Which you are interested and what i know in group theory
#help-36
restive sinew
AS I HAVE SAID i have found order of 5 in z30 no?
Same for 3
Did i ask any help?
Nooo
desert mantle
the problem we have is that $\bZ_{30}\otimes \bZ_{12}$ is not a thing in general notation
restive sinew
I am only asked and the conversation went in wrong direction by stating definition
soft zealotBOT
Denascite
desert mantle
we doubt that your course means the tensor product of those two rings
but we dont know what your course means
and we have been asking you several times to tell us
restive sinew
So it is asking ring groups
I will leave it and back to it when i read it
Thanks for helping all the people
desert mantle
that is not an answer we can work with
final saddleBOT
@restive sinew Has your question been resolved?
fleet cove
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fleet cove
dy/dx = siny/cosy
$$\text{I am getting } cosy = \frac{1}{sqrt(1-e^2x)}$$
$$\text{I am getting } cosy = \frac{1}{\sqrt(1-e^(2x))}$$
soft zealotBOT
hoax
steep nest
how are you getting cos(y) = 1/(sqrt(1-e^(2x)))?
fleet cove
from x =ln(siny)
e^x = siny
siny = e^x/1
oh shit
i made the wrong right triangle
i made e^x the opposite side and 1 the adjacent side instead of the hypotenuse
.close
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modest sequoia
can someone help figure out where my mistake is?
final saddleBOT
can someone help figure out where my mistake is?
vital crag
can someone help figure out where my mistake is?
show the original question and the actual answer
modest sequoia
literally the question is just to evaluate the given integral
and i asked WA, the answer should be 81
modest sequoia
ahhh i see
ill keep this channel open while i do it
now im getting 6z^2 + 6z dz when simplifying, which gets me to 108
nevermind im stupid
that moment when youre doing multivariable calculus and forget how algebra works:
anyway thanks for the help appreciate it
vital crag
you should also practice grouping terms together before integrating
modest sequoia
like just combining like terms? i did do that
or do you mean integrating from factored form
modest sequoia
i see
yeah that makes sense i guess i just try to make it as simple as possible since im very confident in my algebra skills
it may not look like it right now though lol just getting back from holiday break
.close
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old quarry
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old quarry
I got confused by the such that 1 belongs to f(A) part
Does this mean like we have to include an element of A as 1?
For eg like this
Sorry it's kind of a dumb question
onyx peak
I got confused by the such that 1 belongs to f(A) part
it means that something has to map to 1
i.e. there is an arrow to 1
onyx peak
why am i getting different answer than them
does many to one mean that it cant be one to one?
lucid nymph
are you accidentally letting some outcomes have an input mapping 2 outputs?
remember that f(A) is a function so it needs to abide by the rules
silver dew
does many to one mean that it *cant* be one to one?
Yeah. Only this way we get ||175-24||
onyx peak
right, then it makes sense, i got just ||175||, i thought that many to one just means that its not necessarily one to one
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@old quarry Has your question been resolved?
old quarry
i.e. there is an arrow to 1
ohhh
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plain rain
,tex
Hi. Suppose we know for a sequence $a_n $ , that $\forall \epsilon >0 $ , and for a constant $a_0 $ there exist infinite terms of the sequence with the form $ a_0 -\epsilon < a_{n_k} < a_0 +\epsilon $. If we let $ \epsilon = \frac{1}{n} $ for all naturals, then is it rigorous enough to say that we can construct a subsequence whose terms follow that inequality? Since we just take a lower epsilon every time. To me, it looks a bit shady, because even our first term of the subsequence can be arbitrarily close to $a_0 $. Is there a better way to prove convergence?
final saddleBOT
,tex
Hi. Suppose we know for a sequence $a_n $ , that $\forall \epsilon >0 $ , a...
soft zealotBOT
fijokazż
rugged merlin
,tex
Hi. Suppose we know for a sequence $a_n $ , that $\forall \epsilon >0 $ , a...
you can use induction
plain rain
im not sure i know how to use induction here
like, we've already proven that for all naturals , a_n_k is within 1/n of a_0
anyway, ill think it thru cuz i dont think its as deep as i thought it was
.close
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ivory sage
hello there is this sequence i dont know how to find the nth term of its 3,10,29,66
final saddleBOT
hello there is this sequence i dont know how to find the nth term of its 3,10,2...
ivory sage
can you help me find it
glossy zephyr
where you got it from?
lime crest
hello there is this sequence i dont know how to find the nth term of its 3,10,2...
try to look at squares or cubes
after that its pretty obvious
ivory sage
i did i ended up finding a wrong sequence
like
the nth term only applied to the first
term
not the rest
lime crest
try subtracting some number from each term
you should get a pretty recognizable sequence
ivory sage
i could do that but
to get the full mark
i need to use a formula or rule which
is applible to evrry sequence
like guessing will give me one or two marks out of 4
i treid using
lime crest
yeah, guessing is only part of the idea
uneven silo
3rd difference is constant so its a cubic
lime crest
yeah, guessing is only part of the idea
when you write the solution you would phrase it as
"Notice that
2 + 1^3 = 3
2 + 2^3 = 10
2 + 3^3 = 29"
etc
lime crest
when you write the solution you would phrase it as
"Notice that
2 + 1^3 = 3
2 +...
