if you try to start proving its surjective its not very clear how to do so
#help-27
normal verge
i'll try
uncut crow
so, find an element in R x R that nothing maps to
uncut crow
well it should be a pair
no, i can think of something that maps to (-1,-1)
namely, (1,-1)
normal verge
it's( -1 , -1)
i mean there no element in R x R-(0) that maps to -1,-1
normal verge
namely, (1,-1)
yeah you'r right
uncut crow
soz i got mixed up ignore my deleted messages lol
uncut crow
right, because ||for (x,y) to map to (0,1) we would need to have that one of x or y is 0 (otherwise xy is not 0). but then we can't have x/y = 1||
normal verge
yes understood
normal verge
right, because ||for (x,y) to map to (0,1) we would need to have that one of x o...
and y can't be 0 so it's not surjective+
uncut crow
normal verge
i have exam on Wednesday and now i'm gonna do that one more time and start with Relations
could you please help with that topic if i have question?? @uncut crow
uncut crow
perhaps
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stray lark
How to approach the rest of this question? I solved a system of equations to get those numbers but idk how to continue
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How to approach the rest of this question? I solved a system of equations to get...
wispy remnant
How to approach the rest of this question? I solved a system of equations to get...
this is optimisation
havent read too closely so i could be wrong but
form equations and then plot
find intersection points
and try solving to reduce cost
stray lark
form equations and then plot
Ok wait
I have three equations
is that where the t also comes in
in the first part
my three equations are
wispy remnant
wait let me read a little more closely
wispy remnant
x y and z are the factories?
yeah just solve these equations then
assuming these are correct
do u know 3 variable simultaneous eq?
stray lark
yes
uhhhh
well I solved them
and that's how I got x = 8, y = 3 and z = 0
I'm not sure how to apply the t tho
you see there's like a ___ t + 8
wispy remnant
you see there's like a ___ t + 8
im not too sure what this even means
stray lark
im not too sure what this even means
me neither
it's just in the question
and I was like
what
wispy remnant
5x+5y+33z = 55
yeah these equations make sense
wispy remnant
How to approach the rest of this question? I solved a system of equations to get...
.
stray lark
is this also optimization too
@wispy remnant do you by chance know the best way to approach this
I thought it would be multiplying a by "x" and then setting that equal to b
wispy remnant
i need to go
wispy remnant
I thought it would be multiplying a by "x" and then setting that equal to b
dont think this is correct
wispy remnant
i need to go
if this channel is still open in 15 ill help u
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stray lark
NOO
stray lark
@wispy remnant 🙏
mortal sable
Have you learned row reduction?
You could of course make this into a system of eq and solve it
But you should probably just get to rref and then you have to interpret it as a solution
stray lark
Have you learned row reduction?
I have learned rref
it's rudimentary but I think I can do it
rref form?
mortal sable
Yes
If you reduce to rref, that is about the same process as reducing a system of equations
stray lark
If you reduce to rref, that is about the same process as reducing a system of e...
how would it solve the t part though
wait which question are you refering to
the first one I posted or the second one
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GGS
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cunning roost
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cunning roost
what do they mean what is the ratio of height to radius that will admit the most sunlight?
I solved for r
DO i solve for L ?
It says in the answer key that height = l + r but howd they g et this information
and that h = r + r = 2r
then did this
kindred agate
the radius is half the height in that photo
versed jetty
It says in the answer key that height = l + r but howd they g et this informati...
true by how the window is arranged - it's a semicircle on top of a rectangle of height l
versed jetty
yes
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gilded raven
can soemone help me understand this i dont know how to approach this
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can soemone help me understand this i dont know how to approach this
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restive river
can soemone help me understand this i dont know how to approach this
It would just be their product
vital trail
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vital trail
Just making sure I am right
shy kestrel
why -inf for the first one?
shy kestrel
should be inf yes
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lament schooner
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radiant anvil
(x-1)(x^2+x+1)
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spice kernel
Why is the radius 1.5 instead of 3
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Why is the radius 1.5 instead of 3
spice kernel
Mb didnt mean to have 2 channels
winter patrol
pay close attention to the scale
spice kernel
Omg you’re right thank you 😭
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empty moth
Sorry just a quick question, how can the bottom not tell us whether the series converges or diverges if the limit is = 0?
