Lectures related to Change of Variables Theorem

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

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

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

Effective Version of the Explicit Formula

Explores the effective version of the explicit formula in Number Theory, focusing on smooth functions and integral calculations.

Green's Theorem: Understanding Rotations and Closed Paths

Explores Green's Theorem, rotations, closed paths, and integral signs.

Curves with Poritsky Property and Liouville Nets

Explores curves with Poritsky property, Birkhoff integrability, and Liouville nets in billiards.

Green's Functions in Laplace Equations

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

Integration by Parts

Covers integration by parts, including practical applications and mnemonic aids.

Integration of CnR Class Functions

Explains the integration of Taylor series for CnR class functions.

Properties of Definite Integrals: Fundamental Theorem of Analysis

Covers the properties of definite integrals and the fundamental theorem of analysis.

Improper Integrals: Convergence and Comparison

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

Curve Integrals: Gauss/Green Theorem

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

Applications of Residue Theorem in Complex Analysis

Covers the applications of the Residue theorem in evaluating complex integrals related to real analysis.

Fundamental Theorem of Calculus Example

Demonstrates the practical importance of the fundamental theorem of calculus through a detailed example.

Fubini Theorem on Closed Rectangles

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

Quantum Field Theory: Fermions and Grassmann Numbers

Explores quantum field theory, focusing on fermions and Grassmann numbers in the path integral formalism.

Curve Integrals of Vector Fields

Explores curve integrals of vector fields, emphasizing energy considerations for motion against or with wind, and introduces unit tangent and unit normal vectors.

Magnetostatics: Magnetic Field and Force

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

Taylor Series and Definite Integrals

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

Part-based Integration Examples

Showcases step-by-step examples of part-based integration for functions like e^x and ln(x).