Implicit and Inverse Function Theorems

[Dahaka14]

What is a good book that gives a clear idea of what these theorems are? I am taking differential geometry, and I would like to try and get a better understanding of them. To be honest, my calculus sequence NEVER went over them, so I need to get these ideas under my belt.

 

[Daverz]

Wikipedia is not too bad for this:

http://en.wikipedia.org/wiki/Implicit_function_theorem

Also https://www.amazon.com/dp/0521797071/?tag=pfamazon01-20.

 

[charng]

The Implicit and Inverse Function Theorems are important tools in differential geometry and related fields such as calculus and differential equations. They provide a way to study the behavior of functions without explicitly solving for their values.

The Implicit Function Theorem, also known as the Implicit Function Existence Theorem, states that if a system of equations has a unique solution for a certain variable in terms of the others, then it is possible to define a function for that variable in terms of the others. This theorem is useful in many areas of mathematics, such as optimization and partial differential equations.

The Inverse Function Theorem, on the other hand, states that if a function is differentiable and has a non-zero derivative at a point, then it is locally invertible around that point. This theorem is important in understanding the local behavior of functions and is a key tool in differential geometry.

A good book that gives a clear idea of these theorems is "Differential Geometry of Curves and Surfaces" by Manfredo P. do Carmo. This book is a classic in the field and provides a comprehensive introduction to both the Implicit and Inverse Function Theorems, along with many other important topics in differential geometry. It also includes many examples and exercises to help deepen your understanding of these theorems. Other recommended books include "Introduction to Smooth Manifolds" by John M. Lee and "Differential Geometry: Curves - Surfaces - Manifolds" by Wolfgang Kühnel.

I would also recommend consulting with your differential geometry professor or a math tutor for further clarification and guidance on these theorems. It's important to have a solid understanding of these concepts before moving on to more advanced topics in differential geometry. Good luck with your studies!