i've finally managed to get that $f(\vec a)\vec e_j$ is the jth partial derivative of $f(\vec a)$, but what exactly allows me to conclude that the corresponding vector column of the parent matrix is actually the jth partial derivative (so as to conclude that the jacobian is the matrix representation of the derivative)? "we can represent the column of the matrix as this column vector" feels super vague to me

I think the statement wouldn't be "the column is the jth partial derivative" but something analogous to "the coefficients of the column are the coefficients of the jth partial derivative"

i wallow in a sea of vagueness

what is $f(a)e_j$

i don't like that notation. or understand it

$f: \mathbb{R}^n \to \mathbb{R}^m$ and is differentiable at $a$, therefore we have $\lim \limits_{\vec h \to \vec 0}\frac{f(\vec a + \vec h) - f(\vec a) - f'(\vec a)\vec h}{||\vec h||} = 0$

we set $\vec h$ to $t\vec e_j$, where $\vec e_j$ is the jth standard vector of $\mathbb{R}^n$

we do some ninja stuff and arrive at the fact that $\lim \limits_{t\vec e_j \to 0} \frac{f(\vec a + t \vec e_j) - f(\vec a)}{t} - f'(\vec a)(\vec e_j) = 0$, effectively showing that $f'(\vec a)(\vec e_j)$ is the jth partial derivative of $f(\vec a)$

oh, with a prime in there it is starting to make sense.

@shintuku is that a statement or question?

now i'm stuck at finding a meaningful statement that identifies this partial derivative with a column of the jacobian matrix

it's a statement

it is essentially the same.

if $x_k \to x$ then $y^T x_k \to y^T x$.

my daughter yelled at us to take her to the park, and when we got there all she wanted to do was climb on a bench and jump off of it. we have a bench at home.

she then yelled at someone for using swings the wrong way (a kid maybe five times her age was standing in the swing).

Throw her in jail.

@copper.hat now, do not underestimate me because you are likely to considerably overestimate me, but i don't get it

You really could learn from my videos, shin.

Less effort than all the typing here.

@TedShifrin i have this amazing screenshot of exactly what I don't understand in the lecture where you mention this

the caption is "why is the smiling man saying this"

@shintuku i misunderstood what you asked originally. however, $f'(a) e_j $ is the $j$th column of the Jacobian.

Torture Teddy

It's all because for. a differentiable function, $Df(a)v$ is the directional derivative $D_vf(a)$.

Although I probably proved this fact separately first.

i am not distinguishing the linear operator from the matrix representation in my previous comment.

this is by the commutativity of letters with other letters.

It’s been 7 years.

@copper.hat what I don't get is how this is happening. we know that $f'(a)$ is a linear map and we're trying to say that it's jth column is the jth column of the jacobian

but $f'(a)e_j$ isn't the jth column, since it's a vector in $\mathbb{R}^m$

if $A$ is a matrix then $A e_j$ is the $j$th column.

The jacobian is the standard matrix of the map.

highlights copper's comment

The column vectors live where?

the most recent pic on my cat's instagram feed is a shot of her reacting to my daughter yelling. her ears are back and her eyes are bugged out. it's classic cat.

the.. column space? i think @TedShifrin

@copper.hat i'm willing to just accept this, but is this a definitional thing?

i also manually did the multiplication

i.e. the matrix as a linear map multiplied by the vector

It’s precisely how matrix times vector is defined.

one of the most useful facts in life is that A = [A e_1 | A e_2 | . . . | A e_n].

and yes it's often the definition

@leslietownes mein gott this is what I need

i have a set of linear algebra notes from a class i taught but they are riddled with typos.

if ted has lectures on it i defer to him because my nonsense would not improve upon it

from the linear algebra perspective, this is just what it means that the matrix represents the linear map relative to the standard basis

@TedShifrin right but how do we go, if $(b_1, ..., b_m)$ is a basis for $\mathbb{R}^m$, from $f'(\vec a)(\vec e_j) = a_{1j}\vec b_1 + ... + a_{mj}\vec b_m$ to a statement about the jacobian matrix?

