Lectures related to Implicit Functions Theorem

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

Explains the Wronskian and its role in determining linear independence of solutions to differential equations.

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

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

Explores integration by substitution with proofs and examples on anti-derivatives and function equivalence.

Covers the properties of harmonic functions and the concept of mollification.

Explores injective functions in linear algebra, demonstrating how to prove injectivity step by step.

Explains the Wronskian and its role in determining linear independence of solutions to differential equations.

Explores the Darboux theorem for continuous functions on closed intervals, emphasizing uniform continuity and function behavior implications.

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

Covers weak derivatives, their properties, and applications in functional analysis.

Covers the Meyers-Serrin theorem in analysis, discussing the conditions for functions in different spaces.

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

Covers the statement of the Bernoulli's Hospital Rule and its application.

Explores distribution and interpolation spaces, showcasing their importance in mathematical analysis and the computations involved.

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

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

Covers Morse theory, focusing on critical points and non-degeneracy.

Covers the extension and periodicity of Fourier series and the interpretation of coefficients.

Covers mathematical concepts from number theory to probability and statistics.