"Hence, it follows that the n-th term of this sequence is […]"
uneven silo
You can also do it without guessing (although calculated guessing will be faster in most cases).
Okay
Okay I can't get that to work
But once you know it's a cubic
Assume the basic cubic equation
Set up the four equations with the n = 1,2,3,4
And solve the equations
This will get you the sequence as well
final saddleBOT
@ivory sage Has your question been resolved?
ornate ember
hi i'm struggling with some proof based calculus tasks and am wondering wether i can move stuff from one side to the other when i can only use axioms e.g. a = b => a-b = 0 ?
final saddleBOT
hi i'm struggling with some proof based calculus tasks and am wondering wether i...
ornate ember
these are the field properties i was given
onyx peak
hi i'm struggling with some proof based calculus tasks and am wondering wether i...
yes, that should be fine
firstly, by E there exists some -b with b + (-b) = 0
if a = b, then a + (-b) = b + (-b) = 0 (by E)
the fact that if a = b, then a + (-b) = b + (-b) is a property of equality
a = b means that a and b are the same thing, so replacing one by the other (such as in a + (-b) and b + (-b)) doesnt change the value of the expression
onyx peak
if a = b, then a + (-b) = b + (-b) = 0 (by E)
and for the opposite direction,
if a - b = 0, then a - b = 0 and so a - b + b = b and again by E and D, a = b
ornate ember
okay so if i understand correctly the first step would be adding -b to both sides and then using property E?
onyx peak
okay so if i understand correctly the first step would be adding -b to both side...
yep
and depending on whether you need -> or <->, you'd need to consider the other direction as well
final saddleBOT
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jagged wave
want to sanitycheck this (from d&f)
final saddleBOT
want to sanitycheck this (from d&f)
jagged wave
"right inverse" should read "two-sided inverse"
strange sparrow
Looks correct to me
a super tiny nitpick might be to write id_Δ or id_Ω explicitly instead of just id
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robust palm
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robust palm
here it would be better to use the answer to part a to solve b?
y^2 dy = t^2 cos^2 t dt
winter stag
yeah they prob want you to use it actually
final saddleBOT
@robust palm Has your question been resolved?
desert mantle
hence
robust palm
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robust palm
idk how 8g * 1.5cos50 gives you the magnitude of rotation in the opposite direction of F
bold turtle
Correction, it's the magnitude of the moment in the opposite direction of the moment caused by F
robust palm
Correction, it's the magnitude of the _moment_ in the opposite direction of the ...
thanks i didn't know how to phrase it
bold turtle
But what specifically don't you get about the calculation there?
I'm guessing the angle?
bold turtle
Yes
But you're given the weight force, which is acting at a different angle
We need to find out how much of that force is acting in that red direction
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robust palm
.reopen
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✅ Original question: #help-36 message
robust palm
We need to find out how much of that force is acting in that red direction
wait one last thing
runic needle
what's 82-15?
robust palm
why did they write it as 8g * 1.5cos50 instead of 1.5 * 8gcos50, kinda weird?
bold turtle
what's 82-15?
I will throw the pink hammer at you if you persist with this.
bold turtle
why did they write it as 8g * 1.5cos50 instead of 1.5 * 8gcos50, kinda weird?
That is a little odd, yh
They probably took F d cos theta as the formulation
runic needle
I will throw the pink hammer at you if you persist with this.
apologies
bold turtle
Then just plugged things in
robust palm
They probably took `F d cos theta` as the formulation
okay thanks
bold turtle
Because usually the formula for moment is "F d" (where F is described as the perpendicular force)
robust palm
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latent dragon
<@&268886789983436800>
I see 3 mod reacts. Did they all dissect the guy at once or did one get him and the other two are lying? 
final tangle
Yes
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crude narwhal
can someone explain this simply
final saddleBOT
can someone explain this simply
opal plinth
Explain what? How it's calculated?
