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Jan
31
comment Example of two sequences $(a_n)$ and $(b_n)$ such that both of them are bounded, neither of them is convergent, but $(a_n + b_n)$ is convergent?
Think $-1$'s and $+1$'s
Jan
11
comment Why is $[\widetilde{v},\widetilde{w}]_p(f)=0$ when $f$ has a critical point at $p$?
@Marc $[\tilde{v},\tilde{w}]$ is a vector field, and $[\tilde{v},\tilde{w}]_p(f)$ means to differentiate $f$ at $p$ in the direction of $[\tilde{v},\tilde{w}]_p$. Since $p$ is a critical point of $f$, the total derivative of $f$ is zero at $p$, and hence every directional derivative is zero as well.
Dec
5
revised Vanishing of Tor sheaf on a union of subschemes with vanishing Tor.
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Dec
4
asked Vanishing of Tor sheaf on a union of subschemes with vanishing Tor.
Nov
28
awarded  Nice Answer
Nov
14
awarded  Popular Question
Oct
19
comment Why in a directional derivative it has to be a unit vector
For the directional derivative in a coordinate direction to agree with the partial derivative you must use a unit vector. If you don't use a unit vector the derivative is scaled by the magnitude of the vector.
Sep
23
comment Is there a theorem or axiom stating that integers added to integers always yields integers?
To rigorously define the integers one can start by defining the natural numbers using peano's axioms (en.wikipedia.org/wiki/Peano_axioms), and then formally adjoin negatives, as is done in "Foundations of Analysis" by Landau. There are also other ways to define the natural numbers and integers that are more set theoretic in nature.
Sep
23
comment Is there a theorem or axiom stating that integers added to integers always yields integers?
This ultimately will follow from the definition of addition of integers. To define addition, of course, we must first define the integers. However, in most introductory proof classes the integers are not rigorously defined and the most basic properties of the integers (such as closure under addition) are taken for granted. So for your case I think it is ok to state it without any sort of justification.
Aug
22
comment Why must two integral domains with $17$ elements be isomorphic?
Finite rings are integral domains iff they are fields. And there is only one field of order $17$.
Aug
21
comment Is this true: Every open set $A$ contains a neighborhood whose closure is a subset of $A$.
If you want a formal proof, take a point $x$ in your open set $A$ and choose an open ball $B_\varepsilon$ centered at $x$ contained in $A$ of radius $\varepsilon$. Then the open ball $B_{\varepsilon/2}$ centered at $x$ with radius $\varepsilon/2$ has closure contained in $B_\varepsilon$ which is contained in $A$.
Aug
21
answered Is this true: Every open set $A$ contains a neighborhood whose closure is a subset of $A$.
Jul
27
comment Find a plane with distance $3$ from $3x-y-z = 0$
@Mr.Fry your comment is definitely correct and it seems to me to be the best way to approach this problem. In fact for this example we can choose $P=0$ and then it is quite easy to calculate an equation for the plane (i also checked that the equation agrees with the current answer, which has 3 upvotes already)
Jul
27
comment Find a plane with distance $3$ from $3x-y-z = 0$
@Mr.Fry why not just post your comment as an answer?
Jul
26
comment Intuition for the Cauchy-Schwarz inequality
@IttayWeiss I'm really thinking about vectors in $\mathbb{R}^n$ here. In this case $\theta$ has a clear meaning without CS and one can prove the above formula using the law of cosine's (as I'm sure you know). I just finished teaching calc 3 so I got into the habbit of thinking concretely about some of these things.
Jul
26
answered Intuition for the Cauchy-Schwarz inequality
Jul
25
revised Using Green's theorem to find an area.
added 4 characters in body
Jul
25
comment Using Green's theorem to find an area.
Thanks, you can upvote and/or accept this answer if you'd like.
Jul
25
revised Using Green's theorem to find an area.
added 141 characters in body
Jul
25
answered Using Green's theorem to find an area.