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Sorry just a quick question, how can the bottom not tell us whether the series c...
empty moth
Doesnt it just say on the top that if the limit is 0 it converges?
hybrid snow
empty moth
so ur saying the nth term approaches 0 but the sum of all those terms does not converge
jaunty mantle
yes
jaunty mantle
1 + 1/2 + 1/3 + ...
the harmonic series does not converge
but clearly the nth term is 1/n
which goes to 0 as n goes to infinity
empty moth
i agree
i think my problem is how can it be that if a series is convergent then its limit will be 0 but the bottom says if its limit is 0 we cannot tell if it diverges on converges
thats what I dotn understand
is it because u dont know if it is of the form n =1?
jaunty mantle
because A => B and B => A are different
A : it rained
B : the floor is wet
if it rains, the floor is wet (A => B)
the reverse direction doesn't work
the floor is wet (but someone poured a bucket on the floor) doesn't imply it rained
so A => B is not the same as B => A
BUT: the contrapositive statement does hold
A => B and not B => not A is the same
so here we have, it converges implies the limit goes to 0
and also the limit does not go to 0 implies it does not converge
empty moth
yes
jaunty mantle
i think my problem is how can it be that if a series is convergent then its limi...
this is saying we don't know that B => A is true, but we do know that not B => not A is true
(given that A => B)
empty moth
jaunty mantle
do you still have any confusion?
jaunty mantle
A => B is a statement
empty moth
yes
jaunty mantle
not B => not A is the contrapositive statement to A => B
empty moth
ok
jaunty mantle
a statement is true if and only if its contrapositive is true
(they are equivalent statements)
jaunty mantle
yes
empty moth
So contrapositive is, that if the series doesnt converge then its limit is nto equal to 0
Which is divergence test
empty moth
oh right
jaunty mantle
not B => not A is the **contrapositive** statement to A => B
notice the negation on both the statements and the order swap
jaunty mantle
So contrapositive is, that if the series doesnt converge then its limit is nto e...
btw this statement is obviously false by counterexample of harmonic series as well
it doesn't converge but its limit is 0
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hollow marsh
Hi there, I need help integrating the equation
1/(x^2+1)^2 , Wolfram substitutes x with tan but I feel there has to be an easier way?
tall knoll
That's the standard way of doing it
swift oasis
Hi there, I need help integrating the equation
1/(x^2+1)^2 , Wolfram substitute...
its the easiest way
x=tanu
1+tan^2(u) = sec^2(u)
floral ridge
hollow marsh
Alright then :/, none of the courses i've taken has brought up sec so i guess that's why I was uncomfortable with that answer from wolfram.. Thank you very much!
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terse tendon
Can someone help me approach this math problem?
So, I have $u_{n+1} = \frac{1}{n+1} + \sqrt{u_n}$ a sequence and I know that $u_0 = a$ where $a \in \mathbb{R}^+$
I need to find
$\lim_{n \to \infty} (\frac{u_n - 1 - \frac{2}{n}}{\frac{1}{n}})$
We know that $\lim{n \to \infty} (u_n) = 1$
And that $t_n$ which is a special case of $u_n$ where $t_0 = 4$, has for limit when $n \to \infty$ the limit: $\lim{n \to \infty} (t_n) = 1$ and $\lim_{n \to \infty} (\frac{t_n - 1 - \frac{2}{n}}{\frac{1}{n}}) =0$
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Can someone help me approach this math problem?
So, I have $u_{n+1} = \frac{1}{n...
woven radishBOT
Smile!
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terse tendon
<@&286206848099549185> please?
urban harbor
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<@&286206848099549185> anyone?
bright bone
How can i help
woven radishBOT
ThM
terse tendon
Yes
It always converges to 1
t(n) is a specific case of u(n) were alpha=4 and we studied it in another part of the exercise
But it s not so important there
If that was your question?
terse tendon
Yes
Yes it is always the same but I need the justification of that which you cant just do by saying ”it s always the same” since the proofs we did for tn depended on framing we cant do with un
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fiery saddle
tiny hearth
Both satisfy the same recurrence relation and approach the same limit (1). So both limits are identical (you could even do a substitution to make it explicit)
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terse tendon
Both satisfy the same recurrence relation and approach the same limit (1). So bo...