unborn forge
x is the solution of 5^x = 5* sqrt(5)
crude narwhal
yes
leaden moon
so $\log_5 5\sqrt{5}$ is basically a number $x$ when $5^x = 5\sqrt{5}$
soft zealotBOT
1 divided by 0 equals Infinity
heady moon
$\sqrt(a)$ can be written as $a^(\frac{1}{2})$
leaden moon
$\sqrt{a}$ can be written as $a^{\frac12}$
soft zealotBOT
1 divided by 0 equals Infinity
leaden moon
can someone explain this simply
anyways
leaden moon
can someone explain this simply
apply the log rules here
$\log_m ab = \log_m a + \log_m b$
soft zealotBOT
1 divided by 0 equals Infinity
leaden moon
**1 divided by 0 equals Infinity**
do we all agree on this
leaden moon
**1 divided by 0 equals Infinity**
and combining with this rule that @heady moon originally said
you can solve for $x$
soft zealotBOT
1 divided by 0 equals Infinity
heady moon
and combining with this rule that <@1368541610891280424> originally said
-# i'm bad with la te
bold turtle
-# (did you really have to texit an x)
crude narwhal
hahah
leaden moon
-# (did you really have to texit an x)
-# yes 😈
soft zealotBOT
Starlord
leaden moon
**Starlord**
nuh uh
crude narwhal
isnt it 1/2
leaden moon
**Starlord**
and also !nosols
leaden moon
**Starlord**
!nosols
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**Starlord**
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
tired walrus
also \log.
leaden moon
yuh-huh, actually, but yh, this is a little fast
yeah
leaden moon
I don’t think that’s what he was asking for though ?
you aren't allowed to spit out solutions immediately
unborn forge
Yes but I don’t think that’s the solution to his question
leaden moon
Yes but I don’t think that’s the solution to his question
you still can't spit out too fast
crude narwhal
anyways guys thanks i get it\
leaden moon
!done
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leaden moon
https://tenor.com/view/the-rock-spit-gif-20218743
-# spit
crude narwhal
.close
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left trail
for the second part I have it set up as $\frac{\sum_{j=1}^{n}f_j^{'}\prod_{\substack{k=1,k\neq j}}^{n}f_k}{\prod_{k=1}^{n}f_k}$. I can see that if we write out the sum we can have a bunch of fractions and we cancel out the common term and we have the end result. However, I don't see how I can show that with the way I have it written
final saddleBOT
for the second part I have it set up as $\frac{\sum_{j=1}^{n}f_j^{'}\prod_{\subs...
soft zealotBOT
BigBen
gusty ravine
**BigBen**
is it neccesary to show or prove by induction??
gusty ravine
for the second part I have it set up as $\frac{\sum_{j=1}^{n}f_j^{'}\prod_{\subs...
yess
@left trail
left trail
is it neccesary to show or prove by induction??
The way I read the problem was that we had to use induction to show how to differentiate g when g is a product of n functions
gusty ravine
The way I read the problem was that we had to use induction to show how to diffe...
oh ohkhohl
left trail
I am just struggling to see how can I break the sum up so that I have a sum of n fractions where each denominator is the product of n functions and the numerator for each is the dervivative of the nth function times the rest of the functions
gusty ravine
idk really then i can get to the second part but the rule is difficult
gusty ravine
loh yes
left trail
no he is talking about that in 2 chapters I think
tired walrus
ok so bigben doesn't have logs
ugh if it was not late at night i would cook sth
alas i am off to sleep
left trail
ok gn
gusty ravine
no he is talking about that in 2 chapters I think
do you know about logarithms??
left trail
Yes but I want to try following the sequence that the books proposes as closely as I can
gusty ravine
Yes but I want to try following the sequence that the books proposes as closely ...
yea so first we find by log and then using induction we prove our point right
thats what i am saying
i mean basic log is introduced in highschool like basic properties
left trail
True. But is there no way to manipulate the sum?
gusty ravine
gusty ravine
True. But is there no way to manipulate the sum?
idk about that really
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@left trail Has your question been resolved?
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prisma crag
The perimeter of a triangle with all sides the same length is 5 cm 4 mm. Construct this triangle on a 2:1 scale. Calculate the perimeter of the triangle you drew.
final saddleBOT
The perimeter of a triangle with all sides the same length is 5 cm 4 mm. Constru...
prisma crag
So 10 cm and 8 mm?
abstract bramble
i suppose
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@prisma crag Has your question been resolved?
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arctic plover
a) l is 9
b) m is 9
c) l is 27
d) m is 16
final saddleBOT
a) l is 9
b) m is 9
c) l is 27
d) m is 16
tiny gorge
any thoughts?
mighty apex
do uk what are one-one fxns?
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@arctic plover Has your question been resolved?
