MATH-106(e): Analysis II

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Lectures in this course (103)

Implicit Functions and Taylor Polynomials

Covers implicit functions, Taylor polynomials, and tangent equations.

Extremum of a Function: Constraints

Explores extremum of a function under constraints and finding candidate points in practice.

Lagrange Method: 2 Constraints

Explains the Lagrange method with 2 constraints and composite functions' derivatives.

Integral Calculus: Introduction and Summary

Provides an overview of integral calculus, including Darboux sums, closed box subdivisions, and integrability of continuous functions.

Demonstration of Theorem on Compact Functions

Explores the demonstration of a theorem on compact functions and non-regular boundaries.

Calculating Multiple Integrals: Methods and Variables

Covers methods for calculating multiple integrals and introduces polar variables.

Matrix Jacobian Function

Explores the concept of Jacobian matrices for differentiable functions and demonstrates their application through a detailed analysis of multiple choice answers.

Integration Domains

Focuses on identifying the correct integral within a given domain and understanding dependencies between variables.

Polar Coordinates: Line Integral

Explores line integrals in polar coordinates with practical examples.

Optimization Theorem: Lagrange and Integrals

Explores Lagrange's optimization theorem, integration bounds transformation, and singularity removal.

Methods of Integrating Functions

Covers methods of integrating functions of several variables and different coordinate systems.

Exam Information and Revision

Provides practical information and tips for preparing for the final exam.

Lagrange Multipliers: Extremum Points

Explains Lagrange multipliers for extremum points and multiple integration methods for volume and area calculations.

Ordinary Differential Equations: Definitions and Methods

Explores ordinary differential equations, proof methods, and historical examples from Euclid, emphasizing logical reasoning and step-by-step derivations.

Methods of Demonstration: Equivalence and Cases

Covers examples of differential equations with methods of demonstration.

General Solution of Homogeneous Second Order Linear Differential Equations

Covers the general solution of homogeneous second-order linear differential equations with constant coefficients and the concept of linear independence of solutions.

Linear Differential Equations

Covers the characterization of linearly independent solutions of a second-order homogeneous linear differential equation.

Method of Undetermined Coefficients

Explores the method of undetermined coefficients for solving non-homogeneous linear differential equations with constant coefficients.

Vector Spaces and Topology

Covers vector spaces, topology, and proof methods like the pigeonhole principle in R^n.

Convergence and Compactness in R^n

Explores adhesion, convergence, closed sets, compact subsets, and examples of subsets in R^n.

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