Yes but we cannot go from a specific case to the general case like that
rose flower
hello
regal moat
z table?
regal moat
I assume this is Statistic?
rose flower
yes
regal moat
viscid phoenix
No population statistics is known, so it should be two-sided t-interval
regal moat
Hmm, lets wait for some Statistic master to rescue
viscid phoenix
Check inference for sample mean
rose flower
Check inference for sample mean
how do i do this
rose flower
that falls under 0.5
viscid phoenix
Tail probability 0.05, assuming infinity (1-0.90=0.1, 0.1/2=0.05)
rose flower
what does 0.5 have to do with teh question though
rose flower
ok that ddoesnt answer my question
viscid phoenix
90% confidence level
There is 5% on each tail
t* is the critical t-value
margin of error=t* times standard deviation divided by sqaure root of n
margin of error=$$t^{*} \times \frac{\sigma}{\sqrt{n}}$$
woven radishBOT
Reading
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wary fractal
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wary fractal
does anyone understand whats going on in the diagram
do we start in the top left, and then t1 is applied, and then the top right is the result?
im so confused
next crown
perhaps the question wants you to find where the transformation equals the matrix, no?
wary fractal
perhaps the question wants you to find where the transformation equals the matri...
then are the pictures useless?
next crown
T1 rotated the shape 90 deg AC so find the matrix which does this with a simpler graph
next crown
then are the pictures useless?
No you need to match them to the matricies
next crown
No worries think you have to close this now
wary fractal
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vernal anvil
anyone can help me with this regression question?
im just not sure what its asking me for; does it want the standard error of b1? b0? not sure
jagged rampart
yo
jagged rampart
im solving this integral
but i dont get why 2x-6 = negative root y
where does that negative come from
literally nowhere
ok i understand some of it now
when u take the root of both sides
you get root y and negative root y
but why do you remove the positive root y?
shy kestrel
given the domain of x in [0, 3], what's the range of 2x-6
jagged rampart
hard to notice first time i've seen this in a question
shy kestrel
just remember to pay more attention when applying an operation that truncates your range
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bronze quail
I need help solving a question on applications of definite integrals
bronze quail
It’s number 33 a)
restive river
are u good at surds
restive river
surds?
bronze quail
yeah
bronze quail
yeah I guess so
restive river
can u help me
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bronze quail
I need help solving a question on applications of definite integrals, it asks to
bronze quail
Number 33. a)
Solving using the cylindrical shell method
which I did try but I can't tell wether or not to integrate wrt y or x and then what happens after that
so its more just understanding those rules that I need help with
when solving questions like these
restive river
which I did try but I can't tell wether or not to integrate wrt y or x and then ...
did you draw a picture?
bronze quail
uh yeah here
Kinda bad and off to the side
But the dotted line is the line of rotation at y = 1
0 to 1 on the y axis
-1 to 1 on the x axis
Our teacher told us to start cylindrical shell questions like this
restive river
If you know the formula is cylindrical shell then I'm guessing you're confused on what r(x) and h(x) should be?
restive river
I honestly forgot how to do these problems.
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desert girder
hello
desert girder
is there someone here in this channel that can help me?
radiant anvil
you
desert girder
alr
left relic
!da2a
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desert girder
in what chat?
exotic aspen
this chat
desert girder
ok
exotic aspen
im a helper
desert girder
wwow
desert girder
nice
A bag contains exactly 50 coins. The coins are either worth 10 cents, 20 cents or 50 cents, and there is at least one of each. The total value of the coins is $10. How many different ways can this occur?
A-2 B-4 C-8 D-12 E-16
<@&286206848099549185>
spare crypt
try finding one way that works (maybe finding the average value per coin), and then if you can find a way to adjust that coin configuration to all the others, you'll have the count
spare crypt
yea, but there's a nice way of moving between them
starting from fifty 20 cent coins, try to find the configuration that has a 50 cent coin and a bunch of 20's, how many 10's do you need to keep the average of 20 cents?
desert girder
hmm
50 20cent=$10
so that means I need to add some 10 and 50 cent coins
so
if I do 10 twenty cent
five 50cent coins
and
spare crypt
eh just try one 50 cent first
desert girder
oh ok
spare crypt
then you'll be able to count them based on how many 50's there are
spare crypt
I don't think that adds up correctly
desert girder
it doesn;t?
spare crypt
the 50 coin is 30 above the average, so you'd need three 10's to keep the average at 20
oh sorry it does add up
just not fifty coins total
so convert some of those 10's into 20's until it's fifty total coins
desert girder
oh yeahh
ok
3 10's 46 20's and 1 50
they add up to $10 and to exactly 50 coins
spare crypt
yea that's good
so there's an exchange of three 10's for one 50, when you start from fifty 20 cent coins
desert girder
whats next?
spare crypt
last step is what is the maximum number of 50's you can have?
spare crypt
little more
desert girder
20
spare crypt
that'd require 60 10's to offset
every 50 comes with three 10's is what you showed before
desert girder
oh wait
ok
so
17?
wait
19?
