But on the function he calculated radius from
#help-4
woven ivy
zinc spear
yeah that's what i would use, though you haven't shared the solution for the first problem
woven ivy
Pi/4 is where come and sphere meet right?
woven ivy
yeah that's what i would use, though you haven't shared the solution for the fir...
I don't got it
💀
I used gpt
A bit
But i got confused to some degree
woven ivy
Pi/4 is where come and sphere meet right?
So for the first exercise
It would go from 0
To 0?
Cuz the cone in the second one
Starts from 0 and meets the sphere on pi/4
The sheet starts on 0 and meets on 0
Ik it's wrong
woven ivy
zinc spear
let me do something in desmos 3d
woven ivy
Yea
The second one has a cone meeting a sphere
So pi /4 is where they meet
In the first they meet at 0
So my brain would goish to 0 and 0
But it's wrong ik
Like you usually put the coordinates
In the function
To see how it goes upwards
For the 3rd integral
Like you convert
But the 3rd one is used to see how it goes upwards
I'm a bit confused on that
First one uses these 3 integrals
But the 3rd one is odd sorta
Cuz it's z=0
And then cylinder coordinates
In the 2nd one he picks the cone
And just puts the coordinates
Without thinking about the sphere
Idk if I'm being clear
Prob not
zinc spear
this is for the first one<
zinc spear
you should decide between the two because it's a bit hard to follow tbh
woven ivy
I'm trying to compare
To see if i get it
Prob not 💀
Because you always basically use cylinder coordinates
Or spherical
And that's clear from what I've been seeing
They're the most common
zinc spear
But the 3rd one is odd sorta
you go from the surface z = 0 to the surface z = 4-r^2
woven ivy
That's the train of through it had
But for the cone and sphere
You got from 0 to pi/4
?
zinc spear
because of spherical coords
woven ivy
Shouldn't it be 0 to pi
zinc spear
there is no z only phi which is an angle between the z axis and the point you want to describe
woven ivy
Since the cone starts from 0
woven ivy
there is no z only phi which is an angle between the z axis and the point you wa...
Oh
That's true
woven ivy
Nw
So does the third integral measure height?
Like total height
Or nah?
But i sorta get it
You get your coordinates
To convert
Put them in your function
And try to get the intersection
Or top
Depending on what's the second
I believe
Lwk starting to learn fluxes from exercises is tough
Ima close thanks for all the help
.closw
.close
zinc spear
in spherical coord instead of summing cubes, the intuition you are summing piece of balls
vale dockBOT
zinc spear
woven ivy
in spherical coord instead of summing cubes, the intuition you are summing piece...
Ohh
That makes sense
But why pi/4
zinc spear
it's a bit hard to describe in words
zinc spear
the picture might help
zinc spear
it's the phi angle to get to the cone
zinc spear
when summing the piece in spherical coords
woven ivy
when summing the piece in spherical coords
Yes
But that means the cone
Surrounds the sphere at the top too
zinc spear
Does it refer for cylinders
in cynlindrical coord the z axis stays like in cartesian, but the x and y axis works like in polar coord / spherical. it's in between
.reopen
vale dockBOT
✅
woven ivy
in cynlindrical coord the z axis stays like in cartesian, but the x and y axis w...
Oh
It's like disks?
Going up maybe?
Since it's 4-r^2
Which isn't really a disk
But reminds me of one
Somewhat
Alr thanks a lot
Now ima go have fun figuring the basic formula
zinc spear
woven ivy
Yea that makes total sense
zinc spear
i tried to do the picture like spherical for cylindrical
woven ivy
Ur an artist
But yea thanks a lot for the help
Alr peace ima go figure out the open thingy formula
🔥
.close
vale dockBOT
vale dockBOT
Available help channel!
chilly jackal
can someone pls explain what a centroid is im so confused- it wasnt explained in class and the notes weren't posted for some reason
chilly jackal
there's a single problem on it and it just says the answer with no work shown
i mapped it out in desmos but it's not clear
ping me if u respond
terse silo
rose raven
vale dockBOT
rose raven
So I have this question on home work and it says to find the value of f(5)
But I don’t understand how to find y
wraith heart
not enough information shown
rose raven
the figure shows the graph of a function f. Find the value of f(5)
the very next question is to find the derivative of f(5)
noble anvil
Are we to assume that the squares/rectangles are unit?
unborn python
is there like a scale? like every square is 1 unit?
rose raven
gotta be honest with you no idea
just x,y x stops at 8 and y stops at 5
no function given
unborn python
just x,y x stops at 8 and y stops at 5
? eloborate?
rose raven
like on the x and y axis
unborn python
then why did you say there are no y value given?
can we see the whole screen please
rose raven
thats literally the whole screen. I have to move left to show you the y axis but there is nothing else there.
i have to load the prior page to go back a question
noble anvil
By that you've led us to believe it's a unit square.