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arctic plover
.reopen
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✅ Original question: #help-36 message
mighty apex
alr, so u know abt one-one fxn?
mighty apex
do u have the formulas
arctic plover
there are no formulae
arctic plover
this is permutation
mighty apex
and n - no of elements in codomain
mighty apex
this is permutation
yea
mighty apex
yea , are u not given the formula list
arctic plover
no
mighty apex
hmm alr
arctic plover
you have?
mighty apex
ok can u make all the possible one one fxn
mighty apex
ok
chilly pollen
tutor also focuses on logic
That is the desired approach
arctic plover
That is the desired approach
yeah can you explain
mighty apex
so, lemme explain why u can use permutation
chilly pollen
You dont have to rote learn a bunch of formulae
arctic plover
so, lemme explain why u can use permutation
ill send one question to be done my permutation
mighty apex
so 1 has 3 possible outcomes to have, then when 1 will have one of them.
then 2 has 2 possible outcomes since it is one one
similarly, 3 has 1 possible outcome
arctic plover
The number of one-one functions that can be defined from A = {4, 8, 12, 16} to B is 5040, then n(B)=
this sample
mighty apex
ok , so since its one-one
chilly pollen
The number of one-one functions that can be defined from A = {4, 8, 12, 16} to B...
You've studied permutations right?
mighty apex
n!/(n-m)!
= n! / (n-4)!
= n x (n-1) x (n-2) x (n-3) x (n-4)!/ (n-4)!
(n-4)! gets cancelled
= n (n-1) (n-2) (n-3) = 5040
i think u can now solve it
arctic plover
You've studied permutations right?
yeah somewhat
chilly pollen
!nosols please @mighty apex
final saddleBOT
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
mighty apex
!nosols please <@962370448740462612>
i didnt give the answr
mighty apex
the ques is diff, this was an example of how permutation can be used
they are self studying
arctic plover
i think u can now solve it
this one was solved
i am thinking
about the other one
mighty apex
yea, so now u can find I
chilly pollen
You need help with the original question?
mighty apex
NOW THERE IS ANOTHER FORMULA
mighty apex
m
chilly pollen
Ok so for the first one, we can map one element to any of the three elements in B, correct?
I am talking about functions in general (not necessarily one to one)
arctic plover
I am talking about functions in general (not necessarily one to one)
yeah
generally yes
chilly pollen
And the second element can also be mapped to any of the three elements
Any one of them
arctic plover
to find I u can use permutation, what do u think u can do
m thinking logically only
mighty apex
m thinking logically only
YEA IM TELLING U THE LOGIC
arctic plover
And the second element can also be mapped to any of the three elements
yes
chilly pollen
yes
And the same goes for the third element of A
so we have 3 times 3 times 3 choices.
chilly pollen
Because for one choice of a1, we have 3 choices for a2 and for each of them we have 3 choices for a3
So, now you know the total number of functions
Can you think of a similar line of reasoning for one-to-one functions. After choosing one element for a_1, we only have 2 remaining elements to choose for a_2
mighty apex
yea using permutation, u get total functions = i = ?
chilly pollen
Can you think of a similar line of reasoning for one-to-one functions. After cho...
Do you see an answer now to your question @arctic plover
arctic plover
yeah yeah
mighty apex
did u get it?
old jungle
How do I create a individual channel like this and not a little group thing
mighty apex
How do I create a individual channel like this and not a little group thing
go to math help (available)
arctic plover
so l is 27
mighty apex
yes
arctic plover
right
old jungle
Okay thank you sorry I’m bad at reading
arctic plover
m should be 3?
chilly pollen
Okay thank you sorry I’m bad at reading
you're good. Welcome to mathcord.
mighty apex
m should be 3?
no..
old jungle
you're good. Welcome to mathcord.
thanks
chilly pollen
m should be 3?
Think this again
mighty apex
if 1 alr has an outcome, then how many possible outcomes does 2 have left?
chilly pollen
you have 3 choices for a_1, arbitrarily, how many choices does a particular choice of a_1 leave for a_2 @arctic plover (By a_1, I mean the first element of A as in f:A -> B)
chilly pollen
good job
Surjective functions are a little more complicated. Anyways, have you got any additional questions you need help with?
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arctic plover
.close
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craggy atlas
what formula do i have to use for this
final saddleBOT
what formula do i have to use for this
ivory vessel
if you have a point on the graph, what does it mean to compute the distance from that point to the x-axis?
tranquil pine
what formula do i have to use for this
condense the sum into single sin or cosine term and find the amplitude
ivory vessel
!nosols
final saddleBOT
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
ivory vessel
i mean what the factoid said
tranquil pine
i deleted the answer
ivory vessel
do you expect me to delete my messages after you deleted yours?
tranquil pine
nope
ivory vessel
if you have a point on the graph, what does it mean to compute the distance from...
