Mathematics - 2018-02-01 (page 1 of 4)

orbit-stabilizer

00:01

got it!

Ted Shifrin

I knew you would, @orbit.

Meow Mix

00:17

hi

Ted Shifrin

Howdy, Meow. You feeling better?

Meow Mix

yeah, no fever

Ted Shifrin

Cool. I wish my neck would stop aching.

You go back to school yet?

Meow Mix

tomorrow

i had a fever this morning

Ted Shifrin

Oh :(

Meow Mix

00:21

anyways i finished (most) of chapter 13 so i read 14

Ted Shifrin

Ah, the cool stuff :P

Meow Mix

there's a few im stuck on

Ted Shifrin

There are a lot of challenging problems, even in 13.

Clarinetist

Question for you guys

There's a MSE user who has his name on this website

I'm pretty sure he lives near me and have seen what I think are his tutoring profiles on other websites

Meow Mix

screenshots.firefox.com/7GxKnB384Q4QRYxO/null

Clarinetist

00:24

Would it be weird if I started a MSE chat room with him and asked him if he'd be willing to tutor stats?

Meow Mix

i dont know how to approach this

Ted Shifrin

Well, don't be too confrontational, @Clarinet.

Meow Mix

at first i htought somehow making it a derivative but it's a right sided limit

Ted Shifrin

@Meow: You can do one-sided derivatives, but you're on the right track. What is going on, intuitively?

Meow Mix

hmm

we're looking at the behavior of $x$ times that integral as $x$ approaches zero from the right

Ted Shifrin

00:29

So $x$ is going to $0$. What is the integral doing?

Meow Mix

approaching infinity

i think it's like an indeterminate form

$0 \cdot \infty$

Ted Shifrin

So does this phenomenon remind you of something you learned in chapter 11?

BTW, Meow, have you seen/learned Ptolemy's Theorem?

Meow Mix

i feel like ive heard of it before

Ted Shifrin

Quadrilateral — diagonals and sides ...

Meow Mix

yes

Ted Shifrin

00:31

I just wrote up a bunch of exercises for my AoPS kids on it (using complex numbers and/or geometry). Are you interested?

Meow Mix

yeah go ahead and send it

Ted Shifrin

You can proofread for me :P

Anyhow, back to your question.

Meow Mix

using the last e-mail address i sent w/

umm 11 looks like just critical values

Ted Shifrin

Huh?

Meow Mix

and MVT

Ted Shifrin

00:37

and ...

Daminark

... and calculus

Ted Shifrin

Demonark is, as always, o so helpful.

Daminark

I do take pride in this

Eric Silva

Sup

Ted Shifrin

howdy, Eric ... I spent the day writing up inversive geometry problems I'm sure my students won't have time to do.

Meow Mix

00:44

inverse geometry is kewl

Clarinetist

I know nothing

Ted Shifrin

Did you get my email, Meow? Is that why you're so quiet?

Eric Silva

I was trying to find diffeomorphisms of the 2 sphere satisfying a thing and was trying to think if inversive geo could give me answers

Meow Mix

i did

Eric Silva

But then my brain exploded

Daminark

00:45

F

Ted Shifrin

Eric, clean up the mess, please.

Eric Silva

Lol

Meow Mix

yeah all i see here is lhopitals, mvt, and critical values

Ted Shifrin

Ahem. All you see is ... what?

Meow Mix

lhopitals, mvt, and critical values

Ted Shifrin

00:46

And remind me what our question looks like?

Meow Mix

its a limit

Ted Shifrin

That looked like ...

Meow Mix

$\lim_{x\to 0^+}x \int_x^1 \frac{f(t)}{t} dt$

Ted Shifrin

I remember the problem, dopey.

Meow Mix

$0 \cdot \infty$?

Ted Shifrin

00:48

This might even be one of the problems I contributed to the book decades ago. I forget.

And $0\cdot \infty$ can be rewritten how ... ?

Clarinetist

Ted, feel free to tell me if I'm wrong:

Daminark

$0\cdot 8^T$

Ted Shifrin

You're wrong.

Clarinetist

Hint: you want $\infty/\infty$. Why?

Meow Mix

$\frac{1}{\infty} \cdot \infty$?

Ted Shifrin

00:49

Which just happens to be ...