If that's the case, and the line given is f(x), can you not find f(5)?
echo hearth
vale dockBOT
wraith heart
did you click the ? next to regression parameters
vale dockBOT
@echo hearth Has your question been resolved?
echo hearth
did you click the ? next to regression parameters
?
echo hearth
I never used it before
vale dockBOT
@echo hearth Has your question been resolved?
vale dockBOT
@echo hearth Has your question been resolved?
heady jolt
How do you find the roots of a polynomial that ends in a prime?
ruby chasm
How do you find the roots of a polynomial that ends in a prime?
Take x common x(x²+8x-23)
merry hamlet
How do you find the roots of a polynomial that ends in a prime?
use the Rational Root Theorem
If a polynomial with integer coefficients has a prime constant term, p, then its only possible rational roots are ±1 and ±p. This gives you a very short list of numbers to test
torn fox
After factoring the x out you have a quadratic
Is there extra instructions that prevents you from using the quadratic formula to find another solution?
midnight pier
How do you find the roots of a polynomial that ends in a prime?
Put it in a calc easy
heady jolt
After factoring the x out you have a quadratic
Is there extra instructions that...
no, thank you.
heady jolt
If a polynomial with integer coefficients has a prime constant term, p, then its...
I'll look at this as well
.close
ashen gust
is this correct
vale dockBOT
is this correct
wraith heart
,rotate
rocky lotusBOT
wraith heart
Doesn't look like you shifted horizontally
slate marten
So you are pretty close but it looks like you missed was the sideways shift
You can see that in the pi/3 inside of the sin parenthesis
charred fulcrum
why is x^pi only defined for x>0?
vale dockBOT
why is x^pi only defined for x>0?
slate marten
For negative bases, fractional indices are not defined
?
toxic ridge
Eg : (-1)^0.5
steady charm
For negative bases, fractional indices are not defined
it can be defined if the denominator is odd, but of course it isn't here because it's irrational anyway
charred fulcrum
For negative bases, fractional indices are not defined
who decided this?
legit never heard of this before
toxic ridge
who decided this?
Ah no Sorry my bad
toxic ridge
it can be defined if the denominator is odd, but of course it isn't here because...
This
steady charm
who decided this?
well for example what do you get if you try to define the square root of -1? you have to start using imaginary numbers
a similar thing happens with irrational powers
steady charm
the exception we worked out only applies to odd roots specifically, so it doesn't apply to irrational numbers
charred fulcrum
oh
toxic ridge
@steady charm hold up, if the only required condition is that denominator be odd, can we not divide the digits of pi into specific parts so that each individual parts are divisible by 4 or other
charred fulcrum
generally speaking x^n is defined for x>0 but when n is odd then it is also defined for negative x
that is the "exception"
quasi valve
well it's defined if n is even too
charred fulcrum
but for x>0
quasi valve
the problem arises if n is not an integer
quasi valve
x^2 is defined for all x
quasi valve
0^(any nonzero number) is 0
but there's no unique or canonical way to define, say (-1)^pi
there are infinitely many candidates and none of them are real
steady charm
expressing 1.5 as a fraction it is 3/2 which has even denominator
toxic ridge
<@654893611934023680> hold up, if the only required condition is that denominato...
Nvm got it
charred fulcrum
expressing 1.5 as a fraction it is 3/2 which has even denominator
I have sometimes seen x^1.5 defined for x<0 also
maybe not in R
slate marten
For any x^n (not including complex answers):
If x is positive, you can have any value of n
If x is negative you can have any integer n or any n with an odd root
charred fulcrum
what is an exponent defined as exactly then?
steady charm
how we define exponentiation depends on whether we are considering just positive whole number exponents, or just integer exponents, or just rational exponents, or any real exponent
quasi valve
in general if z is a real power, then (-1)^z is actually a set of infinitely many complex values
for specific choices of z (integers, or rationals with odd denominators), one of those complex values is real
but if z is irrational, none of them are
viscid sedge
$$a^{b} = \prod_{i=1}^{b}a$$
rocky lotusBOT
fluX
charred fulcrum
how can I multiply pi times
viscid sedge
well then you use the logarithm to define an exponent
steady charm
you don't, that definition only applies to positive integer exponents
quasi valve
you express your base in polar form $r e^{i \theta}$ and then you take that to your desired power, say $\pi$
charred fulcrum
in general if z is a real power, then (-1)^z is actually a set of infinitely man...
ok I see
rocky lotusBOT
Bungo
quasi valve
but the catch is that $e^{i \theta} = e^{i(\theta + 2n\pi)}$ for any integer $n$, and that's why in general you get infinitely many solutions
rocky lotusBOT
Bungo
steady charm
it's possible to define $e^x$ and $\ln(x)$ with only positive integer exponentiation, in which case we can define
[ x^\pi = e^{\pi \ln x} ]
rocky lotusBOT
cloud
viscid sedge
but then this is not defined for when x is 0
it is
steady charm
well you can make a separate definition that it is 0 if x is 0
charred fulcrum
but the catch is that $e^{i \theta} = e^{i(\theta + 2n\pi)}$ for any integer $n$...
