Mathematics - 2024-06-24

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Xander Henderson

00:38

The Princeton Companion to Mathematics

The Princeton Companion to Mathematics is a book providing an extensive overview of mathematics that was published in 2008 by Princeton University Press. Edited by Timothy Gowers with associate editors June Barrow-Green and Imre Leader, it has been noted for the high caliber of its contributors. The book was the 2011 winner of the Euler Book Prize of the Mathematical Association of America, given annually to "an outstanding book about mathematics". == Topics and organization == The book concentrates primarily on modern pure mathematics rather than applied mathematics, although it does also cover...

Novice

there is also an applied version, but these books may not be what croco wants

Xander Henderson

But, really, Google.

3 hours later…

Thomas Finley

03:31

1

Q: Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between $z=0$ and the paraboloid $4az=x^2+y^2$ equals $\frac{3\pi a^3}{8}.$

Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between $z=0$ and the paraboloid $4az=x^2+y^2$ equals $\frac{3\pi a^3}{8}.$ I tried solving this problem as follows: The equation of the cylinder is $x^2+(y-a)^2=a^2.$ So, the region $E$ as given in the problem is $$E=\{(x,y,z):...

integration multivariable-calculus surfaces volume

Can I consider my solution to be valid?

7 hours later…

CroCo

10:05

@Novice @Novice This book is excellent, although not exactly what I need, it contains valuable information.

CroCo

10:16

@leslietownes @leslietownes yes I totally agree.

@leslietownes To be more precise, I'm looking for an encyclopedia that contains details about major mathematical fields(e.g., calculus, topology, linear algebra, etc.) without filling the book with extensive information. For example, the book "Lectures on Real Analysis" by Finnur Lárusson is a good example, as it is rigorous and short but only about fundamental analysis. On the other hand, books by James Stewart tend to be loquacious (i.e. they are suitable for first reading).

@Jakobian this is a good answer.

@XanderHenderson this is nice book that I definitely read but I'm afraid it is not rigorous enough.

@XanderHenderson I've read several books on linear algebra. Some of them were not worth the paper they were printed on. Google will bring anything up. I always fall into the trap of books with excellent titles. Some authors have a tendency to mimic each other.

1 hour later…

Xander Henderson

11:37

@CroCo My point is not that you should use Google to fund books, but that you use Google when you need to look something up.

Because, frankly, I didn't think the book you want exists.

Binky McSquigglebottom

$\int 2 \cdot \sqrt{x-x^2} - 4x\cdot \sqrt{x-x^2} dx$

How do I do it in the shortest way possible?

@XanderHenderson

Hi :)

Binky McSquigglebottom

12:17

user image

Its correct ?

Xander Henderson

12:34

@BinkyMcSquigglebottom I don't really see what you are doing there, and the mixture of $a$s and $x$s confuses me. It seems like it would be easier to make the change of variables $u = x-x^2$...

Then $\mathrm{d}u = 1 - 2x \,\mathrm{d}x$, hence $$ \int 2\sqrt{x-x^2} - 4x\sqrt{x-x^2}\,\mathrm{d}x = 2 \int (1-2x)\sqrt{x-x^2} \,\mathrm{d}x = 2 \int \mathrm{d}u\ \sqrt{u}. $$

Binky McSquigglebottom

13:13

Thanks

Soumik Mukherjee

13:33

@BinkyMcSquigglebottom No, as you took $x$ as a constant in the third line. You have to write $x$ as some function of $a$ if you want to integrate w.r.t $a$

Binky McSquigglebottom

13:43

Thanks

$$\int_{0}^{1} \int_{-\sqrt{x-x^2}}^{\sqrt{x-x^2}} (1 - 2x - 3y) \, dy \, dx$$ this integral comes out to me as equal to 0, but how can it be 0 if the area exists?

Also on wolfram Is 0

nickbros123

14:17

What are the minimum conditions for an open ball (in a metric space) to have it's closure as the corresponding closed ball ?

Sahaj

@nickbros123 I'm all good; how are you ??

EE18

I am unable to follow the following example and am looking for some help clarifying. In particular, I don't understand the step where the "$x^i$" come out of the $n$-lienar function $D$? $n$-linearity means that we can pass scalars through, but the $x$ aren't scalars...they're elements of the vector space $F[x]$?

user image

Maybe I need to use that $F[x]$ is in fact a linear algebra and prove some lemma to this effect. But this is not immediate is it?

nickbros123

14:37

@Sahaj decent. Rudin is taking a toll on my mental health though

Sahaj

14:55

Baby rudin or papa rudin?

Xander Henderson

@BinkyMcSquigglebottom What is the integral of $\int_{-1}^{1} x \,\mathrm{d}x$?

Remember that you are computing a signed area / volume. If you have both negative and positive area / volume, they will cancel each other out. Perhaps that is what is going on here, n'est-ce pas?

nickbros123

@Sahaj baby rudin

Soumik Mukherjee

15:36

83

A: When is the closure of an open ball equal to the closed ball?

