Differentiability: Functions and Matrices

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Differentiating vector fields along curves: Examples and finite differences

Covers the differentiation of vector fields along curves with examples and finite differences.

Functions and Coordinate Changes

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

Coordinate Changes: Spherical System

Explores differentiability, coordinate changes, Jacobian determinant, gradient transformation, and chain rule.

Derivative Chain Rule

Covers the derivative of the composition of two functions and the chain rule theorem.

Differentiable Functions in R²

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

Chain Rules

Explores the extension of the chain rule to higher dimensions and compositions of functions.

Optimization Basics: Norms, Convexity, Differentiability

Explores optimization basics such as norms, convexity, and differentiability, along with practical applications and convergence rates.

Curve Integrals of Vector Fields

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

Differentiating Vector Fields: Definition

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

Comparing Tangent Vectors: Parallel Transport

Explores parallel transport along loops and covariant derivatives induced by metrics.