I think I understand it in the complex plane
but in real numbers not that much
viscid sedge
pretty close to 0, i'd say
unreal flame
If y is ever equal to zero, shouldn't that mean that y' should also equal zero and should result in y never changing? If thats the case, I'm not sure how the piecewise definition for y makes sense, as y would never change if it is equal to zero. Please let me know if I am misunderstanding anything.
zealous marten
If y is ever equal to zero, shouldn't that mean that y' should also equal zero a...
yerda
quiet marlin
y' should also equal zero and should result in y never changing
specifics of the problem aside, this isn't the case. Any nonzero second derivative results in the function value changing in the future, for example
unreal flame
>y' should also equal zero and should result in y never changing
specifics of th...
For regular calculus problems, I feel like that would make sense as the derivative is defined in terms of another "independent" variable (like y=x^2)
but in this case, y' = y^(1/3), so it's defined with itself
quiet marlin
y is a function of t so if you chose a particular y then you could get it back to the form you're referring to
unreal flame
like conceptually I just can't understand how y can be zero for some set arbitrary period of time and then just diverge from zero
especially since if y = 0, the derivative is zero so y should have no reason to ever change from zero
quiet marlin
again, more things than just the first derivative influence the behavior of a function
quiet marlin
like conceptually I just can't understand how y can be zero for some set arbitra...
many such cases. Take the rectifier function https://en.wikipedia.org/wiki/Rectifier_(neural_networks)
quiet marlin
especially since if y = 0, the derivative is zero so y should have no reason to ...
the derivative at y=0 is 0. Nowhere does it say it needs to be zero everywhere
unless your claim is that if a function has a 0 derivative at some point, it has a 0 derivative everywhere
unreal flame
unless your claim is that if a function has a 0 derivative at some point, it has...
thats more of what I'm saying, yeah
quiet marlin
,w derivative of x^2 at x=0
rocky lotusBOT
quiet marlin
,w plot x^2
rocky lotusBOT
quiet marlin
doesn't look 0 everywhere to me
unreal flame
again, the issue I have with this is the fact that y' is defined in terms of y
regular calculus makes sense to me
for y=x^2 for example, y' = 0 at x = 0
however, y'' is a constant so y' changes away from 0
but in this case, y' is defined in terms of y right? not in terms of t.
if you can excuse my terminology, I imagine it as a kind of closed system. y' depends only on y, and so does y''. if there is no t, how does y ever change if it equals zero?
quiet marlin
y is a function of t
unreal flame
y is a function of t
ok?
quiet marlin
again, more things than just the first derivative influence the behavior of a fu...
all I can do is refer you back to this
a derivative of 0 at a point does not mean the function never changes
unreal flame
is there no more rigorous way to explain this?
I don't think using regular classical calculus examples to justify this makes sense
unreal flame
a derivative of 0 at a point does not mean the function never changes
thats really not what I'm saying
in this very specific example, it seems like it
quiet marlin
thats more of what I'm saying, yeah
it was a few minutes ago
unreal flame
it was a few minutes ago
mb, I didn't read your comment correctly
not any function, just this specific one
of course in many functions, it can have a derivative of zero and still change. Because the derivative depends on some independent variable
for example, y'=2x
x changes so y' changes
if y'=y^(1/3) and y(some arbitrary t) = 0, how can y change?
sorry if I'm being confusing
quiet marlin
i'm still not sure why you think it wouldn't
we go back to my original counterexample, y=x^2 (and assume x>=0 for simplicity). I can write y' = 2sqrt(y), no more x in sight
y'(0) = 0 yet y' isn't 0 everywhere
unreal flame
apologies, how would this look?
quiet marlin
what's "this"
unreal flame
what's "this"
y' = 2sqrt(y)
quiet marlin
i'm not sure what you mean
unreal flame
y'(0) = 0 yet y' isn't 0 everywhere
how can you prove or show this?
quiet marlin
by solving the IVP and showing it's not a constant function
unreal flame
by solving the IVP and showing it's not a constant function
I see what your saying
I think part of my confusion is that there is the constant function y=0 as the solution as well as y=t^2
quiet marlin
perhaps it would help if you didn't think of the equation y'=y^1/3 as a single function, but as the family of functions it really is (since there are many such y and all of them could be written in terms of t to get in the form with which you're more familiar)
unreal flame
to sort of explain my current line of thinking, a function like y' = 2sqrt(y) is not describing a single function but rather a family of functions. So while there are solutions that are not described by the independent variable (like y=0), there are solutions that do (like y=t^2).
