Comments on: The implicit and inverse function theorems http://www.tangentspace.net/cz/archives/2006/09/the-implicit-and-inverse-function-theorems/ somewhere near the beginning. Tue, 09 Mar 2010 11:23:12 +0000 http://wordpress.org/?v=2.9 hourly 1 By: Alex http://www.tangentspace.net/cz/archives/2006/09/the-implicit-and-inverse-function-theorems/comment-page-1/#comment-59871 Alex Sun, 17 Sep 2006 05:37:07 +0000 http://www.tangentspace.net/cz/archives/2006/09/the-implicit-and-inverse-function-theorems#comment-59871 I agree with your sentiment-- some writers do seem to drown simple ideas in layers of formality-- but I don't think this is because the authors are confused; it's probably that they (wrongly) value conciseness and exactness over simplicity and clarity. All are desirable, but the latter should <em>always</em> prevail over the former. I agree with your sentiment– some writers do seem to drown simple ideas in layers of formality– but I don’t think this is because the authors are confused; it’s probably that they (wrongly) value conciseness and exactness over simplicity and clarity. All are desirable, but the latter should always prevail over the former.

]]> By: ObsessiveMathsFreak http://www.tangentspace.net/cz/archives/2006/09/the-implicit-and-inverse-function-theorems/comment-page-1/#comment-59751 ObsessiveMathsFreak Sat, 16 Sep 2006 18:10:47 +0000 http://www.tangentspace.net/cz/archives/2006/09/the-implicit-and-inverse-function-theorems#comment-59751 It's typical in modern mathematics for otherwise straitforward concepts, theorems and definitions to be garbled beyond all recognition and left incomprehensible to all but those who already know their "true" meaning. Sometimes I think that these "correct" definitions allow misconceptions to creep in and find refuge in the minds of mathematicians. The inverse function theorem is a good example of this. The inverse function theorem does not in fact give conditions for our ability to find an inverse function. As the text states, it places conditions on our ability to find a diffeomorphism between the domain and the range, i.e. find an inverse function which is itself differentiable. For example, the inverse function for x^3, i.e. x^(1/3), exists everywhere, including at x=0. It's not differentiable at x=0, but it is an inverse function. After reading the inverse function theorem, a lot of people might in fact think it wasn't. I think a lot of people don't really understand the theorems they put in their books. It’s typical in modern mathematics for otherwise straitforward concepts, theorems and definitions to be garbled beyond all recognition and left incomprehensible to all but those who already know their “true” meaning. Sometimes I think that these “correct” definitions allow misconceptions to creep in and find refuge in the minds of mathematicians. The inverse function theorem is a good example of this.

The inverse function theorem does not in fact give conditions for our ability to find an inverse function. As the text states, it places conditions on our ability to find a diffeomorphism between the domain and the range, i.e. find an inverse function which is itself differentiable.

For example, the inverse function for x^3, i.e. x^(1/3), exists everywhere, including at x=0. It’s not differentiable at x=0, but it is an inverse function. After reading the inverse function theorem, a lot of people might in fact think it wasn’t.

I think a lot of people don’t really understand the theorems they put in their books.

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