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Schrodinger
Could someone try to explain what a Manifold is to me? I've tried to, but haven't been able to find a consistent set of qualities that I could really make a definition into.
In essence it means "many". What context are you using it in?
Manifold 4 Definition: A cylindrical pipe fitting, having a number of lateral outlets, for connecting one pipe with several others. Manifold 5 Definition: The third stomach of a ruminant animal. Manifold 6 Definition: To take copies of by the process of manifold writing; as, to manifold a letter. manifold 7 Definition: a pipe that has several lateral outlets to or from other pipes manifold 8 Definition: a set of points such as those of a closed surface or and analogue in three or more dimensions manifold 9 Definition: a lightweight paper used with carbon paper to make multiple copies; "an original and two manifolds" manifold 10 Definition: combine or increase by multiplication; "He managed to multiply his profits" manifold 11 Definition: make multiple copies of; "multiply a letter" manifold 12 Definition: many and varied; having many features or forms; "manifold reasons"; "our manifold failings"; "manifold intelligence"; "the multiplex opportunities in high technology"
In basic language it refers to a surface of arbitrary dimension that can be treated as flat over a very small distance. An example is the earth. It seems flat because you're really close but in reality its curved. A more rigorous definition is that it is a continuously differentiable region, which employs a differential metric, that is space must be measured in small units in order for the rules of calculus to work. A local set of coordinates can be set up at any point and "regular" euclidian geometry can be applied there. For example all smooth continuous curves are manifolds. For example the parabola: it looks curved but in the limit of two points coming together along it you'll eventually have a straight line, which is the derivative (or slope) at that point. Their analysis uses a more abstract entity called a tensor. Think of these as generalized vectors, to higher dimensions. This is a complicated field of math but I hope my simplified description helps.