I think my confusion came from imagining y as solely dependent on y', so it was kind of like a chicken and egg situation where one could not change without the other changing first.
plugging in y=t^2 makes it clear that y' can change from 0 (y' = 2|t|).
if you have nothing to add, thanks for the help, I really appreciate it
.close
vale dockBOT
vale dockBOT
Available help channel!
fluid acorn
vale dockBOT
copper stump
do i do a cummulative frequency to work out IQR?
you already know Q1 and Q3, why do you even need to do anything at all to find IQR?
fluid acorn
you already know Q1 and Q3, why do you even need to do anything at all to find I...
ya i got the answer in the end
but i got a new problem it is the final one
im not sure what
S_xx means
is it the numerator?
copper stump
is it the numerator?
wdym numerator?
fluid acorn
fluid acorn
but isnt this for standard deviation?
fluid acorn
ya, mb
copper stump
?
a.i just wants you to find the mean
ii needs you to find sd
and for that you need to take the sqrt
fluid acorn
sorry but idk how to do a i
fluid acorn
a.i. means you need to find the x bar value. You need to use both the quantities
shall i do
133 = S(X-x)^2
and find small x?
copper stump
You can expand the summation for S_xx
and use the values of sum of X^2 in second expression to sub it in
and then solve it to find the x bar
fluid acorn
and then solve it to find the x bar
i got it
copper stump
nice
copper stump
no...
fluid acorn
o wait
copper stump
use this
copper stump
,calc sqrt(133/7)
rocky lotusBOT
Result:
4.3588989435407
yep
copper stump
I think sd would scale by 100
so yea
1260
for a moment I read the original as 1.26 lol
vale dockBOT
copper stump
npnp
vale dockBOT
Available help channel!
fresh pecan
vale dockBOT
fresh pecan
Someone help with part a pls
vale dockBOT
@fresh pecan Has your question been resolved?
noble anvil
Someone help with part a pls
What ideas have you had so far?
cloud coral
erm
i think part a is bugged
0 <= x <= 1 yet sqrt(x-1)?
probably sqrt(1-x^2) was intended
If you want to try: can you construct an appropriate right angle triangle?
noble anvil
i think part a is bugged
Lol, just saw that domain 
unreal jolt
🙋♀️ .
unreal jolt
what is the order of transformations in quadratics?
for reflection, horizontal and vertical shifts and vertical, horizontal stretch/shrink
stark wedge
there is not really a set in stone "order of transformations" for any class of functions
noble anvil
what is the order of transformations in quadratics?
You need to know what you want to transform them to
stark wedge
when even slightly in doubt, verbalize all the transformations you want to apply, in order, and write down the equation of the graph after each step
unreal jolt
ok what if there a parent function f(x) and a transformed function g(x) and it says to graph the transformed one.. is there an order I need to follow to graph it or just do it randomly
like vertical shift first then reflection, then compress etc
pliant nova
like vertical shift first then reflection, then compress etc
ah let me guess you learned a(x - h)^2 + k
unreal jolt
ah let me guess you learned a(x - h)^2 + k
yeah
pliant nova
okay well you see the parentheses there that means you have to apply the horizontal shift first
its just order of operations really, if you do it that way
unreal jolt
its just order of operations really, if you do it that way
alright so () first then the "a" then "k" ?
pliant nova
yeah dont forget the exponent too
pliant nova
alright what about reflection btw?
y axis reflection would just be making the entire thing negative
-(a(x - h)^2 + k) => -a(x - h)^2 - k
x axis reflection is a little more complicated since you have to squeeze in the negative inside the parentheses
a(-(x - h))^2 + k => a(-x + h)^2 + k
if thats what you mean by reflection
unreal jolt
y axis reflection would just be making the entire thing negative
-(a(x - h)^2 + ...
no I mean whats the order for that? do I graph the reflection last? like after all the translations and stuff or before
pliant nova
wouldnt the problem tell you what order to do the transformations in
unreal jolt
wouldnt the problem tell you what order to do the transformations in
uhm not really they just give the function for example g(x)= -2(x-1)^2 +2
pliant nova
ok when a or x is negative then theres a reflection. if you have a doubt just do the math and get the correct y value
vale dockBOT
vale dockBOT
Available help channel!
woven ivy
Yo
woven ivy
So basically it's about flux
Ima put both since numbers are messed in translation
But
Why does he pick 0 3 for radius
And not the intersection
I don't get it
Like how would the integrals look if i only consider the upper flux
vale dockBOT
@woven ivy Has your question been resolved?
leaden cargo
Like how would the integrals look if i only consider the upper flux
Questo non è un flusso, è un integrale triplo
“Flusso” è per i campi attraverso una superficie, qui non c'è alcun campo
open niche
💀 what level is this?
leaden cargo
💀 what level is this?
Calculus in 3d
open niche
Calculus in 3d
Damn i got a long way fr
leaden cargo
There's always more to learn, it never ends
leaden cargo
We're integrating over a ball of radius 3
open niche
We're integrating over a ball of radius 3
How did we find that radius is 3?
woven ivy
Questo non è un flusso, è un integrale triplo
Ci sono
open niche
It says ✓93
woven ivy
Hi
It doesn't count as flux?
