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==Tangent bundle== [https://www.youtube.com/watch?v=XZcKSoI17r0&list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic&index=10 Lecture 10] cotangent space and tensor space at a point of a manifold; differential of a smooth map; gradient of a real function on a manifold; dual coordinate-induced basis and gradients of coordinate functions; push-forward and pull-back of smooth maps at a point; push-forward of tangent vectors and pull-back of covectors; immersions and immersed submanifolds; embedding and embedded submanifolds; Whitney's theorem; definition of tangent bundle; proof that the tangent bundle is a smooth manifold.
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