Lectures related to Remarks and Demonstrations

Advanced Analysis II: Hessians and Directional Derivatives

Explores Hessians, directional derivatives, and function continuity in advanced analysis.

Differentiable Functions: Definitions and Properties

Covers the definition and properties of differentiable functions, focusing on differentiability, limits, continuity, and partial derivatives.

Optimization Techniques: Gradient Descent and Convex Functions

Provides an overview of optimization techniques, focusing on gradient descent and properties of convex functions in machine learning.

Fourier Series: Convergence and Dirichlet Theorem

Covers Fourier series convergence, Dirichlet theorem, and applications in signal processing.

Proximal and Subgradient Descent: Optimization Techniques

Discusses proximal and subgradient descent methods for optimization in machine learning.

Existence of Solutions for Poisson-Dirichlet Problem

Covers the existence of solutions for the Poisson-Dirichlet problem, focusing on showing that certain conditions hold for locally bounded and Hölder continuous functions.

Tangent to Graph of a Function

Explores finding the equation of the tangent line to a function's graph at a point.

Monotonicity Criteria in Differentiable Functions

Explores monotonicity criteria, L'Hopital's rule, and Lipschitz continuity in differentiable functions and deep neural networks.

Functions and Coordinate Changes

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

Rolle's Theorem and Mean Value Theorem

Covers Rolle's Theorem and the Mean Value Theorem with illustrative examples.

Differentiable Functions: Definitions and Interpretation

Covers the definitions of differentiable and derivable functions and their geometric interpretation.

Differentiable Functions in R²

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

Rolle's Theorem

Explores Rolle's Theorem for continuous and differentiable functions on closed intervals.

Closed Intervals: Definitions and Remarks

Explains definitions and remarks about closed intervals in functions.

Curve Length and Function Definition

Explores curve length, function definition, continuity, derivatives, integrals, and graphical representations of functions in two variables.

Functions of Class Ck: Terminology and Definitions

Covers the terminology and definitions of functions of class Ck on real number intervals.

Derivative and Linear Approximation

Covers the concept of approximating a function around a point xo using linear approximation.

Integration by Parts: Techniques and Examples

Explores integration by parts technique through examples involving trigonometric functions and exponentials.

Osculating Planes and Parameterized Curves

Covers the concept of osculating planes and parameterized curves, discussing the speed of points and linear independence.

Derivability of Reciprocal Function

Covers the derivative of the reciprocal function and its properties.