Vector Fields and Potentials: Theorem Proof

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Explores curve integrals of vector fields, emphasizing energy considerations for motion against or with wind, and introduces unit tangent and unit normal vectors.

Vector Fields Analysis

Explores vector fields analysis, covering curvilinear integrals, potential fields, and field connectivity conditions.

Divergence of Vector Fields

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

Differential Operators: Theorems and Proofs

Covers the concept of differential operators and presents theorems and proofs related to scalar and vector fields.

Local Invertibility: Vector Fields

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

Gradient: Scalar Field

Explores gradient in scalar fields, directional derivatives, and level sets.

Connections: motivation and definition

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

Curve Integrals: Gauss/Green Theorem

Explores the application of the Gauss/Green theorem to calculate curve integrals along simple closed curves.

Vector Calculus: Line Integrals

Covers the concept of line integrals and their application in vector fields.

Divergence Theorem: Green Identities in R²

Explores the divergence theorem and corollaries related to Green identities in the plane, demonstrating their application through examples.