@craggy atlas
tranquil pine
sorry
tranquil pine
<@810450364557950986>
well it means that u have a point on the graph y = whatever, u have find the max height of such point from x axis or to say maximum y value!!
tranquil pine
oh right sorry again wont happen again
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@craggy atlas Has your question been resolved?
tired walrus
OP gone?
ivory vessel
@craggy atlas pls stop your genshin impact and engage with your channel
tired walrus
damn fungus you're putting her on blast here 
granite tapir
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slender shard
what question do you have
granite tapir
what question do you have
I am trying to understand the question
strange sparrow
I think it's just asking you to show that epimorphisms and surjections coincide in Set
granite tapir
epic is more general arrow, surjective is a function,
a function is a morphism of sets in Set
granite tapir
I think it's just asking you to show that epimorphisms and surjections coincide ...
epimorphisms are in Set?
are all surjective functions surjective homomorphisms?
strange sparrow
A morphism in Set is just an ordinary set function
so an epi in Set is a function that is right cancellable
granite tapir
I think it's just asking you to show that epimorphisms and surjections coincide ...
and homomorphism is a morphism of Grp?
so the comment on surjective homomorphism is epimorphism, is about how
epimorphisms and epi are iso? or something like that
𝒞 ~ Set
morphism ~ function
epi ~ surjection
Grp ~ Groups?
morphism ~ homomorphism
epimorphism ~ surjective homomorphism
strange sparrow
yes
But note that in general epimorphisms need not have a right inverse
this fails in eg the category of abelian groups
granite tapir
But note that in general epimorphisms need not have a right inverse
it says epimorphisms are surjective homomorphisms
ah okay, like in linear algebra
maybe
strange sparrow
in general epimorphisms may not be surjective nor have a right inverse
I guess the book is talking just about epis in Grp
granite tapir
I guess the book is talking just about epis in Grp
I am a bit confused just yet, but I find it's good to think about these similarities/differences, here, maybe this is what you mean:
from Topoi by Goldblatt
Thanks for your help @strange sparrow 🎵
strange sparrow
I am a bit confused just yet, but I find it's good to think about these similari...
yes it is in general hard to decide whether epis are surjective or right-cancellable
A fun fact I like is that every epimorphism is surjective in the category of planar graphs, but this result is actually equivalent to the four-colour theorem
granite tapir
A fun fact I like is that every epimorphism is surjective in the category of pla...
that's awesome, thanks for sharing this
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latent gazelle
Hi people
latent gazelle
Double and triple integrals in calculus
I do not understand how to decide the bounds
I can sketch the region usually
but finding the bounds is painful and excruciating, and everytime I have asked my teachers, classmates or on other forums I get more and more confused
My brain is hardwired to think "the bounds are edge to edge"
but that doesn't work for curved regions
but I don't understand what wroks for curved regions
Everytime, I end up guessing one side of the bound correctly, then everything else is wrong
lucid nymph
could you give an example?
latent gazelle
Yes
let me send a pic
ignore the bottom part
im just trying to figure out what the hell is going on there
it probably doesnt make any sense
but the upper part is from an old exam
that region is sketched accurately i think
i hope
soft zealotBOT
latent gazelle
yes we have that
glossy zephyr
Yeah, you drew it correctly.
glossy zephyr
Okay, technically its right
glossy zephyr
But integrating over those bounds will be a pain in the ass
latent gazelle
but for y i have no clue
glossy zephyr
gimme a sec.
latent gazelle
okay okay
but yeah either way, even if it is technically right, i wouldnt be able to figure out the y part anyway
is that the divergence theorem as your name?
it looks familiar
but im horrible with names
latent gazelle
yes
but okay listen
i know of a trick where you can swap to polar coordinates
and i know how to do that, i mean swap
but is that really the intended way to solve this
how can i solve this without doing that
glossy zephyr
yeah, theres where im heading, im just trying to explain why we will be swapping
glossy zephyr
You can technically avoid swapping, but the integration becomes a problem
latent gazelle
okay im really curious about avoiding swapping, can i say what i would put as my bounds
if i swapped
glossy zephyr
when you started to learn double integrals you prob first saw you learned the idea of choosing a "ceiling and floor" functions
for the region
Right?
latent gazelle
honestly no, im a bad student so i didnt attend the lecture
the online lectures i watched
basically said
"bro just swap to polar"
latent gazelle
thank you and sorry if im being difficult
like idk what ceiling and floor functions is
but i imagine y=-x is the floor function
and y=x is the ceiling function
but wha the circle would be
i have no clue
the wall "function"
latent gazelle
yes i have seen an explanation where
glossy zephyr
Where the region is a rectangle, the bounds are totally straight
Here we will call c or ceiling,
latent gazelle
the square gets sides r and theta
glossy zephyr
Ignore the transformation for now
glossy zephyr
Here we will call `c` or ceiling,
c does necesarilly have to be a constant, in fact, it can be a function.
latent gazelle
despite my struggles
okay
and 0
could also be a function
or let c and 0 be constants, then let a and b be functions?
that also works right
glossy zephyr
yep
This works precisely because the region has constant bounds on one term and not-constant bounds on the other
So its basically still a rectangle, but with varying heights.