you don't just want any basis

hm i expected as much but for the life of me i can't find any definitions. axler, hubbard and wikipedia are failing me

what do you need a definition for

well, at least I'm guessing I'm missing some definition or something: i'm looking to make a meaningful statement starting from the proposition "$f'(\vec a)(\vec e_j)$ is the jth derivative of $f(\vec a)$" to a statement about the jacobian matrix, like this vector is the column vector, or shares column coefficients, or something, with the jacobian matrix. something like leslie mentioned would be it

how do you define the Jacobian

where's grandad?

well, i'm working with this:

but my issue is getting a statement about identity of columns from this

so the ith column is $(Df)(x)(e_i)$, I don't see what more you want

@shintuku the standard basis is $e_1=(1,0...)^T, e_2=(0,1,...)^T,...$.

$a= \sum_k a_k e_k$.

oooh, someone flew over in a nice homebuild t18

grandad is swimming in the water barrel for a change

@shintuku just deal with $m=1$ first.

@TedShifrin not in a row of flats, that's for sure...

i'm still very tempted to make it my avatar but i like the symmetry of the automatically generated thing.

@Thorgott at least on the matrix, isn't it the coefficients instead, so only $f'(\vec x)$? The coefficients are linear maps, but $f'(\vec a)(\vec e_j)$ is a vector, and so are its components.

i suspect you might get some complaints :-)

@leslietownes you can save that and return to it later.

as always you have identified the solution.

@shintuku take $m=1$ and get comfortable with that. you cannot do $m>1$ if you are not ok with the basic.

@copper.hat was just reading your comment, alright I'll meditate on this a bit

what coefficients?

think about what the derivative is. $f(a+t e_k) \approx f(a) + t f'(a) e_k$

a column of an $m\times n$ matrix is a vector in $\mathbb{R}^m$

which is ${\partial f(a) \over \partial x_k}$.

ignoring little things like limits

$(Df)(x)$ is a linear map $\mathbb{R}^n\rightarrow\mathbb{R}^m$ an $e_i$ is a vector in $\mathbb{R}^n$, so $(Df)(x)(e_i)$ is a vector in $\mathbb{R}^m$ as well

i would suggest forgetting vectors for the moment.

so $f(a+ t_1 e_1 + \cdots + t_n e_n) \approx f(a) + t_1 f'(a) e_1 + \cdots + t_n f'(a) e_n$.

$= f(a) + f'(a) \sum_k t_k e_k$.

i've updated the profile. it might take a while for it to roll out to new users.

looks the same to me even after a refresh.

@copper.hat right no problem with that idea

math.stackexchange.com/users/18076/leslie-townes?tab=profile ought to have the new one.

so you see that ${\partial f(a) \over \partial x_k} = f'(a) e_k$?

thanks, irish water safety association.

:-) millions will be grateful

@copper.hat right, that's the statement $f'(\vec a) \vec e_k$ is the kth partial derivative of $f(\vec a)$

i forgot the fractal statistic, but ireland's costal length to area is pretty large

@shintuku so you so see that $f'(a)e_k$ is the $k$th entry of $f'(a)$?

england's is also pretty good.

they must have copied

copied and exported

@leslietownes I refreshed your chat profile from your main profile and the avatar is now visible

@copper.hat yep, no issue there

thanks. this is a great comfort to me.

I'm sure

you will be getting a ceased to exist letter

@copper.hat I think sometimes I've ceased to exist

@shintuku for $m>1$ just stack the $f_j$ on top of each other.

i am just a bot $\bot$.

off to walk the dogs in the oven called outside

we're all bots.

it's uncomfortably hot outside right now.