19 50's 2 20's 1 10
oh I forgot that it doesnt add up to 50 coins
my bad
spare crypt
yea it's 50 total coins, and each 50cent you add comes with three other coins
desert girder
wdym by 'comes with 3 other coins?'
spare crypt
it looks like this
we keep adding three 10's and a 50
but eventually you'll hit the fifty coin maximum, or put another way the 20cent amount reaches 0
spare crypt
yea work with whatever pattern you see
desert girder
3 50's 38 20's and 9 10's
but obv we can do more fifty's
12 50's
39 10's 2 20's and 12 50's
wait a minute
oh mb
wait
spare crypt
yea 12 is good
desert girder
36 10's 2 20's and 12 50's
spare crypt
now you can answer the question
desert girder
oh soo its D
spare crypt
yea I'd have to draw this in a table to get anywhere haha
spare crypt
sure
desert girder
The digits 1 to 9 are used exactly once each to make three 3-digit numbers. The second number is 3 times the first number. The third number is 5 times the first number. What is the second number?
spare crypt
no idea seems like you just have to try things
desert girder
ok...
spare crypt
like the first step is you know where the 5 goes, but everything else is a little jank
desert girder
example 123
spare crypt
you can also tell the first digit of the first number, but after that you more or less just try examples of the first number and check :/
desert girder
123 cant be the second number
wait
there are so many numbers u can create with 9 digits
123
321
231
213
spare crypt
yea but 200 is too big for the first number
spare crypt
it's even worse you can use any digit 1-9, like 198, 345, 782
desert girder
6 combinations for one number
ye ig
its no point doing this way then
wait I got an idea
123 cant be the number because the '1' digit yk
so basically
none of the numbers can have the '1' digit
azure jolt
The digits 1 to 9 are used exactly once each to make three 3-digit numbers. The ...
If the third number is 5 times the first number, the only numbers that those conditions work with for 1-9 is 1 and 5, so you can immediately figure out the 1st and 3rd number
desert girder
If the third number is 5 times the first number, the only numbers that those con...
I still dont get it
azure jolt
u need to use all digits
oh wait nvm im wrong then
desert girder
yea
it says 'the digits 1 to 9 are used exactly once each to make three 3-digit numbers'
once
spare crypt
you can figure out these two red slots, but the rest is a lot of guessing
desert girder
umm
biggest number is 987 and lowest is 123
hmmm
I got an idea
I'll check what numbers are divisible by 5
obv we need the 5 digit
so we need to make the biggest number possible with the 5 digit
I got
955
oh wait u can only use a digit once
BRU
exotic aspen
no idea seems like you just have to try things
U DUMBASS
im joking
i love u
hey plurmorant
@spare crypt
can we be friends
i can teach u if u want
spare crypt
how can I say no
spare crypt
do you know what 5x has to end with?
desert girder
um
spare crypt
and can the first number be greater than 200?
desert girder
nope
spare crypt
other than that it's brute-forcey
multiples of 5 are {5,10,15,20,25,30,35,40,45,...}
the last digit is an easy pattern
desert girder
bru
spare crypt
idk if I can even explain anything else in it, here's the page-long answer https://www.quora.com/The-digits-1-to-9-are-used-exactly-once-each-to-make-three-3-digit-numbers-The-second-number-is-three-times-the-first-number-The-third-number-is-five-times-the-first-number-What-is-the-second-number
spare crypt
you mostly just plug in like 30 numbers once you know x starts with 1 and 5x ends in 5
just a random calculator question idk why it exists
exotic aspen
not year 8
desert girder
7?
exotic aspen
prob year 9
desert girder
@exotic aspen teach us your way of solving
exotic aspen
im like terence tao
desert girder
nice
exotic aspen
it's hard to teach a baby to make a car in 1min
spare crypt
uh this one is just writing out x, 3x, 5x for {123,127,...197}
desert girder
uh this one is just writing out x, 3x, 5x for {123,127,...197}
why does the last digit need to be an odd number?
