanding the "space" a little bit better. (I guess a way Einstein might put it). Sorry for not answering your question I am sure someone will come on here with a lot better answer than mine, till then cheers mate :D
saminator910 said:One can do calculus on a differentiable manifold, what does that mean? Does it mean you can use differential forms on the manifold, or that you can find tangent vectors, What is certified as "calculus on a manifold".
saminator910 said:Do you mean Riemann surfaces, the complex manifolds, or Riemann manifolds, manifolds with Riemann metrics, if you do mean Riemann surfaces, how accessible is the topic to someone with a minimal background in complex analysis?
You can do your derivative there. Integration is a rather different beast.Jim Kata said:Calculus on a manifold means you can pullback to a subspace of R^n and do your integral there
Calculus on Manifolds is a branch of mathematics that studies smooth surfaces, or manifolds, and their properties. It involves using multivariable calculus techniques to analyze functions defined on these surfaces.
Calculus on Manifolds is an extension of regular Calculus to higher dimensions. While regular Calculus deals with functions on the real number line, Calculus on Manifolds deals with functions on more complex surfaces. It also introduces new concepts such as tangent spaces and differential forms.
Calculus on Manifolds has many applications in physics, engineering, and computer graphics. It is used to study the motion of objects in space, optimize the shape of structures, and model fluid flow. It is also used in computer graphics to create realistic 3D images.
Some key concepts in Calculus on Manifolds include differential forms, tangent spaces, vector fields, and integration on manifolds. These concepts are used to understand the behavior of functions on manifolds and to solve problems in various fields.
There are many textbooks and online resources available for learning Calculus on Manifolds. Some popular books include "Calculus on Manifolds" by Michael Spivak and "Differential Geometry of Curves and Surfaces" by Manfredo P. do Carmo. Online resources such as lectures, notes, and practice problems can also be found on sites like Khan Academy and MIT OpenCourseWare.