I don't think it will get better for quite a while

hmm, i forgot the Latex for bot

i am waiting to go for a cycle.

my wife is camping at donner lake, son is off to santa cruz, daughter asleep hopefully and i am fixing test bugs. somethings wrong here

no i didn't, for some reason \bot does not render here $\bot$.

hmm, strange.

now it does.

i was sweaty and disgusting after we walked home from the park and my daughter said i looked like a coworker who had been on a zoom with slightly unkempt hair. i told her, please don't tell her that.

works for me.

my daughter told me of the existence of a bf today for the first time (i mean the first time she is explicitly telling me). i told her that my only concern is her well being and the relevant party needs to understand the physical implications of that concern.

i don't know how i'm going to handle dating and all of that, at all. i know what men are like. i kind of hope it isn't men.

as long as they understand that i (and siblings) are slightly unhinged...

and persistent.

i'm very more permissive and accepting than my wife about some of this stuff. we'll stay friends but if she winds up with a POS bf i might have "commentary" on that

that is a tough one, but you need to be very careful about how you express concern :-)

one of my best friends had her life literally wrecked by having a boyfriend who was a heroin addict who got her addicted to heroin. she lost 10 years of her life to this dumb a-hole.

omg. thankfully i don't think that is on the cards

if people are just jerks i'm just OK with that, different strokes for different folks.

for good or for bad, i think my offspring have enough of my genes to avoid major stuff like that

i certainly can't say that i've made all of the best life decisions. i have navigated to calm waters.

i am still trying to figure out what i want :-)

speaking of, a cycle to tilden now i think

if there's a duck pond, say hi to the ducks.

it is really sad now.

maybe 75x30 yrds^2

oof

there goes my earthquake emergency supply

later!

cheers

If $x$ is irrational, can there be made conclusions about $\sqrt{x}$? Assuming x is not negative.

it's not going to be rational. do you have finer considerations in mind?

sqrt(irrational) is basically all positive real numbers so that might inform the thought process.

found a way around my problem

define a matrix that just so happens to be the jacobian

then prove it is the derivative

and never mention jacobian

then happily go on, ignoring linear algebra for a better world

who knows what identity between a vector and the column of a matrix means: no one needs to know! lalala

what's the problem with linear algebra?

couldn't find a statement for $\vec v$ is identical to the column of the Jacobian matrix

what do you mean find a statement?

do you understand the relationship between a linear operator, basis and matrix representation?

yes (hopefully): the matrix representation of a linear operator is the matrix such that the multiplication of a vector by this matrix results in the same result as the application of the linear operator on this vector

well, there is an underlying relationship between points in the linear space and the elements of $\mathbb{R}^n$ as well.

once you pick a basis for the domain & codomain then you can just work with matrices and elements of $\mathbb{R}^n$ (or whatever field you are using).

unless you have a basis, talking about a column of a linear operator makes no sense.

(in general)

right, noted

@leslietownes imgur.com/LejZGFd there was a duck somewhere but i had was back on my way and did not have the energy to return...

beautiful. probably a mallard.

i miss northern california.

at our pond it is mostly mallards. canada and egyptian geese are seasonal, as are coots. terns drop in from time to time. one time i saw a wood duck but never again.

herons used to hunt frogs in the fields near to my parents house. i could watch that for hours.

i'm very fond of my new avatar.

How to make a tag block?

math.stackexchange.com/questions/tagged/probability

[math.stackexchange.com/questions/tagged/probability]

Doesn't work :-/

Cool avatar, Leslie :-D

@Wolgwang Are you trying to make something like probability

Yes.

Thanks :-)

I believe the meta tags are like mathjax

evidently that's correct $\ddot\smile$

robjohn did you hear the sonic boom on friday? my wife and i felt like it might have been an earthquake but it was just one big boom. someone testing an aircraft apparently.

@leslietownes there was an earthquake at about 3:40 PM. When was the sonic boom?

in the morning. about 9:30.

Did not hear it, but there are many sources of noise here with 2 dogs, 3 cats, and a rabbit, plus my son and a lot of vehicles that pass by not too far away.

if i'd been watching something or listening to something i would have missed it.

the USGS put out a report about it just to relieve the public that it was not a geological event.

ah

when it happend i asked my wife "what was that? earthquakes don't just go boom. did something explode?" but news was silent for many hours.