Meow Mix

$\frac{\infty}{\infty}$

lol

Ted Shifrin

I think you skipped too many of the Chapter 11 problems.

Clarinetist

What book is this?

Ted Shifrin

Spivak's Calculus

Clarinetist

Yeah...

I never read this book in depth, but this is really important

(some day, when I have time... I will)

Ted Shifrin

00:53

Don't forget to learn my book, too.

Clarinetist

Of course!

I definitely will

Ted Shifrin

LOL

Yeah, right :P

Clarinetist

I'm planning on Spivak > your book > Pugh when I manage to finally get some free time

I bought the Stein-Shark-Ican'tspellthelastnamefrommemory series 2 weeks ago

Ted Shifrin

You'll actually know most of Pugh if you do the first two books right.

I'm disappointed in the Stein-Stakarchi complex analysis book. I don't know the others well.

Clarinetist

That's good to hear (about Pugh)

@MeowMix Here's a hint: chapter 11, exercise 56. This is really important to know when it when you see it

Fourier analysis is starting to come up in my studies... so I figured I might as well get the whole set while I'm thinking about it

Ted Shifrin

00:59

I guess you're wealthy these days, Clarinet :)

Clarinetist

I haven't spent that much at once in a long time

But then again...

I did meet with the insurance agent last week. I think my book collection is worth 60k, roughly

I'll be tallying up the actual figure this weekend

Ted Shifrin

I gave up figuring things like that decades ago. And I gave away most of the books in my office.

Clarinetist

Yeah, what the heck am I going to do with these books when I don't want them?

I thought about that a few days ago

I guess I have... at least another 30-40 years to decide that

orbit-stabilizer

Build a house of books

Xander Henderson

@TedShifrin My plan is to die before retiring, and force my wife and/or daughter to sell/give away/burn my books.

Ted Shifrin

01:06

Naive plan.

Xander Henderson

But then I hope to die of a stroke 5 minutes before office hours, and thereby traumatize the poor undergraduate that finds my still warm body.

Ted Shifrin

One of my colleagues had this plan, and then recently he got very ill with cancer ... and his poor students didn't fare too well during the semester. He really should have retired before the downhill trek started, but he refuses.

I decided it was better to go out while still a good teacher than to deteriorate in the job.

Xander Henderson

See, men in my family have a habit of stroking out while they are still otherwise more-or-less healthy. My plan might work!

Ted Shifrin

heya Cookie

@Meow: Did you fall asleep?

Xander Henderson

But yeah, I hope to retire and/or die before my instructional ability deteriorates.

My wife has long standing instructions to leave a bowl of "candy" out on the table when/if I start losing it.

(By "candy", we mean "very strong opioids in overdose quantities")

Ted Shifrin

01:10

The issue with Alzheimers and dementia is that one tends to be oblivious and in denial by the time it is really an issue.

Xander Henderson

@TedShifrin Yar... that's what really scares me about dementia.

Ted Shifrin

I hope that by the time your time comes that such things aren't going to land her in jail, Xander.

Clarinetist

Wow, what a question:

142

Q: Professor creates assignment making students advocate for a bill being presented to Congress. Is this legal?

I attend a public university in Kentucky (USA). My professor has assigned an advocacy assignment that requires students to write three letters of support for a certain House bill that will soon be voted on (H.R. 592 / S. 109). The professor has given us all a template where we get to "customize"...

Xander Henderson

But, like I said, men in my family tend to stroke out while still in complete control of their mental faculties, and dementia is unheard of.

Ted Shifrin

This sounds highly illegal, Clarinet.

I wish I were as confident as you, Xander. All power to you.

I figure that my cancer might come back in a while and "save" me.

Xander Henderson

01:12

@Clarinetist That looks totes illegal

Clarinetist

143 upvotes. I think that's the most I've ever seen on Academia SE

Xander Henderson

@TedShifrin I'm not sure that I am "confident" so much as "really f'n' afraid of dementia"

and in denial

Meow Mix

maybe not

Ted Shifrin

@Xander: I'd say you're a little too young for this right now.

@Meow ???

Meow Mix

sorry i was doing a bio assignment

Ted Shifrin

01:14

Oh.

Meow Mix

im done

okay so l'hopitals here

Xander Henderson

@TedShifrin You are probably right, though I am rather old for a grad student, and only 10 years short of my grandfather's first stroke

Ted Shifrin

Finally. :)

Damn, Xander. What is rather old? Over 30?

