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In this proof of Kepler's 2nd Law (Equal Areas in Equal Times: Variable-Speed Model), there are two (2) derivations, both utilizing vectors and calculus (differentiation and integration), but for the first derivation it is done so in rectangular or Cartesian coordinates with matrices, and for the second derivation it's accomplished in Polar coordinates; whereby both arrive exactly at the same final result for Kepler's 2nd Law.
Of course, the underlying proof is provided by Newton's Universal Law of Gravity together with Newton's 2nd Law of Inertial Motion, which is historically accurate as well. That is, Kepler devised his laws but it took Newton to prove his laws!
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