CalTech put out a report, too, it seems. If they get enough inquiries about an earthquake and nothing shows up on the seismographs, they assume it is a sonic boom.

1.2 miles south of San Dimas 9:20 AM

The earthquake I was thinking about was on Thursday.

when i was growing up you could often hear the sonic boom from concord in the distance.

i am sorry i did not get to fly on the concord. i did however share the traffic pattern with the concord once at shannon airport where they came to train from time to time.

Hi

Is converse to L’Hopital’s rule true?

That is, given $f(x), g(x) \to 0$ as $x\to a$ and $\frac{f’(x)}{g’(x)}\to l$, it follows that $\frac {f(x)}{g(x)}\to l$

Hmm, try sometime simple like $f(x) = x, g(x) = c+x$.

But does $\frac {f(x)}{g(x)}\to l$ imply limit of derivatives is also l?

why would it. you need to try some simple examples. $f(x) = 1+x, g(x)=1-x$ at $a=0$.

@Koro Hi Koro!

@copper.hat f, g here both don’t approach 0 so this isn’t an example. 😕

I’ll explain my point:

sry, what you have above is just L’Hopital’s rule (as long as $l \neq 0$).

Let $f$ be differentiable on (a,b). Let $t\in (a,b)$ so $f’(t)=\lim_{h \to 0}\frac {f(t+h)-f(t)}{h}$ and RHS here is of the form 0/0 and applying L’Hopital’s gives $\lim _{h\to 0} f’(t+h)$ which is equal to $f’(t)$ if we assume L’Hopital’s converse to be true and it follows that f’ is continuous at t. This proves that f’ is continuous on (a,b) which we know is not true in general.

@copper.hat: this is the problem we get if we assume L’Hopital’s converse to be true so the converse is not true. Am I right?

That is the example, I had tried.

sry, i don't really follow what you are asking.

i don't know what you mean by the converse.

I’m sorry @copper, my question was not clear enough. Let me state more clearly: My question is: Let f and g be two differentiable functions on (a,b) and let g’ be non zero. Suppose that f(x) and g(x) both have limit 0 as $x\to p$, where p is limit point of (a,b) and it is given that $\frac {f(x)}{g(x)}\to L$ as $x\to p$ then is it true that $\frac{f’(x)}{g’(x)}\to L$.

This is what I meant by converse.

The converse of If A, then B is If B, then A @Koro

@user178758: here A=existence of limit of ratio derivatives and B= existence of limit of ratio of f and g

@Koro It is not the converse, but it is true. write $f(x) = f(p)+f'(p)(x-p)+o_f(x-p)$ and the same for $g$ with appropriate changes. Divide, substitute and then divide by $x-p$ and take limits

I see so if $f(p)=g(p)=0$ then the result is true.

Well, yes, that should be clear from what I wrote?

If not, let me know what I should elaborate

Yes copper, that is absolutely clear from what you wrote. Thank you! Now I would request you to let me know what went wrong in my “all derivatives will be continuous “ comment.

Applying L’Hopital’s rule here is implying that derivatives are continuous. I don’t understand why that’s happening.

Please let me know if my question needs more details.

Sry, my proof was not correct.

My proof showed that ${f'(p) \over g'(p)} = L$.

If f(p)=g(p)=0 then your proof was correct. @copper.hat

Not that the limit is $L$

So what

For that one would need continuity. Good catch (sort of :-))

There is a difference between ${f'(x) \over g'(x)} \to L$ and ${f'(p) \over g'(p)} = L$,

sloppy workn on my part

Ohh yes. Right, that was wrong!! Sloppy observation on part too!

🤭

So my above linked comment makes sense?

My opinion is: yes it does and I have provided my explanation above. Do you have any objections to the explanation?

sorry, can you write what it is you are presuming and concluding?

I mean “sloppy observation on my part too!” chat.stackexchange.com/transcript/message/58588248#58588248

i regularly make mistakes. i am running tests for work in another window, so this is not getting my full attention

@copper.hat I’m assuming that converse (its meaning is what I have stated already which is according to you not exactly converse) is true and then I am showing how it results in contradictory results by showing that all derivatives will become continuous as a result of the assumption.

please don't use converse for that. i find it really confusing. it is a very peculiar assumption.

