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Lecture
Submanifolds: Locally Deformable into Linear Patches
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Related lectures (48)
Smooth Manifolds: Diffeomorphisms
MOOC: Introduction to optimization on smooth manifolds: first order methods
Explores smooth manifolds through diffeomorphisms and embedded submanifolds in a linear space.
Smooth Manifolds: Setup
MOOC: Introduction to optimization on smooth manifolds: first order methods
Introduces smooth manifolds, emphasizing the importance of submanifolds of linear spaces.
Geodesic convexity: why, and what we need
Explores geodesic convexity and its extension to optimization on manifolds, emphasizing the preservation of the key fact that local minima imply global minima.
Retractions vector fields and tangent bundles: Tangent bundles
MOOC: Introduction to optimization on smooth manifolds: first order methods
Covers retractions, tangent bundles, and embedded submanifolds on manifolds with proofs and examples.
Linear Algebra: Subspaces and Transformations
Explores subspaces in linear algebra and transformations, including kernels and images of linear transformations.
Comparing Tangent Vectors: Parallel Transport
Explores comparing tangent vectors and parallel transport on manifolds.
Dynamics of Singular Riemann Surface Foliations
Explores the dynamics of singular Riemann surface foliations, focusing on vector fields, linear parts, and coordinate changes.
Tangent spaces: Linearization of Embedded Submanifolds
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Explores tangent spaces as free movement directions on submanifolds, offering a geometrically satisfying linearization notion.
What is a (sub)manifold
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Introduces the concept of a submanifold in a linear space, defining it as a set smoothly embedded in the space.
Lines Spaces and Equivalent Matrices
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Covers the concept of lines spaces and equivalent matrices in linear algebra.
Gradients on Riemannian submanifolds, local frames
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Discusses gradients on Riemannian submanifolds and the construction of local frames.
Linear Maps and Matrices
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Covers linear maps, matrices, and applications, including exercises on bases and invertibility.
Convexity: Functions and Global Minima
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Explores convex functions, global minima, and their relationship with differentiability.
Linear Algebra: Matrices and Vector Spaces
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Covers matrix kernels, images, linear applications, independence, and bases in vector spaces.
Linear Applications: Properties and Examples
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Explores properties of linear applications, including symmetric matrices and linearity in analysis.
Kernel, Image and Linear Maps
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Explains kernel, image, and linear maps, illustrating concepts with examples.
Hessians, Symmetry and Examples: Sphere, Stiefel
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Covers Hessians, symmetry, and examples related to vector fields, functions, and manifolds.
Tensor Products and Symmetric Power
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Covers tensor products, symmetric power, and exterior power of vector spaces, including properties and applications.
Linear Transformations: Matrices and Applications
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Covers linear transformations using matrices, focusing on linearity, image, and kernel.
Finite Element Method: Quadrangular Elements
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Explores basic functions and criteria in quadrangular finite elements using the finite element method.
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