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Author Archives: Behnam Esmayli
What is a Manifold? (6/6)
In posts 13 we were able to reduce all of the geometry of a curve in 3space to an interval along with two or three realvalued functions. We also discussed when two sets of such data give equivalent (overlapping) curves. This … Continue reading
What is a Manifold? (5/6)
In our last post, we invented a new geometry by rescaling the inner product of the usual Euclidean plane. This modification did not change any of the angles in our geometry, in the sense that if two curves intersected in a particular Euclidean … Continue reading
What is a Manifold? (4/6)
After our luxurious treatment of 1d manifolds, we turn to 2d manifolds. My story of surfaces starts in a beautifully weird morning when I got up to realize that life in the usual Euclidean plane had changed dramatically. Vectors had shortened, areas … Continue reading
What is a Manifold? (3/6)
Intrinsic descriptions One immediate benefit of considering coordinatefree descriptions of geometric objects is that we may talk about “curves” that are not a priori embedded in . In other words, we don’t have to start with a subset of to … Continue reading
What is a Manifold? (2/6)
We continue from Part One of this journey our attempt to illustrate how one can start with calculus and arrive at the definition of a 1dimensional manifold. In the previous segment, we concluded with the fact that a curve in may … Continue reading
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