Clarinetist

When I was in Madison, I met a professor on the bus who was reading an advanced linear algebra textbook. I asked him if he was a grad student. He was flattered.

Ted Shifrin

Hopefully you take better care of yourself (exercise, healthful eating) and have access to statins and other drugs if necessary.

Xander Henderson

01:16

I'm almost 40

Ted Shifrin

tries to remember 40

Xander Henderson

heh

Meow Mix

tries to remember 4

Xander Henderson

And yeah, I take care of myself, but so did my grandfather (who was out cross-country skiing three days before his first stroke)

Ted Shifrin

did he have annual physicals, etc.?

smacks Meow

Xander Henderson

01:17

but I am super-paranoid about my cholesterol (which is, thankfully, good)

Ted Shifrin

I think living a life in paranoia is not healthy, so quit it.

The paranoia, not the living.

Clarinetist

^Been there, done that

Xander Henderson

@TedShifrin My cholesterol is good, so I'm not that worried about it; and I used to get very regular physicals, but since the small child was born, I've had less time to fly, and therefore no need for biennial flight reviews and the associated physicals

Ted Shifrin

Once a year will suffice.

Xander Henderson

heh... It's been almost 7 years now (but the blood work gets done by Red Cross, or whatever the organization that steals my blood is)

Ted Shifrin

01:20

No, get annual physicals. Independent of your paranoia.

Health insurance covers that automatically (usually).

Meow Mix

01:38

the proof of lhopitals rule in chap 11 is kind of confusing

i mean i understand it but it's hard to know where these decisions came from

like why was it chosen to use cauchy mean value theorem

Clarinetist

5

Q: In the conventional less-than-rigorous calculus course, is Velleman's new notation useful only once?

Daniel Velleman has written Calculus: A Rigorous First Course. It seems to me that it's questionable whether such a course should be used for all but a small number of students, not just because most will never appreciate logical rigor, but because there's so much stuff they may appreciate that ...

calculus notation

Ted Shifrin

@Meow: That's the only correct proof I know.

You have to end up evaluating $f'(c)$ and $g'(c)$ at the same $c$ or else you're up the crik without a paddle.

@Clarinetist: I doubt I'd even be interested in trying to use Velleman's book for a standard calculus course.

Clarinetist

I've never even heard of the text. All I noticed is that the notation looks strange.

Ted Shifrin

Is it even published? Or is it on line?

Clarinetist

Yep, it's published: amazon.com/Calculus-Rigorous-Course-Aurora-Originals/dp/…

Ted Shifrin

01:52

Two fabulous mathematicians at Berkeley, Marsden and Weinstein, wrote a rather non-standard calculus book (but it wasn't pretending to be ultra-rigorous, but they had a new way of thinking about derivatives geometrically). It never, never caught on.

Clarinetist

Here's my opinion on Calc. I, given that I spent June through December teaching someone that material:

Either:

1) teach it using Spivak

Ted Shifrin

Nah. I only would ever use Spivak with a superb student who's super into math.

Nor am I advocating my text as a text for the multivariable masses. Far from it.

Meow Mix

im flattered /s

Clarinetist

or 2) if you're going to use Stewart or something similar, toss out Rolle's Theorem, the Mean Value Theorem, linear approximation, and bring in partial derivatives and double integrals

Ted Shifrin

The complaints in the post are that Velleman teaches at Amherst, which has an above-average level of intellect amongst its student body.

Clarinetist

01:54

I did this for a friend of mine. Turned out well.

Ted Shifrin

I disagree. I hate Stewart, but I disagree.

Linear approximation is the whole point of derivatives, man.

It's my whole motivation for multivariable derivatives.

Clarinetist

Okay, I get that. But I don't understand the point of having someone compute, by hand, $\sqrt{4.01}$. Sure, neat.

Ted Shifrin

I would agree that we should omit the parts of the course that are about theory for the sake of mathematical masturbation. If it's totally obvious, which a lot is not, then don't make a big deal out of it if you're not going to do a more rigorous course.

@Clarinet: In this day and age with calculators it's stupid. But apply it with an implicit differentiation problem and then it is not. (Yes, graphing calculators can do that, but regular calculators can't.)