Ok. Let’s forget I ever used the word converse. You may replace converse by this: chat.stackexchange.com/transcript/message/58588168#58588168

this seems like some version of l'hopital's rule. not exactly, but related. the usual proofs might shed light on this. rudin has one.

I have got my answer now. The conclusion is not true in general. For example: $f(x)=x^2\sin (\frac 1x)$ when $x\ne 0$ and $f(x)=0$ when $x=0$. Clearly $\lim f’(0+h)$ does not exist as $h\to 0$

@Koro that is false

I mean the statement before that

x^2 sin 1/x strikes again.

Haha

@copper: Did you mean this?

I am well aware that all derivatives are not continuous. The above linked comment only shows that “peculiar “ statement is not true in general.

By arriving at “all derivatives would be continuous had the “peculiar “ statement were true”

@leslietownes what’s that in your profile photo?

sry, i really cannot get beyond converse.

it's grandad, or what's left of him.

he's joined the choir invisible.

it's a still from an irish public service announcement about water safety. the link is one of the featured comments.

copper's.

one of my comments is a british ad about the dangers of water.

i don't think i ever saw an american version of this message.

i lived by a creek and people used this ratty old rope to swing from one side to the other. it might have been a 20 foot drop and into very fast flowing water in the rainy season.

it was horrible design. if you didn't use the rope or hazard the currents it was like 4 more blocks of a walk to school. the rope cut that down significantly.

@Koro the word "converse" has a natural meaning in english of the opposite of "verse" that is what the "con" prefix means. Think of it as "with terms the other way round" in a standard "If-Then" statement.

Try writing out L’Hopital’s Rule as an If-Then statement first.

perhaps, that may help to clarify your thinking :-)

an old habit of mine is that when i encounter something non essential that makes no sense to me i stop reading and move on to something else. only so many hours in the day.

generally, at work & in life, fixing things as early as possible avoids many downstream issues.

i am out of white wine apart from some 2BC junk for cooking. may hit a bordeaux. i don't like drinking on my own and i hate to waste a bottle just for one glass.

anyone want to come over :-)

thanks for the invite :-)

i can get some gruner and be there in about, i dunno, 8 hours.

i love to watch the sun come up with a glass of wine in my hand. i don't know about you.

Feynman used to see the universe in a glass of wine.

i was in cvs the other day and a college kid was getting three large bottles of moscato. i said, you know, there's other stuff here that is better than that and she said "i just need some cheap s--t because i'm throwing a party." CVS moscato it is.

there are two kinds of parties, the kind where you put out the best stuff, and the kind where you fill expensive bottles with cheap stuff because you have no idea who your guests are going to be

growing up my college (and sometimes hs) friends generally wanted to get blotto which had no appeal to me. i would rather nurse a glass of something nice.

cheap wine can be good but cheap bad wine is disgusting. which is why i tried to stop her.

i am alone tonight. my usual cohort are traveling or out of the town.

but if the spirit was, f- my guests, i can't argue with that.

my son went away with some friends to an airbnb in santa cruz.

they didn't invite me, wonder why

airbnb should be airb

i loved santa cruz the last time i visited it, it was still very grizzled and had hard edges. i assume it's gentrified now.

one thing ireland used to do really well was b&b

no, its still crap. really

proximity to google land has not improved things

i guess it's too far from anything else to be gentrified.

a gazzilion years ago i checked out a job in scotts valley

fun ride on my bike then. strangely, showing up on a bike impressed folks back then

fun fact, one which i wish iknew in advance: riding a motorbike for hours with acute appendicitis is painful.

oh god.

long boring story best told over bordeaux

i like the area. living here for 5 years i've seen a lot of seaside dumps be transformed into stuff that looks like it could be right outside the orange county airport.

my closest american friend throws me out of the house when he is ready for me to leave

i want seaside dumps.

