Lectures related to Analysis III: Third Class

Differentiability of Functions of Several Variables

Covers the differentiability of functions of multiple variables and the significance of directional derivatives and gradients.

Differentiable Functions and Lagrange Multipliers

Covers differentiable functions, extreme points, and the Lagrange multiplier method for optimization.

Derivatives and Functions: Definitions and Interpretations

Covers the definition of derivative functions and their interpretation, focusing on differentiability, velocity, and curve lengths.

Improper Integrals: Convergence and Comparison

Explores improper integrals, convergence criteria, comparison theorems, and solid revolution.

Curvature and Inflection Points

Explores curvature, inflection points, and angular functions in plane curves, highlighting the importance of inflection points.

Curvilinear Coordinates: Calculations and Examples

Covers curvilinear coordinates and area calculations using double integrals with examples of different curves.

Residue Theorem: Calculating Integrals on Closed Curves

Covers the application of the residue theorem in calculating integrals on closed curves in complex analysis.

Geometrical Aspects of Differential Operators

Explores differential operators, regular curves, norms, and injective functions, addressing questions on curves' properties, norms, simplicity, and injectivity.

Fundamental Groups

Explores fundamental groups, homotopy classes, and coverings in connected manifolds.

Meromorphic Functions & Differentials

Explores meromorphic functions, poles, residues, orders, divisors, and the Riemann-Roch theorem.

Complex Integration and Cauchy's Theorem

Discusses complex integration and Cauchy's theorem, focusing on integrals along curves in the complex plane.

Convex Functions

Covers the properties and operations of convex functions.

Continuity and Limits

Covers continuity, limits, and different approaches to approaching points in various coordinate systems.

Implicit Functions Theorem

Covers the Implicit Functions Theorem, explaining how equations can define functions implicitly.

Field Derivation: Potential and Convexity

Covers the derivation of a field from a potential and explores convexity.

Level Curves and Graphs of Functions

Covers level curves and graphs of functions in two and three dimensions.

Integral Techniques: Integration by Parts

Explores the integration by parts technique through examples, showcasing its step-by-step application to functions like cos(x) and sin(x.

Limits and Continuity: Analysis 1

Explores limits, continuity, and uniform continuity in functions, including properties at specific points and closed intervals.

Length of Curves and Tangent Vectors

Covers the calculation of the length of curves and the properties of tangent vectors.

Applications of Theorems

Demonstrates the practical application of theorems in calculus through two clever examples.