Clarinetist

I understand how important linear approximation is conceptually, but in the time I tutored calculus... it was always taught with stupid exercises like the one above. At the end of the day, it was "memorize this formula"

Ted Shifrin

There are tons of stupid exercises. That doesn't make the concepts stupid.

Clarinetist

01:57

True

Ted Shifrin

I agree with you totally on those.

Meow Mix

so umm ted can you confirm my understanding of this

Ted Shifrin

I'm not willing to bastardize the course and make it purely computational with no concepts. The whole point of teaching math is to make people analyze and think.

Meow Mix

nevermind i got it

Clarinetist

I think my bias toward stats definitely influenced how I taught it

There were a few things that I brought in theory-wise that I've never seen mentioned in a Calc. I course, but not much time, unfortunately

e.g.

$$\lim_{x \to a}[f(x) + g(x)] = \lim_{x \to a}f(x) + \lim_{x \to a}g(x)$$

Ted Shifrin

02:00

@Meow: Always glad to be useless.

Well, that's a stoopid statement as it stands, @Clarinet.

Clarinetist

@TedShifrin I should clarify: a counterexample to my edited statement above

Ted Shifrin

You have to stipulate that the limits of $f$ and $g$ exist.

No, there's no counterexample if you are careful to state hypotheses, which all mathematicians MUST be.

Clarinetist

@TedShifrin True! But I've rarely seen students pay attention to that assumption in Calc. I tutoring

Ted Shifrin

Well, that's part of the tutoring.

Seriously.

Clarinetist

Hard to tutor that when 99% of the time, the limits do exist for exercises in the course

Ted Shifrin

02:03

Same thing with $x\sin(1/x)$ and using the "product rule" for limits.

So maybe we shouldn't worry about this with the generic engineering calculus student.

I'm OK with not discussing craziness in a standard calc class. I certainly taught multivariable (except for my particular course) without worrying pedantically about what differentiable meant. I'm fine with that.

There are plenty of interesting concepts, regardless, and in Calc I I want to emphasize the "applied" word problems anyhow.

Clarinetist

How does one balance (for lack of a better word) "thought" and computation in a calc course? There doesn't seem to be a good answer to that. Perhaps it's because I just left a state government job, but I blame most of not being able to teach math rigorously on K-12 education regulations

Ted Shifrin

The Common Core thing is trying to make students use and understand concepts ... but parents and (unprepared) teachers are undermining it.

Not to mention our idiots in charge of education in the country.

Clarinetist

My job was funded by federal dollars toward school accountability. Did it for two years. It was eye-opening.

Ted Shifrin

I'm about done with anything bureaucratic or government-based in this day and age.

@Meow: Are we OK? Did you solve that integral question?

Clarinetist

@Ted The last job I was in was very interesting. Most, if not all, of my colleagues had Ed.D.s or were former teachers, and... it was a really strange contrast: knowing that everyone cared about serving public education, yet at the same time, federal and state laws prohibit them from doing specific things to fulfill the mission

and people would complain about that in private. But there's nothing that could be done because of law.

Ted Shifrin

02:12

I'm cynical about Ed.D.'s, but that's another matter.

The fact that the UGA school of ed inculcated in their students the attitude that any good teacher would be giving all A's (after all, anything less than an A means the teacher didn't do a good job) made me extremely cynical and angry.

Clarinetist

Wow

Ted Shifrin

Education is a mess, yes.

But our current administration is all about dismantling education to get more votes.

beyond furious

Clarinetist

My perspective is this: the administration is trying to do what happened in WI at a national level

Ted Shifrin

yes, and that idiot governor in WI was sooooo competent.

Clarinetist

The dynamic in WI was really bizarre with the governor

Ted Shifrin

02:16

I sure wouldn't want to be young in today's world. So depressing.

Clarinetist

Coincidentally, the department of ed in WI was the only state agency in WI which was, by the WI constitution, free from the governor's control

Ted Shifrin

well, Trompolini is trying to erase law in the US and be a dictator. Very clearly.

Clarinetist

DeVos is an idiot. There was a question that Al Franken asked her that I knew the answer to better than she did

I don't even know what's going on in higher ed, but I'm still keeping up with what goes on in K-12

Here's what I imagine will happen 20-30 years from now:

Daminark

gets popcorn

Clarinetist

1) Most public K-12 institutions will have either merged into much larger conglomerates or will be gone

2) Most kids won't be attending K-12 institutions anymore, but religious "choice" schools

In WI, 1) is already happening

Ted Shifrin

02:20

I have to hope the current administration will be short-lived and things will (slowly) revert to sanity.