You're right
But i have a doubt about the radius
In fluxes
leaden cargo
It says ✓93
The equation of a ball centred in (0,0,0) of radius r is x²+y²+z² = r², and our equations has x²+y²+z² = 9, so r² = 9 and r = 3
woven ivy
So basically
leaden cargo
It doesn't count as flux?
flux is for vector fields through a surface
woven ivy
If i had to calculate the flux tho
leaden cargo
there's no vector field here
woven ivy
Just the integrals
leaden cargo
If i had to calculate the flux tho
flux of what?
leaden cargo
Cannot forget the vector if you want to talk about flux
woven ivy
Focus on the 3 integrals
open niche
The equation of a ball centred in (0,0,0) of radius r is x²+y²+z² = r², and our ...
Oh i was looking at the left one nvm yea i see
woven ivy
It's just to understand how those work
The vector doesn't influence the integrals
From what i saw
Only the solids do
leaden cargo
I don't know what you are talking about now
how can the flux of a vector field not be influenced by the vector field?
The flux depends on 2 things, a vector field and a surface
woven ivy
I agree
But the integrals
That you use to calculate it
The range
They're within
Those ranges are influenced by the solids shapes
If i have a 2 solid intersection
The radius
leaden cargo
but that's not a flux
you're asking about integrating over a surface, which you can talk about without mentioning fluxes
leaden cargo
Ci sono vari tipi di integrali e non devi mischiarne il significato
Puoi parlare di integrali su di una curva, su di una superficie e su di un solido
I flussi sono particolare integrali su superficie, allo stesso modo in cui il lavoro è un particolare integrale su curva
woven ivy
but that's not a flux
you're asking about integrating over a surface, which you ...
But you do use it for flux?
Ok
Ma siamo d'accordo che se ho un qualcosa di chiuso
Posso usare il teorema della divergenza
Se ho un campo vettoriale
Ecc..
leaden cargo
Sì
woven ivy
Quindi dovrò fare 3 integrali
leaden cargo
Ma in questo caso la richiesta è già per un integrale su un solido
woven ivy
In uno di questi 3 uso il raggio
woven ivy
Ma in questo caso la richiesta è già per un integrale su un solido
Ok
Ma perché uso 0 e 3
leaden cargo
Quindi non lo sai se quella funzione è la divergenza di un qualche campo
woven ivy
Non si intersecano diversamente
leaden cargo
Quindi il teorema della divergenza non c'entra niente
woven ivy
Aspetta ho 2 domande
Se vado sul teorema della divergenza
E guardo il calcolo del raggio per l'integrale
Vado a eguagliare i due solidi sostituendo i loro z
leaden cargo
Per parlare di divergenza ti serve un campo (divergenza di chi?) In questo caso non abbiamo alcun campo in partenza, abbiamo semplicemente un integrale triplo
woven ivy
L'uno con l'altro
È una cosa concettuale
Solo per l'integrale
Se ho due solidi che si intersecano e formano qualcosa di chiuso
Dovrò inserire un raggio nel mio integrale
(poi torno all'esercizio)
Però voglio essere sicuro di questa cosa
leaden cargo
Ma se non stiamo parlando dell'esercizio, che vuol dire “raggio”?
“inserire un raggio nell'integrale” non vuol dire niente
leaden cargo
Raggio di chi se non parliamo dell'esercizio in questione?
leaden cargo
È una cosa concettuale
Sì ma le parole che stai usando nel vuoto non hanno significato
leaden cargo
ok
leaden cargo
No aspetta, hai descritto una superficie
leaden cargo
L'oggetto che hai descritto è un po' strano e non penso sia quello che hai in mente
leaden cargo
È un paraboloide infinito che parte da (x,y) in un cerchio di raggio 1
woven ivy
Scusa
-x^2-y^2=z
Consideri l'insieme E compreso tra i due
Fai la divergenza del campo vettoriale
Il primo integrale e fra 0 e 2 pi greco
Il secondo invece
Devo fare -x^2-y^2=radice x^2+^y2
E poi convertire in coordinate cilindriche
?
E quello è il mio estremo?
leaden cargo
-x^2-y^2=z
Neanche si toccano gli insiemi che hai descritto
leaden cargo
Eh ma non lo fanno, io non lo so che immagine hai in mente
leaden cargo
Ma non so neanche che intendi tu con incrocio chiuso
Ma ti serve per spiegare questa cosa che devi prendere la zona tra 2 superfici?
woven ivy
Ti ricordi l'esempio di ieri
Con la cupola
E il tappo
Qualcosa di simile con un cono e una cupola
La cupola orientata verso il basso
Il cono verso l'alto
Avranno entrambi funzioni con z dentro
E per fare il secondo integrale con estremo raggio
Eguaglio le 2
leaden cargo
Stai calmo, allora
Hai un cono, tipo z² = x²+y², e un paraboloide del tipo z = 4-x²-y²
Ora di quale integrale stai parlando?
(Flusso? Superficie? Volume?)
woven ivy
Ok aspetta ti do qualcosa di concreto
Guardiamo questo
Per ora
È un elissoide
Ma come si calcola 1/3 radice di 2x-...
