Proof of the implicit and inverse function theorems

In summary, the conversation discussed revising knowledge from multivariable calculus and encountering difficulty remembering the proofs of two theorems. The speaker then consulted Rudin and found clarity, except for two functions that they didn't fully understand. These functions were later revealed to be standard constructions used in the implicit function theorem and a standard quotient.
  • #1
r4nd0m
96
1
Today I revised my knowledge from multivariable calculus and I found that I couldn't remember the proofs of these two theorems. Then I looked in Rudin, and everything was clear.
Except one thing, which probably made me forgot the proofs. There are two weird functions in these two proofs: [tex]\phi(x) = x + A^{-1} (y-f(x))[/tex] and [tex]F(x,y) = (f(x,y),y)[/tex].
I can see how they are used and that it really works, but I don't really understand what do the functions really say, if you know what I mean. How did someone figure out that these are the right functions we should use in the proof?
 
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  • #2
Hard to tell what you exactly mean without seeing the proofs. However, ##F(x,y)=(f(x,y),y)## is a standard construction when dealing with the implicit function theorem, i.e. it is the implicit function theorem!

The first function is something like a derivative: ##A= \dfrac{y-f(x)}{\phi(x)-x}##, so again a standard quotient to consider.
 

1. What is the purpose of the implicit and inverse function theorems?

The implicit and inverse function theorems are used to prove the existence and differentiability of solutions to equations involving multiple variables and functions. They provide a powerful tool for solving problems in higher mathematics and physics.

2. How do the implicit and inverse function theorems differ?

The implicit function theorem deals with finding solutions to equations in terms of a dependent variable, while the inverse function theorem deals with finding solutions in terms of an independent variable. In other words, the implicit function theorem solves for y in terms of x, while the inverse function theorem solves for x in terms of y.

3. What are the assumptions for the implicit and inverse function theorems to hold?

The implicit function theorem requires that the function in question is continuously differentiable and has a non-zero derivative at the point of interest. The inverse function theorem requires that the function is continuously differentiable, has a non-zero derivative at the point of interest, and is invertible.

4. How are the implicit and inverse function theorems used in real-world applications?

The implicit and inverse function theorems have many real-world applications, such as in optimization problems, economics, and engineering. They are also used in physics to solve equations involving multiple variables, such as in the study of fluid dynamics and electromagnetism.

5. What are some common misconceptions about the implicit and inverse function theorems?

One common misconception is that these theorems only apply to linear equations, when in fact they can also be used for non-linear equations. Another misconception is that the inverse function theorem can always be used to find an explicit solution, when in reality it may only provide an implicit solution. It is important to carefully consider the assumptions and limitations of these theorems when applying them to a problem.

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