desert girder
oh yeah
spare crypt
but then you just spam x 3x 5x into a calculator till you see 192 works idk
desert girder
idk
desert girder
ill try my best
waiting for this mathematician
desert girder
yea
I get it now
u gotta find the lowest possible number
for the first number
which is 123
then as u said
123,125,127,129
desert girder
o hok
alr
u just gotta multiply 3 for each number and make sure the digits are different
123 x 3= 369
cant use the digit 3 again so it cant be 123
desert girder
I left out 125 because 5 was already in 5x
cant be 125 cuz of that
same for 127 and 129
not 129
there we go we found the first number 129
desert girder
then 129 x 3
exotic aspen
it must be odd since the numbers are from 1-9
desert girder
=387\
exotic aspen
it cant be 0
exotic aspen
bc 0 doesnt exist within these parameters
exotic aspen
yes
spare crypt
I mean I see the answer but I hate the question
desert girder
ikr
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desert girder
@spare crypt do u have any amc year 9 test papers
amc-australian maths comp
<@&286206848099549185>
does anyone have?
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quasi creek
I count 20
Nvm ignore what I said
It's prob more
I'm sleep deprived. Good night sry
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naive saddle
Whats the smallest n bigger than 1 such that n!>=10^(n-1)?
devout snowBOT
Whats the smallest n bigger than 1 such that n!>=10^(n-1)?
weak cove
bruh just compute them
naive saddle
Fr?
naive saddle
I have to like, actually compute
No quick trick?
Like, I know it has to be either 22! or 23!
But idk which one
weak cove
how do you know that sir
naive saddle
I did some uh, calculations
weak cove
it's too late sorry
naive saddle
What?
weak cove
just kidding
weak cove
No I calculated with some rough estimations
okay so if you've narrowed it down (I'm assuming accuretely (?)) to 22 or 23
just compute n=22
and then if it works
ur done
if it doesn't
ur done and its 23
naive saddle
just compute n=22
How 💀
weak cove
22!
naive saddle
What the hell how am I supposed to compute 22!
weak cove
22*21....*2*1
naive saddle
There gotta be some prettier solution
weak cove
well
yes
once you start trying to do it actuall
you can see like
22=2*11 ~ 2*10
likewise for 21, 20
pick up some extra factor of 2,
do some other shennanigans
find the amount of powers of 10
there are in 22!
by doing this
and if there is idk say 21 of them
then you're done
and if there isn't
then n=22 is not true
Like, it's not easy to explain but once you start trying to work out 22! just by this factoring (not finding the real value) to get it into scientific notation form
it will be easy to do that
naive saddle
Ok I'll try
2^19 x 3^9 x 5^4 x 7^3 x 11 x 13 x 17 x 19
Ok so, 7^2 is basically 10x5
11 is just 10
weak cove
11=10 is a well known fact yes
weak cove
agreed
naive saddle
So 3^8=10000
2^9 is just 500
19x5 is just 100
3x7x5=100
25x13x17 is just 5000
Ye I think this will do
10x10x1000x10000x100x100x100x1000x125
1250000000000000000000
Uh, this doesnt seem to be right
Wait no this is
This has 22 digits
Oh wow I didn't know you can do this
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solid osprey
$$m+n+\frac{n}{m}=343$$
$$\frac{n(m+n)}{m}=2022$$
find the maximum value of $m+n$
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$$m+n+\frac{n}{m}=343$$
$$\frac{n(m+n)}{m}=2022$$
find the maximum value of $m+n...
woven radishBOT
Skill_Issue
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fallen harness
is the x column just the numbers?
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is the x column just the numbers?
brittle burrow
yes x is the scores
fallen harness
and then there’s just a single value for the second and third column?
fallen harness
average
brittle burrow
and then there’s just a single value for the second and third column?
so why would they all be the same value for x-x bar?
brittle burrow
once should be fine
what does your teacher do
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echo cedar
hey
echo cedar
i need help in linear algebra with geometrically describing solutions to systems of linear equations. given this system in the screenshot, I got:
[x,y,z,w] = δ [1, -5, 3, 0] + λ [-4, 2, 0, 3]
as the general solution
I dont really know how to think about describing any solution, and especially as its in 4 dimensions i dont know how to start
appreciate it
fiery moth
is the x column just the numbers?
dr du has live homework help you know
fallen harness
dr du has live homework help you know
i know but don’t talk here
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echo cedar
<@&286206848099549185>
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mortal sable
Hi @echo cedar
Are you familiar with the dimension of a vector space
And basis?