Sono confuso
Perché
Se lui mi può chiedere il flusso di una superficie
woven ivy
Ok
leaden cargo
Flusso di un campo attraverso una superficie
woven ivy
C'è differenza
Senza attraverso una superficie
Sono due casi diversi?
Conta che sto imparando i flussi dagli esercizi
leaden cargo
No, non vuol dire dire niente
woven ivy
Ok
leaden cargo
Flusso riguarda 2 oggetti
Un campo
Una superficie (orientata)
leaden cargo
Dove quell'orientata è descritto come uscente/entrante
leaden cargo
Invece l'integrale di una superficie?
Integrale di superficie riguarda una funzione e una superficie
woven ivy
E fai doppio integrale di 1
leaden cargo
Senza orientaziome
leaden cargo
Stai confondendo un po' cose
woven ivy
Probabilmente
woven ivy
Integrale di superficie riguarda una funzione e una superficie
Ok
È questa la formula
Aspetta
Questa è la per la superficie
In 2d?
leaden cargo
Questo è l'integrale per l'area di una superficie descritta come grafico di una funzione
È un particolare integrale di superficie
woven ivy
Ha senso
leaden cargo
Noncè che sono tutti così
leaden cargo
Questo è l'integrale per l'area di una superficie descritta come grafico di una ...
Ho corretto, avevo visto male
woven ivy
Ok grazie
Aspetta stiamo tirando tante robe fuori
Quale la formula dellintegrale di superficie
In 2d e 3d
leaden cargo
Le superfici sono oggetti 2-dim e basta, c'è una sola formula
woven ivy
Ma la superficie non è lo stratto più esterno?
Ho capito cosa intendi
Puoi renderle piatte
Non so se mi sono spiegato bene
rocky lotusBOT
Luigi
Compile Error! Click the 
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woven ivy
Niente paura
leaden cargo
$$
\int_S f dS = \iint_D f(\phi) |N_{\phi}| dxdy
$$
Questa è la formula per l'integrale di superficie
Dove f è una funzione scalare e phi è una qualunque parametrizzazione di S
Pessima connessione
rocky lotusBOT
Luigi
leaden cargo
Ho di nuovo connessione stabile, credo
woven ivy
Ok
leaden cargo
Allora
woven ivy
Ho capito la divergenza
leaden cargo
Stavo per dire prima, devi distinguere i vari tipi di integrali che ci sono
leaden cargo
Hai a che fare con integrali singoli, doppi, tripli
poi integrali su curva, su superficie
poi integrali di lavoro e flusso
woven ivy
Hai a che fare con integrali singoli, doppi, tripli
poi integrali su curva, su s...
Lavoro vero
Che usa il rotore
leaden cargo
Non usa il rotore
leaden cargo
No, il teorema del rotore vale per curve chiuse
come la divergenza vale per superfici chiuse
woven ivy
Sennò usi doppio integrale di F(g) per g primo
woven ivy
No, il teorema del rotore vale per curve chiuse
Esatto
leaden cargo
Però in generale quelli sono integrali che hanno significato diverso
E sono legati da varie formule
woven ivy
Vero
Devo ordinare un po'
Ora ho ordinato la divergenza
Dovrebbe essere apposto
Però il flusso
Ha il caso in cui sono aperte
leaden cargo
il flusso è un particolare integrale su superficie
woven ivy
Ok
leaden cargo
Il flusso di un campo F attraverso una superficie S orientata con normale unitaria $\nu = N/|N|$ è dato da
$$
\int_S F\cdot \nu , dS
$$
woven ivy
Chiaro
rocky lotusBOT
Luigi
leaden cargo
Quindi è un particolare integrale di superficie
woven ivy
v è un vettore normale
leaden cargo
Si integra la funzione F⋅ν
woven ivy
Ok
Mi può chiedere l'integrale di superficie di una sfera?
In quel caso
Farei un integrale doppio di F per vettore normale
leaden cargo
Poiché $\nu = N/|N|$, quando usi la definizione di integrale di superficie ti esce
$$
\int_S F\cdot\nu,dS = \iint_D F\cdot N/|N| \cdot |N| , dxdy = \iint_D F\cdot N , dxdy
$$
woven ivy
No spe
leaden cargo
Quindi la definizione è più snella
rocky lotusBOT
Luigi
woven ivy
S è il bordo
leaden cargo
No S è una superficie
woven ivy
Ah
leaden cargo
Non c'è nessun bordo qui
woven ivy
Ok ok
leaden cargo
Nella scrittura sto sott'intendendo che la superficie S è parametrizzata da una certa funzione di dominio D
woven ivy
Ha senso
leaden cargo
Quindi se vuoi calcolare un flusso, prima devi parametrizzare la superficie, poi trovare una normale coerente con l'orientazione, e poi inserire nell'integrale
leaden cargo
Qui S è una superficie particolare perché è il bordo di E, quindi si può applicare il teorema della divergenza
leaden cargo
Lui mi dava già la parametrizzazione?
