Lectures related to Local Invertibility: Vector Fields

Covers local invertibility of vector fields and the Inverse Function Theorem.

Differentiability: Functions and Matrices

Explores partial differentiability, the chain rule, and invertibility of vector fields.

Explores stability in linear 2D systems, covering fixed points, vector fields, and phase portraits.

Differentiability: Functions and Matrices

Explores differentiability for functions and matrices, covering partial differentiability, the Jacobean matrix, and the chain rule.

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.

Implicit Differentiation: Solving Equations Locally

Explores implicit differentiation and the Implicit Function Theorem for finding unique functions locally.

Vectors: Coordinate Calculations

Covers calculations in coordinates for vectors, including bases, scalar product, and determinants, with geometric interpretations and examples.

Differentiating vector fields: Why do it?

Explores the importance of differentiating vector fields and the correct methodology to achieve it, emphasizing the significance of going beyond the first order.

Differentiating Vector Fields: Definition

Introduces differentiating vector fields along curves on manifolds with connections and the unique operator satisfying specific properties.

Linear Algebra: Matrices and Linear Applications

Covers matrices, linear applications, vector spaces, and bijective functions.

Local Frames MOOC: Introduction to optimization on smooth manifolds: first order methods

Covers the concept of local frames, their construction, and limitations.

Functions of LR: Differentiability Log in to Mediaspace to watch this video

Explains differentiability in LR functions and introduces the Jacobian matrix.

Functions and Coordinate Changes Log in to Mediaspace to watch this video

Explores bijective functions, coordinate transformations, and graphical representations in two dimensions.

Derivability and Composition Log in to Mediaspace to watch this video

Covers derivability, linear applications, composition of functions, and the gradient vector.

Spherical Coordinates: Determinant of Jacobi Log in to Mediaspace to watch this video

Covers spherical coordinates and the determinant of Jacobi in linear algebra.

Inverse Function Theorem Log in to Mediaspace to watch this video

Discusses inverse functions and local invertibility based on the determinant of the Jacobian matrix.

Differentiable Functions in R² Log in to Mediaspace to watch this video

Explores differentiable functions in R², including vector fields and coordinate transformations.

Connections: motivation and definition Log in to Mediaspace to watch this video

Explores the definition of connections for smooth vector fields on manifolds.

Differentiation and Coordinate Changes in Multivariable Functions Log in to Mediaspace to watch this video

Discusses differentiation of multivariable functions and coordinate transformations, including polar and cylindrical coordinates, along with the Laplacian operator and its applications.

Divergence of Vector Fields Log in to Mediaspace to watch this video

Explores divergence of vector fields, rotational definitions, and integral derivation applications.