Which is really easy to describe for us
latent gazelle
wait
glossy zephyr
By first integrating through the bounds of y, we get a function of x that is added to the last part of the integral
latent gazelle
to me, my brain has trouble understanding
what is what in this picture
like i think x is the variable with constant bounds
and y is varying
glossy zephyr
latent gazelle
but it is already a struggle to see that
glossy zephyr
yep, the "height" varies as a function of x.
latent gazelle
like it is very taxing
for me
idk why
i had to really think about it
even if it looks obvious to other people that is the truth
glossy zephyr
The problem comes with this kind of things
There is no "rectangular" shape that is immediatly obvious
latent gazelle
agreed
glossy zephyr
So you have to do the following:
latent gazelle
that is what my problem looks like to me
like that old exam problem
i know it isnt very similar, but it looks like that in my head
i dont know what to do with it
glossy zephyr
You have to cut up the region
Which makes everything a mess
for this particular case you would have to solve 7 different integrals.
latent gazelle
OH MY GOD
but even after cutting those up
i would have to cut horisontally?
because they are still not nice shapes
glossy zephyr
you can choose to cut up horizontally.
latent gazelle
wouldnt i do both here
glossy zephyr
When given an arbitrary region with no order, you can choose
latent gazelle
like turn the non-nice triangles into right angled ones
latent gazelle
i meant like, do both, so you get more integrals
but they are all easier
because they are over squares or triangles
latent gazelle
i actually did sketch that
glossy zephyr
In red, the vertical cut, in purple the horizontal
glossy zephyr
You can do this. the straight lines are easy
latent gazelle
and not a nice right angled triangle
glossy zephyr
But the curved bounds are a pain to integrate over
latent gazelle
yes i could even tell that
glossy zephyr
thats why we do polar
glossy zephyr
Check the angles.
glossy zephyr
its not only [0, pi/4]
glossy zephyr
youre still missing the bounds, lets go over that then
glossy zephyr
oh, i already understood
no, you wouldnt do that
because then in polar it becomes a mess
latent gazelle
i am not doing that
glossy zephyr
you just solve the whole shape.
latent gazelle
i am saying that is true
glossy zephyr
Ok
latent gazelle
i am trying to communicate
what i can see
because i cant see the right way to do it
im hoping stuff i can see is helpful to finally understanding how to do this
how would you even integrate over the curved bound
would you try to look at it like x^2 but shifted down
somehow
i mean without swapping to polar
like what do i do if im given this
glossy zephyr
there you only have straight lines
in the horizontal cuts, it means that all bounds are of the form x = my + b
Anyways, as for the polar transformation. This is what is basically happening
latent gazelle
right
glossy zephyr
The transformation itself transforms the quarter circle in a "rectangle" in another coordinate system.
glossy zephyr
just r.
glossy zephyr
Can you find the bounds?
glossy zephyr
yep.
glossy zephyr
the bounds are numerical here.
latent gazelle
r goes from 0 to r, and theta goes from 0 to 2pi
okay
if it is the unit circle maybe it would be 0 to 1, and theta from 0 to pi
2piÄ
glossy zephyr
you got the bounds of r correct.
Nowhere near for theta.
Try to look at the quarter circle on the left.
glossy zephyr
The bounds of A' are defined by the bounds of A
latent gazelle
yes i am stupid
but i could tell that
i literally just didnt think of A' as being defined by A
i was just thinking generally of that transformation
for some different more general circular thing
glossy zephyr
,tex you basically get this:
$$\iint_A y,dx,dy \implies \iint_{A'} f(r,\theta), dr, d\theta$$
soft zealotBOT
glossy zephyr
you know polar transformations to find f, right?
latent gazelle
im not trying to catch you out, but im just wondering if there should be an r in there
or is that implied somehow
or not necessary
you didnt type equivalent so
oh
glossy zephyr
implied, yeah. I was hoping you were able to construct the whole thing
latent gazelle
wait that is what your question means
sorry
yes sure i can try
double integral over A' of r^2sin(theta)drd(theta)
latent gazelle
dude you are being so helpful
thank you so much
genuinely
even if im not all the way there sincerely
thank you
glossy zephyr
np.
glossy zephyr
yeah, go ahead.
latent gazelle
okay okay
like i know how to do this step
but im slow
the problem has just been the bounds
glossy zephyr
Consider that the bounds of A' are independent from each other (all constants)
and that f(r,theta) is a separable function
aka, it can be transformed into f(r) * f(theta)
latent gazelle
sorry what were you trying to explain
i didnt understand at all where you were going and
i dont really understand what you are saying
so we know A' bounds are constants
idk what a seperable functio nis
function is*
is it just that we can get
y on one side
all the y terms
and all the x terms on another side
or in this case
all the r terms
and theta terms
maybe terms isnt even the correct word
to use
glossy zephyr
separable functions + constant bounds allow you to do this
latent gazelle
i have only seen seperable used to describe seperable differential equations before
in singlevariable calculus
glossy zephyr
i have only seen seperable used to describe seperable differential equations bef...
