Tangent spaces: Linearization of Embedded Submanifolds

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Introduces the kernel of a linear application and its properties.

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Explores comparing tangent vectors and parallel transport on manifolds.

Introduces smooth manifolds, emphasizing the importance of submanifolds of linear spaces.

Explains the Rank Theorem for linear applications in 3D space and its implications.

Explores linear applications in 3D, emphasizing images, kernels, and solution uniqueness in systems.

Covers the basics of linear algebra, including linear maps, bases, and matrix operations.

Explains kernel, image, and linear maps, illustrating concepts with examples.

Explores properties of linear applications, including symmetric matrices and linearity in analysis.

Explores how submanifolds can be deformed into linear patches using diffeomorphisms and the inverse function theorem.

Discusses gradients on Riemannian submanifolds and the construction of local frames.

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