Take a plane in 3 dimensions that goes through the origin. This is a 2D vector space in 3D.
You may assign the plane a basis, which must consist of two 3D vectors (because it is a 2D subspace of 3D space)
An example for you to work out is to construct a basis for the plane defined by the equation z=x+2y
Going back to the problem, your solution is the span of two vectors in 4D.
Therefore, what is the dimension of this subspace?
And what shape is it geometrically
echo cedar
Hi <@328864217287622657>
hi thanks so much for responding
i dont think i have a good conceptual idea of the dimension of a vector space
but i do think i understand basis
echo cedar
Therefore, what is the dimension of this subspace?
as my solution is the span of 2 vectors, does that mean its simply a 2D vector space in 4D? so also a plane going through the origin
echo cedar
An example for you to work out is to construct a basis for the plane defined by ...
for this i got [x,y,z] = δ [1,0,1] + λ [0,1,2]
echo cedar
as my solution is the span of 2 vectors, does that mean its simply a 2D vector s...
so if this is correct, would a solution with 3 vectors then be a 3D vector space in 4D? a hyperplane going through the origin?
and then span of 4 vectors be the whole 4D space itself
mortal sable
as my solution is the span of 2 vectors, does that mean its simply a 2D vector s...
💥 yes exactly
All of that is true
mortal sable
Yes
echo cedar
so if i have a solution of span of 5 vectors in 100D then i have a 5D vector space in 100D where its a 5D hyperplane going through the origin
mortal sable
The definition of hyperplane is different, but a 5D space, yes
Hyperplane is a space of n-1 dimension in an n dimensional vector space
echo cedar
ah ok thats my bad
but cool thats good to know
alright thanks so much for your help
mortal sable
Of course!
mortal sable
if you're done with the channel make sure to .close
echo cedar
if you're done with the channel make sure to .close
Ah yep my bad sorry just saw
.close
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rotund lotus
I need help confirming the answer to a question
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rotund lotus
Let a bowl contain 15 chips of the same size and shape. Only one of those chips is red. Continue to draw chips from the bowl, one at a time at random and without replacement, unitil the red chip is drawn.
a) Find the probability mass function of X, the number of trials needed to draw the red chip.
b) Compute the mean and variance of X.
c) Determine P(X > 10).
graceful cosmos
rotund lotus
i essentially put the pmf into excel with the x values of 1-15 and subtracted the sum from P(X=1,2,3,..,10) from 1
the main thing that i was hestitant on was whether i was correct in using the probability mass function of the geometric distribution
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cunning roost
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cunning roost
howd they find domain?
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regal moat
howd they find domain?
Domain restrict the variable from making the function undefined
cunning roost
Domain restrict the variable from making the function undefined
Oh but
Whyd they do 2pi r = 8
Instead of
4x = 8
regal moat
Like the wire is only 8 meter long
cunning roost
Oh ok
green edge
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green edge
can someone explain how this was rounded up like this?
am i wrong? it's supposed to be 0.79?
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atomic vapor
Hi
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icy barn
show ur work
humble ferry
calculus
humble ferry
wicked turtle
what is your doubt?
humble ferry
idk how to do question 2
and can u review my ans for q1 see see if it looks right
2(b)
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sick cradle
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tender cobalt
- move the - 1 to be a + 1 on the other side
- multiply both sides by s
- divide both sides by (s(s - 4))
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sick cradle
1. move the - 1 to be a + 1 on the other side
2. multiply both sides by s
3. div...
But where does the extra 1/s-4 comes from?
tender cobalt
- multiply 1/s by s
- divide 1/s by (s(s - 4))
also thats not an "extra" 1/(s-4)
before there were three terms: (s - 4)Y(s), -1, and 1/s
after there are still three terms: Y(s), 1/(s - 4), and 1/(s(s - 4))
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jaunty mantle
I may not know enough background, but what sort of thing is supposed to be in the 2nd slot of Gamma?
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I may not know enough background, but what sort of thing is supposed to be in th...
jaunty mantle
Is it similar to TM but instead of attaching to each point in M a tangent space you instead attach a tensor space to it?
