Lección 3 - El teorema de la función inversa e implícita | Funciones de Varias Variables II | UNED

Introduction to Key Concepts in Multivariable Functions

Theorems of Inverse and Implicit Functions

• The session focuses on two fundamental concepts: the Inverse Function Theorem and the Implicit Function Theorem, essential for understanding differentiation and differentiability in multivariable functions.

Analysis of Functions

• To explore inverse functions, the discussion begins with analyzing single-variable functions, specifically exponential, cubic, and quadratic functions.

• Three types of functions are considered: f_1(x) = e^x , f_2(x) = x^3 , and f_3(x) = x^2 , each exhibiting different behaviors regarding invertibility.

Injectivity and Global Inverses

• The function f_1 is injective with a global inverse given by its logarithm; similarly, f_2 is also injective with an inverse as the cube root.

• Conversely, f_3 , being non-injective (not one-to-one), lacks a global inverse but can have local inverses near certain points like 5.

Derivatives and Local Inverses

• It’s noted that while some functions have derivatives that allow for local inverses (like around 5 for f_3 ), others do not possess this property at specific points (e.g., at zero).

Definitions of Local and Global Invertibility

• A function F is defined as locally invertible at a point if it is injective within a neighborhood around that point.

• Global invertibility requires injectivity across the entire set; local conditions can lead to defining an inverse function locally.

Theorems on Differentiability

Conditions for the Inverse Function Theorem

• For a function to be globally invertible, it must be differentiable at a point where its Jacobian determinant is non-zero.

• If these conditions hold true, there exist open sets such that the function maps bijectively between them.

Properties of Inverse Functions

• When evaluating derivatives of inverse functions using Jacobians reveals relationships between original and inverted derivatives.

Consequences of Differentiability

Class C1 Functions

• If both original and inverse functions are continuously differentiable (class C1), then their properties extend to ensure smoothness in behavior across transformations.

Book's Approach to Systems of Equations

Solving Systems via Inversion

• The book simplifies discussions by focusing on solving systems of equations rather than delving into deeper theoretical constructs surrounding inverses.

Introduction to Implicit Function Theorem

Overview of Implicit Functions

• Transitioning from explicit definitions to implicit ones involves considering higher-dimensional mappings where variables depend on each other through equations.

Conditions for Existence

Requirements for Implicit Solutions

• A function must equal zero at specific points while having non-zero partial derivatives concerning dependent variables. This ensures unique solutions exist locally around those points.

Generalization to Multiple Variables

Systematic Approach

• Extending results from single-variable cases allows us to analyze systems involving multiple equations with several unknown variables effectively.

Conclusion on Both Theorems

Summary

• With established conditions met under both theorems—Inverse Function and Implicit Function—their applications provide robust frameworks for handling complex multivariable calculus problems.