Here is a characterization that is straight from the definitions, but which it seems may be useful when verifying that a particular space has the property. For any metric space $(X,d)$, the following are equivalent: For any $x\in X$ and radius $r$, the closure of the open ball of radius $r$ arou...

Jakobian

@XanderHenderson uno flamingo por favor

Xander Henderson

@Jakobian Sorry, I don't speak Klingon.

Jakobian

I thought I was speaking French!

Xander Henderson

@Jakobian Oh, well, I don't speak that, either.

Jakobian

@nickbros123 I think a more interesting question would be, what are the conditions for such equivalent metric to exist?

Soumik Mukherjee

15:44

@EE18 F[x] is not the vector space here, rather the ring of scalars

Jakobian

Every metric space is a subspace of normed space, so what kind of subspaces of normed spaces work?

Might be too general of a question

There are definitely spaces where such metric doesn't exist, like any discrete space

nickbros123

@SoumikMukherjee thanks! That's an interesting and intuitive characterisation. I sort of had it in mind that "if I'm on the 'edge' of a given opn ball, I would require an epsilon ball to have a point 'on the side' of the original ball

Jakobian

Moreover if your space has isolated points then such metric can't exist

nickbros123

@Jakobian you mean for what types of metric functions this property is guaranteed?

Jakobian

What's a metric function

nickbros123

15:58

d(x,y)

Jakobian

?

nickbros123

$d: X \times X \to \mathbb{R}^{+} \cup {0} $ ?

Jakobian

Huh?

nickbros123

The distance? Metric? Whatever you call it

Jakobian

You just call those metrics

And what I mean is that we are starting with some metrizable topological space and treat this like a topological property i.e. admissible metric with given property exists

psie

16:05

> Definition 4.18: $E(X\mid Y=y)$ is any random variable on $\mathbb R$, where $Q(B)=P(Y\in B)$, satisfying $$\int_B E(X\mid Y=y) dQ = \int_{\{Y \in B\}} X dP,$$ for all Borel sets $B$.

I am paraphrasing from Breiman's book Probability. Isn't it weird to call $E(X\mid Y=y)$ a random variable?

Jakobian

If your metric space X contains some non-empty proper compact open set U, then such metric also can't exist

So this means disconnected compact metrizable spaces don't have this property (and actually larger amount of spaces)

Any such space is either connected, or not compact

nickbros123

@Jakobian I see. I'm not familiar with general topology, can you not ask the question for an arbitrary set X where u can define a metric?

Jakobian

What question

You had some question like that. But I'm saying that sounds boring

So I introduce equivalent metrics - this is in grasp of someone with understanding of only metric spaces

Something holding for an equivalent metric is a way to talk about topological properties without explicitly mentioning topology

Here I mean weak or topological equivalence, and not uniform equivalence

(Some authors use equivalence to mean the latter, especially in context of normed spaces)

psie

16:31

@psie maybe the "random variable on $\mathbb R$..." part actually makes sense. In Breiman's book, he says it is a random variable on $(\mathbb R,\mathcal B)$. If $X,Y$ are random variables on $\Omega$, then $E(X\mid Y=y)$ can be viewed as a random variable on $\mathbb R$.

Thomas Finley

0

Q: Where is the mistake in this solution?

Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between $z=0$ and the paraboloid $4az=x^2+y^2$ equals $\frac{3\pi a^3}{8}.$ I tried solving the problem as follows: The equation of the cylinder can be rewritten as $x^2+(y-a)^2=a^2.$ Now, the paraboloid and the cylinder inters...

multivariable-calculus surfaces volume fake-proofs

Need some help with this.

EE18

@SoumikMukherjee how’s that? Isn’t F the field of scalars over which we take the vector space F[x]? That part I’m pretty sure about. I think I need to use the linear algebra structure of F[x] though

Binky McSquigglebottom

@XanderHenderson 0

@XanderHenderson but the first integral does not equal 0 even if the limits are opposite

It is 0 only there is x, for example 2x, x etc but already 1-2x is not 0

Maybe we are not calculating the area, but we are evaluating the volume ?

Xander Henderson

16:47

@BinkyMcSquigglebottom That whole integral can be thought of as a (signed) volume, yes.

By why would you expect that the "first" integral must be zero in order for the double integral to be zero?

Binky McSquigglebottom

@XanderHenderson you gave me an example previously

" If you have both negative and positive area / volume, they will cancel each other ou"

But for example

$\int^1_{-1} 1-2x dx$ Is not 0

Xander Henderson

Who claimed it was?

Binky McSquigglebottom

I didn't understand what you mean by negative and positive areas, they cancel each other then

Xander Henderson

If you imagine that an integral represents an area (or a volume, or whatever), then you need to keep in mind that the area is signed. You can have positive and negative areas. If you add together positive and negative areas of the same magnitude, they cancel each other out.

So it is not impossible for an integral to be zero, even if the curve being integrated is not zero.

