Lectures related to Calculating Integrals in Multiple Dimensions

Unclosed Curves Integrals

Covers the calculation of integrals over unclosed curves, focusing on essential singularities and residue calculation.

Improper Integrals: Convergence and Comparison

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

Generalized Integrals and Convergence Criteria

Covers generalized integrals, convergence criteria, series convergence, and harmonic series in analysis.

Differentiating under the integral sign

Explores differentiating under the integral sign and continuity of functions in integrals.

Generalized Integrals: Type 2

Covers the integration of limit expansions and continuous functions by pieces.

Differential Forms Integration

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Derivative of an Integral with Parameter

Covers deriving integrals with parameters and their derivatives, including special cases and proofs.

Differentiating under the integral sign

Explores differentiating under the integral sign and conditions for differentiation, with examples and extensions to functions on open intervals.

Integration Techniques: Change of Variable and Integration by Parts

Explores advanced integration techniques such as change of variable and integration by parts to simplify complex integrals and solve challenging integration problems.

Curve Integrals: Gauss/Green Theorem

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

Angle Calculation on Regular Surfaces

Covers the calculation of angles between curves on regular surfaces and the concept of curvilinear abscissa.

Demonstration of Uniqueness Theorem

Presents a detailed proof of the uniqueness theorem for functions f and g.

Integration of CnR Class Functions

Explains the integration of Taylor series for CnR class functions.

Rings and Modules

Covers rings, modules, fields, minimal ideals, and the Nullstellensatz theorem.

Analysis IV: Convergence Theorems and Integrable Functions

Covers convergence theorems and integrable functions, including the Lebesgue integral and Borel-Cantelli sets.

Improper Integrals: Fundamental Concepts and Examples

Covers improper integrals, their definitions, properties, and examples in two and three dimensions.

Green's Functions in Laplace Equations

Covers the concept of Green's functions in Laplace equations and their solution construction process.

Fundamental Theory of Integral Calculus

Covers the fundamental theory of integral calculus, integration methods, and the importance of finding primitive functions for integration.

Analytic Continuation: Residue Theorem

Covers the concept of analytic continuation and the application of the Residue Theorem to solve for functions.

Comparison Series and Integrals

Explores the relationship between series and integrals, highlighting convergence criteria and function examples.