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l manifolds embedded in higher-dimensional Euclidean spaces. Parametrization: Curves and surfaces are often described via smooth maps: Curves: \( \alpha: I \subset \mathbb{R} \rightarrow \mathbb{R}^3 \) Surfaces: \( X: U \subset

motions of celestial objects under the influence of gravitational forces. It combines principles from classical mechanics, Newtonian physics, and, in more advanced contexts, Einstein's theory of general relativity. Its primary goal is to predict the positions and velocities o