Lecture 37(C): The Implicit Function Theorem

[@HyazintheLindemann-lu3vm]

You are saving my bachelor degree, sir !

[@ArizonaMathCamp]

Well, I think your own hard work is the real key. But if my videos are helping, that's great! Thanks for the positive feedback!

[@virmaq5187]

This is such a good explanation. Thank you so much!

[@ArizonaMathCamp]

I'm glad it was helpful. Thanks for the good feedback.

[@ilkemgok3373]

This video has been very helpful for me to understand the IFT. Thank you!

[@ArizonaMathCamp]

Great! I'm glad it was helpful!

[@murielfang755]

Thank you, sir! The geometric demonstration is really helpful!

[@ArizonaMathCamp]

Great! I'm glad you found it helpful.

[@Eco-nk1xe]

I cannot thank you enough. You've boosted my interest to be a theorist. I have a question, though. You refer to the domain as open sets in a couple of videos. Isn't that a problem to generalise the various theorems for compact sets to the solutions of the maximisation problem (which would require you to work with closed sets instead).

[@ArizonaMathCamp]

I'm glad you liked the videos! I'm a little confused by your question. It's correct that maximization problems often have a closed (and often not compact) domain. Can you point me to examples of the theorems with open-set domains and the ones with compact domains -- there are several, but point me to ones that ones don't seem to apply to these closed-domain maximization problems and I can probably give you an answer.

[@Eco-nk1xe]

@@ArizonaMathCamp Many thanks for your reply. For instance, with the following theorem, we are certain that a solution exists for the maximisation problem: Suppose S is a compact and u : S → R is continuous. Then the set of solutions {x ∈ S | ∀y ∈ S : u(x) ≥ u(y)} is non-empty and compact. My intuition is this certainty about the solution comes from the fact that the domain is closed/bounded (on top of the continuity of U). Can we also be certain about the solution of the problem will be non-empty when we have an open set? Pardon me if my explanation lacks clarity (a confused PhD student lolll).

[@ArizonaMathCamp]

@@Eco-nk1xe Yes, the solution is guaranteed by the compactness of the domain (closed and bounded if in Euclidean space). For a simple example when the set is *not* closed, let f(x)=2x and let the domain be the half-open interval [0,1). This is bounded and fails to be closed only because the right-hand endpoint is missing. Clearly no maximum of f exists. Alternatively, let the domain be R_+, the set of nonnegative numbers, which *is* closed but fails to be compact only because it's undbounded. Again, no maximum exists. Finally, let f(.) be the quadratic function f(x)=(.5-x)^2. This *does* attain a maximum on either of the two domains -- this shows that compactness is not *necessary* for a maximum to exist, it (along with continuity) is only a *sufficient* condition -- but one that cannot be dispensed with if you want to guarantee the existence of a solution.

[@Eco-nk1xe]

@@ArizonaMathCamp Many thanks. Be sure that many out there value your work. All the best sir...

[@ArizonaMathCamp]

@@Eco-nk1xe Good luck in your PhD program. Which program are you in?

[@alijoueizadeh2896]

Thank you.

[@denisespinoza2597]

Excellent explanation.

[@ArizonaMathCamp]

Glad it was helpful! Thanks for the positive feedback.

[@footballmint]

Thanks from an economics student at UEA :)

[@ArizonaMathCamp]

Glad it's been helpful. UEA is East Anglia? I haven't been to Norwich, but other parts of E Anglia and they were very nice places.

[@footballmint]

​@@ArizonaMathCamp Yes Norwich is lovely, I'd recommend a visit if you're ever in the area again.

[@Iamawandererthatiam]

I don't think Simon and Blume has a formal proof of the Implicit Function Theorem (just a sketch). Is there a notationally similar proof online somewhere you recommend?

[@ArizonaMathCamp]

I recommend de la Fuente's book, a very good exposition and proof, but unfortunately not online. You could try Kim Border's Caltech notes, online. (I assume by notationally similar you mean x for variables to be determined instead of for parameters. Both of these references do that.)

[@annahollands8604]

This was perfect thank you so much!

[@ArizonaMathCamp]

Glad it was helpful!

[@ameliali9489]

yay