same idea btw.
latent gazelle
for ODEs
okay yeah i have never seen that trick
but if one of the bounds were variables
this wouldnt have been okay
this only works when everything is constants?
glossy zephyr
Tbh, if the bounds of theta where functions of r or viceversa
You can still do it, just make sure you dont integrate the other part before
latent gazelle
okay and
say i changed my original problem
maybe so that
x^2+(y-1)^2=9
so now it is centered around (0,1)
and with radius 3
now it is more hellish but
let me try to work out bounds anyway
glossy zephyr
the problem with shifted circles is that they arent that well behaved in polar coordinates.
latent gazelle
i had some intuition that told me that but
what if i
swap coordinates to unshift it
then i swap to polar
is that allowed
glossy zephyr
yeah, you can
latent gazelle
or do i have to swap back at the end
glossy zephyr
Btw they obviously didnt explain it
but shiftings are the same thing as polar transformations
latent gazelle
jacobian
glossy zephyr
beyond the obvious fact that they are different in how they transform.
latent gazelle
they did explain this
but it is freaky
how the hell do i make my own jacobian again
nabla something
partial derivatives of the whole function
i dont remember
how to determine the jacobian
glossy zephyr
quick sec.
latent gazelle
actually no
this was too ambitious
i think
and i am getting more confused
how come i dont have to also change the lines that bound the weird circle part
or am i secretly doing that, and somehow it still makes sense
rsin(theta)=rcos(theta)
and rsin(theta)=-rcos(theta)
glossy zephyr
,tex \newcommand\pd[2]{{\partial #2 \over\partial #1}}
$\begin{vmatrix} \pd ux&\pd vy \vspace{3pt}\ \pd uy& \pd vy \end{vmatrix}$
soft zealotBOT
glossy zephyr
this is the formula for the jacobian determinant
latent gazelle
hermmm
hermmmm
is my mind playing tricks or why does it remember there being an i, j and k in there
or is there no i j k
glossy zephyr
not here, no.
latent gazelle
ok ok
then my mind is playing tricks
so it is 2x3 matrix!?
oh wait
my god
it is 2x2
whoopsies
the fractions confused me
oooooh
wait what
glossy zephyr
try to find the determinant of the polar coord transformation yourself.
latent gazelle
ad-bc
glossy zephyr
consider u = r, v = theta.
latent gazelle
oh
glossy zephyr
first try to compute the derivatives btw
latent gazelle
sry i got a surprise improtant call from my father
i will come back to this after
so sorry
thank you so much again
final saddleBOT
@latent gazelle Has your question been resolved?
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old quarry
final saddleBOT
old quarry
how to integrate this
tired walrus
maybe try decomposing (2c+1)/(c+2)^2 into partial fractions
old quarry
2/cosx+2 -3/cosx+2 square
tired walrus
are you allergic to parentheses 😭
2/(cos(x)+2) - 3/(cos(x)+2)^2
hmmmm
okay maybe this is no good
WA gives a very neat answer but honest to god i dont know how to get there.
old quarry
oh
old quarry
what if you expanded the denominator and then split the top
🤔 will that help?
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thin ice
helooo
i need to find the sum of different values that x can take
i just need a tip on how to start, i searched for examples with x^lnx but i couldnt find something exactly like that. my guess is that it equals 1? but i dont have any reason for it to be 1
final saddleBOT
helooo
i need to find the sum of different values that x can take
i just need...
tired walrus
take ln on both sides
barren hound
no that's right
thin ice
nvm i think its correct
ye :3
im gonna try from now on, give me a sec and thanks!!
i got it, thank you!!
.close
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minor sandal
Whats the use of m?
final saddleBOT
Whats the use of m?
glossy zephyr
an arbitrarily big quantity as far as i can tell.
minor sandal
Whats the "use" of m
weak laurel
effectively saying that after some point m, all xn are arbitrarily close to x
i believe so anyway
glossy zephyr
As an arbitrarily big quantity, it marks a point at which all elements of index >= n of both sequences satisfy the left equation.
minor sandal
effectively saying that after some point m, all xn are arbitrarily close to x
Thats what I thought too
glossy zephyr
effectively saying that after some point m, all xn are arbitrarily close to x
more like they start closing in, but yeah
minor sandal
But if that is the case , shouldn't m be a function of ε
minor sandal
more like they start closing in, but yeah
So all n≥m , satisfy
|xn - x| < ε
glossy zephyr
not quite
thats why we use C(a_n)
C(a_n) is like a sequence we "compare" to.
if all elements of |x_n - x| are smaller than C(a_n) on their respective index n >= m, and a_n approaches 0
then x_n has to approach x.
yours is more restrictive
minor sandal
I m thinking m to be a number , after which ....the terms of sequence are inside ε distance from limit of sequence
glossy zephyr
tbh i wouldnt advice you think of epsilon as part of the statement.
minor sandal
Sorry I didn't understand
glossy zephyr
I dont think this will actually help, anyways
we cant make any statements about the sequence x_n on indices n<m
Assume |x_n - x| to be other sequence we construct.