Which would also mean for my initial question that you need to stick some sort of tensor space in general, and the TM is just a special case of Gamma (M, V)
wooden veldt
Yeah TM is a special case of a vector bundle
uncut crow
why does this sound like a dn joke
jaunty mantle
A bundle of deez nuts
jaunty mantle
Yeah TM is a special case of a vector bundle
When you say this, you mean that Gamma(M, • ) just takes some vector space in •?
wooden veldt
The dot would be replaced with a vector bundle, the gamma is denoting the space of sections
jaunty mantle
Oh by a vector bundle
Do you write a vector field over M to be an element of Gamma(M, TM)
No that would be too specific
wooden veldt
Yeah or just Gamma(TM)
jaunty mantle
Yeah or just Gamma(TM)
See this I’ve seen but the extra entry is confusing me
wooden veldt
It's bc you already have the M from TM so the extra M is redundant
jaunty mantle
I may not know enough background, but what sort of thing is supposed to be in th...
So this 2nd slot can have absolutely nothing to do with the tangent bundle
wooden veldt
But in general the bundle doesn't need to be related to M specifically so you wanna specify you're considering a bundle on M
jaunty mantle
Right
So you just take a smooth manifold
And then attach to every point some tensor space
wooden veldt
Like you can take a manifold and your vector bundle can just be attaching R^n at every point
jaunty mantle
So a tensor field could be very useless
If you don’t define what your vector bundle is saying
Or what information it contains
wooden veldt
Yeah like you can literally take any vector bundle and create a tensor field out of it
wooden veldt
But presumably something like Gamma(TM^p tensor (TM)*^p) is something useful
Where the powers are tensors
jaunty mantle
Yeah it seems
wooden veldt
We dropped the first M again bc it's already there
jaunty mantle
Right
I’m looking at differentials and it’s stated to just be a totally anti symmetric (0,n)-tensor field
So that field can have absolutely nothing to do with M
wooden veldt
Hmm
My understanding of differentials is that the V in your definition is the cotangent space
jaunty mantle
Oh I lied
wooden veldt
Differentials eat tangent vectors
jaunty mantle
It clarifies that it takes arguments from Gamma(TM)
wooden veldt
Yeah
jaunty mantle
Ok so it’s forced to be something that is connected to the M
And in particular the tangent bundle of M
wooden veldt
Yeah they're just giving you a general defn of tensor field bc diff forms are a particular case
jaunty mantle
or do I say in particular the vector bundle of tangent spaces of M
Or is that the same thing
wooden veldt
Need a dual somewhere but yeah basically
jaunty mantle
Yeah they're just giving you a general defn of tensor field bc diff forms are a ...
Yeah right that picture I had was just from Wikipedia on tensor fields
Hmm
jaunty mantle
It clarifies that it takes arguments from Gamma(TM)
This needs to say Gamma(T*M)?
wooden veldt
I've literally started learning this stuff last week lol I recommend a book by Morita "geometry of differential forms"
wooden veldt
This needs to say Gamma(T*M)?
Oh no sorry you're talking about the arguments it takes than yeah what you said is fine
jaunty mantle
Well presumably a (0,n)-tensor field only has space to eat n vectors aka things in Gamma(TM)
wooden veldt
ω \in Γ(Alt^n(T*M))
jaunty mantle
Alt?
jaunty mantle
Ah
Something something this is a determinant
Soon I’ll figure out what’s supposed to be there instead of something something
🙏
wooden veldt
Lol yeah I think actual computations of them result in dets
jaunty mantle
It states that the top form is the determinant of a map or something
wooden veldt
Oh Michael penn has a playlist on this stuff too
jaunty mantle
I’m thinking of it as if I slapped the column vectors of my matrix in as the arguments for the (0,n)-thing I’d get the determinant
Ok anyway reaching too much now
wooden veldt
It's not quite that idt but you'll see
You form a matrix of certain elements from a list of n vectors from the tangent space
The matrix will be nxn and you take det of that
jaunty mantle
It’s quite nice to have a not laplace expansion definition of the determinant
wooden veldt
Do you have the sum over perms defn
jaunty mantle
No
jaunty mantle
It’s quite nice to have a not laplace expansion definition of the determinant
I just want to delete this from my brain so the cross product can’t terrorise me
jaunty mantle
For determinant?
wooden veldt
Yeah
wooden veldt
That is laplace expansion no?
wooden veldt
It’s quite nice to have a not laplace expansion definition of the determinant
Did I misinterpret this msg
I read this as you now have a diff defn
jaunty mantle
Did I misinterpret this msg
Oh, the top form thing