No
leaden cargo
Integrale di superficie di quale funzione? Qui non c'è nessuna funzione scalare
woven ivy
Allora dovevo fare (x, y, 4-x quadro - y quadro)
leaden cargo
C'è solo un campo, quindi puoi calcolare il flusso
woven ivy
Ah
leaden cargo
Allora dovevo fare (x, y, 4-x quadro - y quadro)
A computer si scrive x^2 per dire x²
leaden cargo
x^2 è più veloce di x quadro
leaden cargo
Per parlare di integrale di superficie ti serve una superficie e una funzione da integrare
woven ivy
Faccio il flusso
leaden cargo
E come se io ti dico di punto in bianco “fai l'integrale da a a b”, non ti viene spontaneo di chiedere, di cosa?
leaden cargo
E è il solido pieno
woven ivy
Decisamente
Richiamo la mia domanda originale
Non fa la divergenza in questo esercizio
leaden cargo
Non c'è nessuna divergenza qui perché è un altro contesto
leaden cargo
abbiamo un solido Ω e una funzione scalare f(x,y,z) = z√(x²+y²)
questo è un integrale triplo
woven ivy
Si
leaden cargo
nessuna superficie, flusso, curva, etc.
woven ivy
Esatto
woven ivy
Perché considera il raggio 3 0
?
leaden cargo
Come perché, l'hai detto tu che si ottiene intersecando un cono e una sfera
woven ivy
E sale su
woven ivy
Conti il più grande dei due?
leaden cargo
Nessuno è più grande
Il disegno che devi avere in mente è un cono gelato con una pallina sopra
leaden cargo
woven ivy
Esatto
leaden cargo
La parte blue a sinistra è il nostro disegno
leaden cargo
Che vuol dire “la parte blue non ha 3 come raggio”
woven ivy
Io considero la parte blu giusto
leaden cargo
Eh
woven ivy
Come omega
woven ivy
Ok
leaden cargo
Se ti muovi lungo quel segmento parti dall'origine per r=0 e ti muovi finché non tocchi il bordo per r=3
leaden cargo
Perché guardi lì?
leaden cargo
leaden cargo
Questo deriva dal fatto che parli senza essere preciso, quando prima mi dicevi “prendi due superfici e guarda il raggio”
la parola raggio non significa assolutamente niente senza contesto
Nel contesto delle coordinate sferiche di centro (0,0,0) il numero r è la distanza dall'origine
woven ivy
Ok grazie mille
leaden cargo
Questa “a” non è la distanza dall'origine
woven ivy
Per il cilindro invece
woven ivy
Questa “a” non è la distanza dall'origine
Giusto
leaden cargo
devi imparare a contestualizzare
leaden cargo
Nel caso del cilindro
Quale cilindro?
woven ivy
Se avessi
woven ivy
Quale cilindro?
Aspetta
leaden cargo
Se avessi un cilindro?
woven ivy
Ti faccio un disegno
woven ivy
Se avessi __un__ cilindro?
Aspetta
Se fosse questo il mio contesto
Il raggio quale sarebbe
Questo?
leaden cargo
Scusa ma leggi quello che dico?
leaden cargo
Questo deriva dal fatto che parli senza essere preciso, quando prima mi dicevi “...
.
La parola raggio non vuol dire assolutamente niente
woven ivy
La parola raggio non vuol dire assolutamente niente
Come devo esprimermi
leaden cargo
Se tu vuoi descrivere quell'insieme blu devi usare delle coordinate
che coordinate vuoi usare?
leaden cargo
queste sono coordinate (x,y,z), e non esiste alcun numero che puoi voler chiamare r
Fai dire a me
woven ivy
E il cilindro x^2+y^2=4
leaden cargo
Si usano di solito 2 tipi di coordinate per descrivere oggetti “sferici/di rotazione”
woven ivy
Con altezza da 0 a 5
leaden cargo
Le coordinate sferiche
{x = x₀ + ρcosθcosφ
{y = y₀ + ρsinθcosφ
{z = z₀ + ρsinφ
In cui (x₀,y₀,z₀) è un punto fisso, ρ è la distanza di (x,y,z) da lui e (θ,φ) sono angoli che misurano longitudine e latitudine
Poi ci sono le coordinate cilindriche
{x = x₀ + ρcosθ
{y = y₀ + ρsinθ
{z = z
In cui (x₀,y₀,0) è un punto fisso, ρ è la distanza di (x,y,z) dall'asse verticale in (x₀,y₀) θ è l'angolo che descrive il punto rispetto tale asse
In questi 2 contesti ρ ha due significati diversi
leaden cargo
Questo qui sarebbe il valore di ρ quando ti trovi alla quota indicata nelle coordinate cilindriche
leaden cargo
Questo è il valore di ρ nelle coordinate sferiche
woven ivy
woven ivy
Questo è il valore di ρ **nelle coordinate sferiche**
Ok
leaden cargo
La risoluzione dell'esercizio è in coordinate sferiche, quindi ρ è la distanza dall'origine
woven ivy
Ma p nelle cilindriche
leaden cargo
Si può fare anche in coordinate cilindriche, ma è terrible in termini di calcoli
woven ivy
Non parte dall'origine
leaden cargo
Non parte dall'origine
In nessuno dei due parte dall'origine, dipende dalla figura specifica quali sono i valori di ρ,θ,φ,x,y,z, etc.
