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Apr
17
comment A diffeomorphism whose tangent map preserves dot products is an isometry.
What is your class's definition of "isometry"? Some contexts, like Riemannian geometry, take "isometry" to mean exactly "tangent map preserves the inner product."
Apr
16
comment What are the conditions for the implicit function theorem to hold?
No! You need your level set to pass the vertical line test. That's why you need that condition on the Jacobian of $f$.
Apr
15
comment What are the conditions for the implicit function theorem to hold?
Another example in a more familiar setting might be to take the distance function $F(x,y) = x^2 + y^2 - 1$ and see where you can apply the IFT to the level set $F^{-1}(0)$.
Apr
15
revised What are the conditions for the implicit function theorem to hold?
added 1130 characters in body
Apr
15
answered What are the conditions for the implicit function theorem to hold?
Apr
13
revised example of not surjective isometry on a bounded not closed set.
added 439 characters in body
Apr
13
answered example of not surjective isometry on a bounded not closed set.
Apr
13
comment Let $S$ be a regular surface and let $F$ be a diffeomorphism. How can I prove that the image $F(S)$ is a regular surface?
Sorry --- is this for a curves-and-surfaces class? The Jacobian of $F$ is an invertible matrix.
Apr
13
comment Let $S$ be a regular surface and let $F$ be a diffeomorphism. How can I prove that the image $F(S)$ is a regular surface?
$F$ is a diffeomorphism, so its differential is an isomorphism on each tangent space.
Apr
13
comment Let $S$ be a regular surface and let $F$ be a diffeomorphism. How can I prove that the image $F(S)$ is a regular surface?
@user111854 What map carries a sphere to an ellipsoid?
Apr
13
answered Let $S$ be a regular surface and let $F$ be a diffeomorphism. How can I prove that the image $F(S)$ is a regular surface?
Apr
12
comment Construct a subset of a rectangle
@Marc The question could also be interpreted as, find $S\subset Q$ with $\bar{S}=Q$ such that no two points of $S$ share any coordinates.
Apr
11
comment If $g$ is continuous then $x^ng(x)$ converges on $[0,1)$
A bit late, but ... +1 for presenting your thoughts and previous progress in the question.
Apr
10
answered Question from a topology textbook regarding the uniform topology
Apr
7
comment Determining the rate of change of a radius as a sphere loses volume
+1 for a question which shows work.
Apr
5
comment Proving $\,2\sqrt 2 + 1\,$ is irrational by contradiction
Well, if you already know that $\sqrt{2}$ is irrational, aren't you done on line 2 ... ?
Apr
4
revised Examples of Sobolev Spaces
added 32 characters in body
Apr
4
comment Examples of Sobolev Spaces
Ah! I missed the ${}_0$! Editing accordingly.
Apr
4
answered Examples of Sobolev Spaces
Apr
3
comment Fractional Laplacian on the torus
In general, the natural domain of $(-\Delta)^s$ is the Sobolev space $H^s$. You might find this helpful: arxiv.org/pdf/1104.4345.pdf