In the case of $(1-2x)\sqrt{x-x^2}$, you might observe that the function is symmetric across the point $(1/2, 0)$. That is, $f(x) = -f(1-x)$. Hence integrating that function across any interval of the form $[1/2-a,1/2+a]$ will give zero.

Binky McSquigglebottom

But therefore I already knew that the integral was 0? because the extremes of integration were opposite

Xander Henderson

17:00

@BinkyMcSquigglebottom No. That makes no sense...

Binky McSquigglebottom

Since there were only x and y, and there were no numbers

Xander Henderson

I'm confused about what your question is.

Binky McSquigglebottom

So in this case we are not calculating the area, but the volume?

In the end it is not a double integral over dxdy , but over a function

Xander Henderson

@BinkyMcSquigglebottom In what case?!

You can think of a double integral as a volume, and you can think of a single integral as an area.

Binky McSquigglebottom

So it makes sense that the integral is 0 and that there are parts of volume below the xy plane and above the xy plane?

Xander Henderson

17:07

The double integral $\int_{a}^{b} \int_{c(x)}^{d(x)} f(x,y)\,\mathrm{d}y\,\mathrm{d}x$ can be thought of as the volume bounded between the $xy$-plane and the surface defined by $f(x,y)$ in the domain specified by the limits of integration.

Binky McSquigglebottom

@XanderHenderson In the double integral

Xander Henderson

If that integral comes out to be zero, then the interpretation is that, thinking of the integral as a volume, the parts above and below the $xy$-plane have the same volume and therefore cancel each other out.

Binky McSquigglebottom

Thanks

ephe

17:44

Where can I read about Hausdorff completion of modules?

Soumik Mukherjee

17:58

@EE18 Check the question, is F[x] the vector space in this context?

leslie townes

18:08

EE18 go back to whatever that source's definition of "n-linearity" is and check whether in context it actually requires "scalars" to come from a field (even if it calls them "scalars," in context that might not bean 'from a field')

my guess is if you just think about what that definition is, there's nothing that would prevent you from considering it for maps from matrices over a commutative ring to the ring, with the ring playing the role of 'scalars'

*might not mean

even if they don't define it, note that you can think of matrices over F[x] as matrix valued functions on F [i.e., evaluate 'x' at some element of the field and it becomes an honest to goodness from-a-field 'scalar' that you presumably know how to pull out]

nothing too shaky going on here

2 hours later…

amWhy

20:38

@robjohn Maybe plan ahead for your July 4th avatar? I'll stay tuned... :-)

user 85795

in this year of election, patriotism may become skewed

amWhy

@user85795 I understand completely. I just like robjohn's creativity. I'm fine with an upside-down parody of uncle sam... or anything

user 85795

20:54

Agreed, his creations are admirably imaginative.

Dimitris

21:32

I want to post the following question. I do not know if the question is suitable for the site and furthermore which tag I should use.
This is the from the course notes 'Mathematical Structures' of Professor Peter Cameron. [I greatly enjoyed his blog and his book on Algebra.]

'There are 26 sheep and 10 goats on the boat. How old is the captain?'

I did a googling and I found this Quora thread and this bbc article. Still, I am wondering if this a suitable question for an intro to math structures. Could it be just the author just wanted to make the reader think outside the box? …

1 hour later…

Jakobian

22:47

@Dimitris I don't think its a good question but if you post it you'll probably get lots of upvotes

Also no, it doesn't fit under mathematical structures umbrella term

As for intentions of the author it doesn't have to be the author wanting you to think outside the box, but simply put it there as a joke question

Generally speaking, mind reading is not possible

Although many people would like to know what someone meant, without elaboration this is meaningless

Xander Henderson

@Dimitris The question itself comes from a chapter which is about the very nature of mathematics. The point of the question is that it is not a well-posed mathematical question. Other work by the same author further emphases that this is an example question which was given to children, who reasoned that the captain must be Noah, Noah lived to be quite old, and $26\cdot 10 = 260$, so the captain must be 260 years old.

I.e. students often just start computing and arrive at nonsensical answers before even determining if the question is mathematical or not.

webspace.maths.qmul.ac.uk/p.j.cameron/slides/talk2.pdf

Jakobian

Best one can do is go through the route of "this makes the most sense to me, so I'll assume author meant this"

Of course there's also the agnostic position of "I don't know what author meant so I'm open minded about possibilities"

Xander Henderson

It should also be noted that the linked "book" probably should not be thought of as an independent document. It is really just a gussied up set of lecture notes, which are used by the author when teaching his classes.

Jakobian

23:06

Of course, if you're in a discussion with someone then assuming their position is generally a bad idea, although to have it you need to assume some position. To compensate, people try to use the so called "steelman" method, where you do assume what is meant, but you try to represent someone's position in the best way possible

Note the difference between discussion and trying to figure what someone you never met meant, though

The latter is generally speaking impossible

Xander Henderson

23:36

How much wood would a woodchuck chuck, if a woodchuck could chuck wood?

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