What Is Differential Geometry?

Differential geometry studies properties of curves, surfaces, and higher-dimensional curved spaces using tools from calculus and linear algebra. Just as the introduction of calculus expands the descriptive and predictive abilities of nearly every field of scientific study, so the use of calculus in geometry brings about avenues of inquiry that extend far beyond classical geometry.

Before the advent of calculus, much of geometry consisted of proving consequences of Euclid’s postulates. Even conics, which came into vogue in the physical sciences after Kepler observed that planets travel around the sun in ellipses, arise as the intersection of a double cone and a plane, two shapes which fit comfortably ...

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