Lectures related to Advanced analysis II: jordan-measurable sets

Fubini's Theorem: Multiple Integrals

Explores Fubini's Theorem for multiple integrals, emphasizing the n=2 case.

Advanced analysis II: properties and applications

Explores properties and applications of Riemann integrals, Jordan-measurable sets, and continuity in advanced analysis.

Multiple Integrals: Definitions and Properties

Covers the definition and properties of multiple integrals, including double and triple integrals.

Fubini Theorem on Closed Rectangles

Explores the Fubini theorem on closed rectangles in R², discussing integrability, iterated integrals, and compact sets.

Double Integrals: Definitions and Properties

Covers the definitions and properties of double integrals over compact regions.

Improper Integrals: Convergence and Comparison

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

Advanced Analysis II: Jordan-Measurable Functions

Explores Jordan-measurable functions and double integrals for volume calculations in 3D space.

Definite Integrals: Properties and Interpretation

Covers the calculation of minimum points and the concept of definite integrals.

Taylor Series and Definite Integrals

Explores Taylor series for function approximation and properties of definite integrals, including linearity and symmetry.

Lebesgue Integral: Comparison with Riemann

Explores the comparison between Lebesgue and Riemann integrals, demonstrating their equivalence when the Riemann integral exists.

Multiple Integrals: Extension and Properties

Explores the extension and properties of multiple integrals for continuous functions on rectangles.

Multiple Integration: Fubini Theorem

Explores multiple integration in R², focusing on double integrals over closed rectangles and the Fubini theorem.

Generalized Integrals: Definition and Applications

Covers the definition and applications of generalized integrals in advanced analysis, including real functions, differential equations, and multiple integrals.

Differentiation under Integral Sign

Explores differentiation under the integral sign, comparing it with the Riemann integral and discussing key assumptions and theorems.

Riemann Integral: Construction and Properties

Explores the construction and properties of the Riemann integral, including integral properties and mean value theorem.

Taylor Series and Riemann Integral

Explores Taylor series expansions and Riemann integrals, including limits, convergence, subdivisions, and sums.

Magnetostatics: Magnetic Field and Force

Covers magnetic fields, Ampère's law, and magnetic dipoles with examples and illustrations.

Lagrangian Multipliers: Extrema and Constraints

Covers Lagrangian multipliers, extrema with constraints, multiple integrals, and Darboux sums.

Riemann Sums and Definite Integrals

Covers Riemann sums, definite integrals, Taylor series, and exponential of complex numbers.

Green's Functions in Laplace Equations

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