leaden cargo
Per esempio se devi descrivere una corona sferica, ρ non parte da 0 ma parte dal raggio minimo
stessa cosa se devi descrivere un cilindro su una corona circolare
leaden cargo
Chi è P?
woven ivy
Non lo so più
leaden cargo
ρ?
woven ivy
leaden cargo
Si chiama rho
leaden cargo
$\rho$
woven ivy
Ma non ho lettere greche
rocky lotusBOT
Luigi
woven ivy
Nella tastiera
leaden cargo
I tipi di coordinate sono modi diversi di descrivere un oggetto
woven ivy
Vero
leaden cargo
Un oggetto geometrico lo puoi descrivere usando (x,y,z)
dette “coordinate cartesiane”
woven ivy
Vero
leaden cargo
O in altri modi, e ce ne sono infiniti
woven ivy
Giusto
leaden cargo
Tra gli infiniti modi, ce ne sono 2 importanti nello spazio per descrivere gli oggetti di rotazione in maniera comoda
coordinate cilindriche e coordinate sferiche
Allo stesso modo in cui nel piano esistono le coordinate polari
woven ivy
Sì
leaden cargo
Un cambio di coordinate ti dice solo come passare da una descrizione in (x,y,z) ad una in (ρ,θ,z) oppure (ρ,θ,φ)
woven ivy
Esattamente
leaden cargo
Ora non è che sono coordinate astrattissime, hanno dei significati geometrici
woven ivy
Sì
leaden cargo
(x,y,z) facile, sono le distanze da ciascuno dei piani coordinate
woven ivy
Esatto
leaden cargo
(ρ,θ,φ) hanno il significato che ho detto sopra
ρ è la distanza dall'origine
θ è la lungitudine
φ è la latitudine
woven ivy
Vero
leaden cargo
per esempio il punto (ρ=2, θ=π/4, φ=π/4) corrisponde in coordinate cartesiane a chi?
leaden cargo
45° non vuol dire niente nello spazio
Ci sono 2 angoli, longitudine e latitudine
Però comunque io ti ho chiesto chi sono le sue coordinate cartesiane
leaden cargo
Devi usare la formula per il cambio
woven ivy
Non me la ricordo
leaden cargo
Le coordinate sferiche
{x = x₀ + ρcosθcosφ
{y = y₀ + ρsinθcosφ
{z = z₀ + ρsinφ
I...
scritta qui
{x = ρcosθcosφ
{y = ρsinθcosφ
{z = ρsinφ
woven ivy
Ah quella
leaden cargo
Poi scrivi in TeX, non ti fa male
tra i dollari inserisci quello che vuoi formattare
radice quadrata si fa con \sqrt
leaden cargo
Lo ho scritto
Le coordinate sferiche uno può centrarle nel punto che vuole, esattamente come le polari in ℝ²
leaden cargo
Però facciamo che sono centrate in (0,0,0)
vale dockBOT
vale dockBOT
Available help channel!
quaint bolt
vale dockBOT
sleek nebula
was thinking LH
somber sequoia
Are you aware of the 1^infinty formula
rocky lotusBOT
dyxn
somber sequoia
English?
quaint bolt
We have x -> +infini
somber sequoia
yep
quaint bolt
somber sequoia
yep
quaint bolt
Then e^[1]=e
somber sequoia
yes
somber sequoia
Just a geometric series
rocky lotusBOT
dyxn
viscid spade
quaint bolt
Just a geometric series
But the sign changes
somber sequoia
its okay
we only need the absolute value to be less than 1
here x = -1/2
|-1/2| < 1
Oh wait
Don't plug and chug
quaint bolt
I need to put( -1)^(n) outside the serie?
quaint bolt
this starts from k = 0, your sum starts from k = 1
Yeah ik
somber sequoia
You can write your series as $\displaystyle \sum_{k = 1}^{\infty} \left ( -\frac 12 \right )^k$
rocky lotusBOT
dyxn
somber sequoia
does that make sense?
quaint bolt
somber sequoia
That's the formula for k = 0
quaint bolt
So i need to add 1
rocky lotusBOT
dyxn
somber sequoia
So i need to add 1
subtract it
somber sequoia
yep
vale dockBOT
ebon glade
!nopdf
vale dockBOT
Please post images (such as PNGs or JPGs) of the question rather than other filetypes such as PDFs which have to be downloaded. Non-image downloads can potentially contain viruses or other security risks.
river shale
guys i made this test is the reasonably hard
Sister, claim another channel
river shale
Here
river shale
Bruh 😂
tribal plover
you got she/her pronouns
i forgo