The smoothness of a surface is determined by how many derivatives you can take before your result is discontinuous. Differential Geometryĭifferential Geometry is the application of Calculus to smooth manifolds. We can use these tools to come up with some laws of Physics that work in all coordinate systems. You might be thinking that we could start doing Physics on the surface of the Earth or maybe some other topics in Physics (and we might later), but I’m more interested in the tools used to study manifolds. It begs the question of why I’m even talking about manifolds. In fact, with the exception of the surface of a sphere in the previous section, we’ve been working entirely with the standard 2D and 3D Euclidean spaces. To be clear, curved coordinates do not always represent curved spaces. Curved Coordinates Don’t Imply Curved Space For example, we can represent the surface of a sphere without distortion in three dimensions with the relationship s( θ, φ) = ( x, y, z) = ( R cos φ sin θ, R sin φ sin θ, R cos θ) where R is a constant. An embedding is how some object fits into a larger object. ![]() The word “isometric” translates to “same distance” and means that nothing is stretched. In math terms, we cannot isometrically embed the surface of a sphere into 2D Euclidean space. Although we can, we have to stretch the surface. You might think that we should be able to represent it in a 2D space, like a piece of paper. While a sphere is a 3D object, its surface is 2D. In other words, a manifold is anything that can be locally approximated by Euclidean geometry. As your points of interest on the surface of a sphere get closer together, Euclidean approximations get better, which makes the surface of a sphere a manifold. So, the surface of the Earth isn’t Euclidean, but it approximates Euclidean space on the scale of cities.
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