If the terms of it are approachin 0, then x_n is clearly approach x, right?
minor sandal
Yes
glossy zephyr
now, past some index "m", i say that all elements of the sequence |x_n - x| are lower than the elements of a_n
and i tell you that a_n limits to 0
do you agree that this also implies that x_n aproaches x?
minor sandal
yes..
glossy zephyr
now, C(a_n) is just the same sequence times a scalar over all terms
So it still limits 0.
minor sandal
Yes , each term just scaled
glossy zephyr
if $|x_n-x| \leq C a_n$ for all terms past a term indexed $m$, then $\lim_{n\to\infty}x_n=x$
soft zealotBOT
minor sandal
I have only studied basics, I only know that epsilon definition so far
glossy zephyr
If you agreed with the 3 things i said previously this is a simple consequence of tying the three together.
$\forall n \geq m$
basically encodes the meaning of "for all terms past a term indexed $m$"
soft zealotBOT
minor sandal
Yup
glossy zephyr
the m here is just some arbitrary big element
Maybe the series start to behave like this after the 1000th element
minor sandal
It wasnt mentioned in the theorem
glossy zephyr
but the idea still remains
glossy zephyr
It doesnt have to be large either, again, its arbitrary
maybe it behaves like this after m = 0
which means the whole sequence behaves like this
minor sandal
Maybe after working out few examples ...i will understand it better
glossy zephyr
in epsilon-delta proofs epsilon is usually meant to be really small.
minor sandal
Yess
glossy zephyr
but nothing is actually stopping you from choosing a really big number
and is nowhere mentioned that it has to be small.
glossy zephyr
np
minor sandal
Can you suggest some youtube channel
From where I can clear my real Analysis concepts
glossy zephyr
tbh i dont know any
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livid oyster
Let g(x) = x^11 − 2x + k − 7 , where k is a constant. If g(x) is divisible by x + 1 , find the
remainder when g(x) is divided by x .
A. −1
B. 1
C. 6
D. x^10 − 2
final saddleBOT
Let g(x) = x^11 − 2x + k − 7 , where k is a constant. If g(x) is divisible by x ...
uneven silo
!status
final saddleBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
1
lucid nymph
Do you know algebraic long division?
livid oyster
no sorry'
uneven silo
If a polynomial is divisible by x+1, what must g(-1) equal?
livid oyster
i have noclue
scarlet sequoia
The division remainder theorem should tell you
If g(x) = (x+1)*...
What is g(-1)
lucid nymph
The rule is that if a function is divisible by x+a, then it is always a root of the function
livid oyster
0?
livid oyster
oh ok
uneven silo
So substitute x = -1 into the equation and set your result equal to 0
livid oyster
.close
final saddleBOT
ornate knot
doesnt that just give the value of k?
we were supposed to find reminder left when divided by x 
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bitter zodiac
we were supposed to find reminder left when divided by x <:thonk:406575732563902...
(The translator makes me feel awkward. Please understand.)
You just have to use Polynomial Reminder Theorem to substitute zero in x as you did before.
final saddleBOT
(The translator makes me feel awkward. Please understand.)
You just have to use ...
ornate knot
(The translator makes me feel awkward. Please understand.)
You just have to use ...
ik , was just saying that finding k was not what the question said
cuz OP closed the channel
bitter zodiac
Ok
bitter zodiac
is 0^n always 1
bitter zodiac
https://tenor.com/view/bosnov-67-bosnov-67-67-meme-gif-16727368109953357722
Brainrot
latent dragon
What’s the question here? 
spark swan
<@&268886789983436800>
small grail
lyric obsidian
.close
final saddleBOT
desert mantle
<@&268886789983436800>
lime basin
Do u know why i cant do it like how i did it on the left since it should work right?
final saddleBOT
Do u know why i cant do it like how i did it on the left since it should work ri...
wary juniper
Whats thats on the 3rd line
wary juniper
If you factor out x, you get 1/(x-4)
lime basin
If you factor out x, you get 1/(x-4)
is it possible if u can right that down
And send it
wary juniper
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loud sundial
<@&268886789983436800> 
twin pivot
$y(t) = log[x^2(t+1)]$
final saddleBOT
$y(t) = log[x^2(t+1)]$
soft zealotBOT
Goofy Joe
tired walrus
what's the question
twin pivot
|x(t)| ≤ M so |y(t)| ≤ |log(M²)|
twin pivot
what's the question
Is y(t) stable?