Clarinetist

One year has elapsed. Three years remaining, at the very least. Honestly, what this last election has shown me more than anything is that if you market things the right way and target people appropriately with good data, you can do anything

Daminark

My guess is that it'll be less obviously out of it but... The waves that brought it don't seem to me to be an 8 year fad or something, I think some (if not a lot) of the influence will persist

William Oliver

Is the tensor product between two vector spaces the same operation as the tensor product between two individual vectors?

Ted Shifrin

@WilliamOliver: You define a vector space by its elements.

Clarinetist

We're in this era where you can collect data on anything and you can use that to influence people. There's no precedent for that.

Ted Shifrin

02:23

This administration is — what — 80% about total lies?

Jenna Sloan

Which administration?

William Oliver

yeah but I often see people use the notation $V \otimes W$ where $V$ and $W$ are vector spaces, and then $v \otimes w$ where $v \in V$ and $w \in W$

Is this really the same operation?

Ted Shifrin

Right, @William, a general element of $V\otimes W$ is a sum of things like $v\otimes w$.

William Oliver

But what does that mean?

Ted Shifrin

It's notation on the sets, an operation on the elements.

Clarinetist

02:25

If there's one thing I wish I could do right now, it would be to pursue a Ph.D. in stats and start teaching students about how data can be used to take advantage of people

Ted Shifrin

It's a formal operation with the properties that I'm sure you know.

William Oliver

Oh okay so they aren't technically the same thing

Ted Shifrin

Yup @Clarinet ... I realized a few years ago that a good stats education is important.

@WilliamOliver: Well, what does $V\times W$ mean?

Clarinetist

BTW, if any of you haven't read this book... it is a really good book on ethics

amazon.com/Weapons-Math-Destruction-Increases-Inequality/dp/…

William Oliver

@TedShifrin Well I'd assume that we represent that usually (in terms of the elements) as $(v, w)$

As opposed to $v \times w$

Ted Shifrin

02:27

Right ... so the notation for the sets is defined in terms of its elements. Same thing with tensor product, except you have to include sums ("generated by ...")

Clarinetist

O'Neil got her Bachelors at UC Berkeley and Ph.D. at Harvard in 1999 (both in math, I believe), left academia 8 years later

William Oliver

@TedShifrin I am confused by what you meant by that

Ted Shifrin

How so?

@Clarinet: I'm sure I'm largely on her side, but I'd have to read it.

William Oliver

Are you saying that the notation $v \otimes w$ is analogous to $(v, w)$?

Clarinetist

@TedShifrin I'll just say that I like her opinions, but she rants quite a bit

Ted Shifrin

02:30

Sorta, @WilliamOliver, but not precisely.

I'm saying that you define a set by giving its elements. The notation you use is arbitrary.

William Oliver

Well I mean notation wise, I know that the operations behave differently

Ted Shifrin

The difference is that direct product has no relations. Tensor product has relations. Relations = rules.

William Oliver

Right okay

Clarinetist

@TedShifrin The thing I found really interesting was reading about one of these topics and seeing a similar situation play out right in front of me in my job from an external consultant

Ted Shifrin

Well, I guess direct product also has. $(v,w)+(v,w') =(v,w+w')$, etc. So it's very similar.

William Oliver

02:33

but really $V \otimes W$ is not the same thing as $v \otimes w$. One defines a vector space, the other defines a vector in that space.

Ted Shifrin

But for products you can do $(v,w)+(v',w') = (v+v',w+w')$ and that won't work for tensors.

Of course, William.

But I'm trying to emphasize that elements of $V\otimes W$ cannot necessarily be written in that simple form, whereas elements of $V\times W$ can.

William Oliver

Hm, I suppose I am confused then as to why they can't

Jenna Sloan

@Clarinetist lol

Ted Shifrin

In $V\times W$, $(v,w)+(v',w') = (v+v',w+w')$, but you can't write $v\otimes w + v'\otimes w'$ as $v''\otimes w''$ in general.

Addition works componentwise, but multiplication doesn't work that way.

