Lectures related to Advanced Analysis II: Derivatives and Functions

Derivatives: Chain Rule and Composed Functions

Covers derivatives, chain rule, and composed functions with examples and applications.

Implicit Functions Theorem

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

Integration by Substitution

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

Bernoulli's Hospital Rule

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

Calculus: Derivatives and Integrals

Covers the fundamentals of calculus, focusing on derivatives and integrals.

Laplacian in Polar and Spherical Coordinates: Derivatives

Covers the Laplacian operator in polar and spherical coordinates, focusing on derivatives and integral calculations.

Derivability and Composition of Functions

Explores derivability and function composition with examples.

Applications of Theorems

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

Taylor Polynomial: Order 2

Presents the solution to finding the second-order Taylor polynomial of a function.

Differential Equations: Solutions and Monotonicity

Explores solutions of differential equations and their monotonicity properties.

Differentiation and Coordinate Changes in Multivariable Functions

Discusses differentiation of multivariable functions and coordinate transformations, including polar and cylindrical coordinates, along with the Laplacian operator and its applications.

Mathematical Notations and Functions in General Physics

Introduces mathematical notations and functions essential for general physics.

Differentiability and Composition

Covers differentiability, composition of functions, partial derivatives, Jacobian matrix, and polar coordinates.

Real Numbers and Functions

Introduces Real Analysis I, covering real numbers, functions, limits, derivatives, and integrals.

Advanced Analysis II: Derivatives and Notations

Covers derivatives, notations, composition of functions, and multiple derivatives in advanced analysis.

Introduction to Mathematics for Engineers

Introduces the purpose of mastering mathematics and calculation tools for engineers, emphasizing the need to think methodically and rigorously.

Limits and Derivatives in Multivariable Functions

Covers limits and derivatives in multivariable functions, focusing on continuity, partial derivatives, and the gradient.

Introduction to Mathematical Concepts

Introduces fundamental mathematical concepts and their applications in equations and inequalities.

Partial Derivatives and Derivability

Explains partial and directional derivatives, and functions' derivability.

Solutions of Differential Equations

Explores finding solutions of differential equations, emphasizing maximal solutions and general solutions with constants.