This subject unifies linear algebra and vector calculus from second year into the sort of calculus used in differential geometry and analysis. The central notion is that of manifolds, which is a generalization of curves and surfaces, and provides the mathematical context for understanding "space" in all of its manifestations. Topics include: the implicit and inverse function theorems, manifolds and tangent space, curvature and torsion for curves, intrinsic and extrinsic curvature for surfaces, and the Theorema Egregium of Gauss.

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