William Oliver

Oh right okay

Ted Shifrin

02:36

That's all I'm saying.

William Oliver

Oh okay that makes total sense

thanks

Ted Shifrin

Sure thing :)

Meow Mix

02:47

bac from shower

William Oliver

One more question, would it be correct to define the tensor product $T$ of two vector spaces $V$ and $W$ as the set of all bilinear maps $t: V \times W \to T$?

Ted Shifrin

Nope.

William Oliver

Lol

Why

Ted Shifrin

$T$ is the set of bilinear maps $(V^*\times W^*)\to\Bbb R$ (assuming they're $\Bbb R$ vector spaces).

Remember that $V^*$ is the set of linear maps $V\to\Bbb R$ and $(V^*)^*$ is the set of linear maps $V^*\to\Bbb R$. When $V$ is finite-dimensional, $(V^*)^* \cong V$.

William Oliver

I see. So actually, can the tensor product itself be thought of as a bilinear map $\otimes: V \times W \to T$ then?

MatheinBoulomenos

02:55

it's even the universal bilinear map

Ted Shifrin

Yikes, no.

Tensor product is not a map.

All bilinear maps $V\times W$ to $Z$ factor through and give a linear map $T\to Z$.

MatheinBoulomenos

I mean, the tensor product itself is a vector space, but the map $V \times W \to T$ should be considered part of what makes $T$ the tensor product

Ted Shifrin

@Mathein: You should read the last half hour's stuff.

William Oliver

But $\otimes$ is an operator right??

Ted Shifrin

operation, not operator

William Oliver

02:57

Very confused

Ted Shifrin

Well, I'm going to leave and cook dinner, so maybe that'll help.

William Oliver

Haha thanks for the help so far!

Jacksoja

I don't see how exactly that would help

Ted Shifrin

Good luck. See you the next time :)

@Jacksoja: Sometimes i just muddle people up.

Bye for now.

William Oliver

@Jacksoja It helps @TedShifrin

Jacksoja

02:59

@TedShifrin Haha that is fine, bye and enjoy your meal

William what you need tensor product for ?

It's a nasty thing

William Oliver

I am just trying to understand it

I have been for so long haha

I am finally starting to get it I think

Jacksoja

if you think about it , it is just a formal symbol

and make sure you understand , that there are like 3-5 different meaning of that symbol

V tensor W, e tensor f , v tensor w , don't mean the same thing

William Oliver

Yeah! Haha thats what I am starting to get.

Jacksoja

draw the analogy on complex numbers, immagine you never knew what i is

and you did all complex analysis with 2 -tupples ,augmented with some element that behaves like i

William Oliver

So $\otimes$, like $i$, is sort of its own object right?

That can't really be defined in terms of other things, but we can say things about its properties.

Is that what you mean?

Jacksoja

03:06

I meant more like (0,1) for the imaginary unit

but we define the complex number arithmatic as we used to

(a+bi) (c+di) = ac-bd+i (ad+bc)

but we ignore the "i" and write it as 2 -tuples

William Oliver

So we ignore the $\otimes$ in $v \otimes w$ and and write it as 2 tuples? Just keeping the operations intact?

Jacksoja

My point is , accept the defintion, play with it some decent amount of time, then it will be come second nature, we worked with numbers since we were kids, but did we really knew what numbers are ?

do we really know now what they are? we just use them because we somehow accepted them and got familiar with em

Leaky Nun

18 mins ago, by Ted Shifrin

Tensor product is not a map.

William Oliver

I know what you mean. I suppose I just don't have something that I need to use them for right now. I don't entirely understand what motivates them, and thats the problem.

Leaky Nun

@TedShifrin the tensor product is the object equipped with two projection maps

Xander Henderson

03:14

When life gives you lemmas, make theoremade.

William Oliver

I think have a vague understanding that they help with invariant properties like the dot product. They help with formalizing them. I have a vague understanding that they help us characterize multilinear maps with n dimensional arrays of numbers.

If I forget about tensor products entirely, and just think about multilinear maps. It makes sense to me

@LeakyNun What do you mean?

Leaky Nun

@WilliamOliver ignore that :P

William Oliver

Haha okay

But like, I think I kind of get that when defining something like the dot product, which is a multilinear map and (I think) is technically a tensor, we can define it among all multilinear maps of that type by defining a 2 dimensional array. Is that correct?