his wife was appalled. i told her it was fantastic, i didn't have to do any silly wondering

whoa, i just got my first upvote on a reddit comment.

i do hate the feeling of wondering if you're overstaying your welcome or being rude by trying to leave early.

i am not a frequent commenter on reddit

i generally prefer to overstay. that way i know i didn't miss any wine

plus it encourages my hosts to plan accordingly :-)

a friend of my brother's would do a quick fake yawn and announce "i'm off to bed so you can go home"

my friend in the berkeley hills used to party basically all night. we'd drive home at 2 or 3 in the morning. i don't know how he did it, he was in his 60s. my wife and i were falling asleep but there was something else to do.

i like company.

i could stay forever if involves company :-). drives my wife nuts

life is short.

i think i see my social genes appearing in my kids as they get older.

i don't regret it either, they were great parties. my wife and i still talk about them.

i have no social skills, unlike my wife who is incredible

however, she is polite, i am not.

that's advantage joe in my mind :-)

we're supposed to do a housewarming party, we moved into our house over a year ago but covid. the place is something of a mess due to covid and toddler.

it's massively better to be impolite. if you're rude but not cruel about it you get better and quicker results.

this is a constant tension between myself & wife. i told her i woudl prefer friends to a tidy house. if it bothers people i don't want to sound mean but i don't care/

i agree

but there are some cultural issues.

some of my closest friends, when i met them, they said something i didn't like and i was rude about it. and 30 years later here we are. politeness is a way of distancing, i think.

but yeah totally cultural.

there are countries where i would rein my personality in.

my wife was interrogating my son about his friends & friend groups the other day. she stopped when she hit about 40 people in a few mins (she was taking notes). i knew at that moment i had done something right :-)

its funny, i often get on really well with people whose world view is entirely inconsistent with my own :)

so my friends are an interesting and somehwhat mutually incompatible group

i'm sometimes on the phone with a friend at work and my wife will ask, who was that? why were you fighting? i wasn't fighting we just yell at each other. it's how we're friends.

i am listening to sinead at the moment. she has a truly awesome voice

my sister played out all of her tapes in the early 90s.

one of my brother's brothers-in-law used to play the uillean pipes for De Dannan

i love the pipes

that might be the most irish thing i've ever seen on the chat.

i:-)

the uillean is amazing. i think it's very technically difficult to play.

no, surely where's grandad?

taps the sign

that is def the most irish thing, just a bit insider

i'm still on the fence about wasting a bottle of bordeaux

maybe i have some port.

one time at christmas i saw a harpist who completely blew my mind. he moved me to tears with his music, which is not something i do. another irish waste of time.

there is something about live music

i have a greek probabilist friend who is also a professional classical guitarist. i love listening to him

a disgusting polyglot

i love guitar music. i resent polyglots.

i resent folks whose skill set just dwarfs anything i have. that means i have a lot of resentment.

i am kidding :-)

there's a guy i work with who gives lectures in other languages a lot and i used to resent him, then i saw a video of one of his lectures in spanish, and his accent was horrible.

i liked him after that.

:-) that is very irish

it's nicer for me if people are just being polite to him than if he actually speaks seven languages.

my greek friend can choose what Spanish locale he wished to appear to be from.

that's frightening.

@copper.hat It's funny; I like to think my German is quite good and I have the accent of someone from quite an obscure place so I pretend to also be able to do this. Unfortunately I can't change the accent I have and my accent is that of a backwards racist farmer. Oh well.

I can only mange English unfortunately :-)

mange = manage or mangle

wow, i am getting a lot of negative fb on my reddit comment

Yeah my home town is also full of backwards racist farmers :(

i went to school with someone who could do almost all of the regional accents of the UK. he was welsh.

i can only do flat, unaccented, maybe event accentless western united states american english.

He just absorbed the energy of sheep from various regions

i can hardly distinguish Scots & N England anymore

if "absorbed the energy" is what the kids are calling it these days.

when i went to law school some people made fun of me for my western accent (it does not distinguish between various vowels sounds that easterners do) and others made fun of me for my traces of a boston accent inherited from my parents.