If we have a multilinear operation $t: V \times W \to R$ and we let $e_i$ be the basis for $V$ and $\epsilon_i$ be the basis for $W$. The multi linearity of $t$ means that we can define $t$ entirely by defining what $t(e_i, \epsilon_j)$ evaluates to. So, for example, if we define $t(e_i, e_j) = 1$ where $e_i$ is the basis for cartesian …

Leaky Nun

the multilinear map is not the tensor

William Oliver

But I thought thats what they were defined to beee

So confusedd

Leaky Nun

03:27

any multilinear map $V \times W \to R$ can "be thought of" as a linear map $V \otimes W \to R$, but the map itself is not the tensor

William Oliver

Whhaaaaaaatt

I am completely and utterly confused now

Leaky Nun

let's look at the cross product

it is a bilinear map $\Bbb R^3 \times \Bbb R^3 \to \Bbb R^3$

so it can be thought of as a map $\Bbb R^3 \otimes \Bbb R^3 \to \Bbb R^3$

William Oliver

Right

Leaky Nun

but the map is not the tensor product right

William Oliver

Ooh

Okay, but the Tensor is the map right?

The tensor product isnt

but the tensor is

Daminark

03:34

2

Leaky Nun

not really

Daminark

Contravariant Functors IRL

Leaky Nun

the tensor is just an object of the tensor product

William Oliver

Well, so what is a concrete example of a tensor then. Is the dot product not an example?

@Daminark Thats how my brainfeels right now

Leaky Nun

so the tensor version of the dot product sends $e_i \otimes e_j$ to $1$

@WilliamOliver tensors only make sense inside the tensor product

and they look like linear combinations of objects of the form $v \otimes w$

Xander Henderson

03:40

Pardon my French, but contravariant functors can suck my kid.

Daminark

@Xander contravariant functors throw me off

William Oliver

Okay, so is $e_i \otimes e_j$ is just its own thing? Like not really definable in terms of anything else, we just know that its a way of pairing things that behave in a multinear way?

Daminark

Covariant functors <3

0celo7

how about no functors

Xander Henderson

Honestly, if you need to use the word "FUNCTOR", you are doing something wrong with your life.

0celo7

03:43

@XanderHenderson amen

Daminark

I never need to use it in caps lock tho

0celo7

@WilliamOliver honestly unless you're trying to be very algebraic just think of tensors as multilinear maps

Xander Henderson

On the other hand, I nearly told someone today that bijections are the wrong kind of morphisms, and that they should really be thinking about the category of metric spaces, where the morphisms are isometries.

(Or maybe bi-Lipschitz maps)

I threw up a little in my mouth.

William Oliver

@0celo7 Is my reasoning here chat.stackexchange.com/transcript/message/42612786#42612786 correct?

0celo7

@WilliamOliver I am not sure what dot product you're talking about.

Daminark

03:47

@0celo7 >not trying to be very algebraic

0celo7

@Daminark problem?

William Oliver

@0celo7 The one that is sometimes defined to be $|v||w|cos(\theta)$

0celo7

@WilliamOliver I'm saying I don't know what the dot product has to do with tensors. The dot product extends to the tensor algebra in a natural way, but I'm not sure what your question is.

William Oliver

Is the dot product an example of a tensor

0celo7

That's a loaded question...a physicist might say yes, I'm inclined to say yes, but the small amount of algebraic training I have forces me to say no.

William Oliver

03:52

I give up haha. I have no idea what a tensor is for.

0celo7

@WilliamOliver It's probably not possible to understand why anyone would care about them without learning geometry.

William Oliver

Differential geometry?

0celo7

Yeah

William Oliver

I need a good resource for that. It seems that all of the resources I have seen try to define a tensor first, without providing motivation. I suppose I should just push through it.

But I still don't understand what it is

Like I feel like I need to know what it is to know what its for, but I need to know what its for to understand what it is

0celo7

@WilliamOliver I struggled for a long time with them until it just made sense

People try to do all of these things to motivate them but they never work

William Oliver

03:56

But how did they originate?

0celo7

It's one of those things that's so natural when you understand it that it's hard to explain to people who don't get them

William Oliver

I don't know how to bridge that gap for myself

Transcript for

$Mathematics$ Mathematics