haha aight
#help-10
restive jungle
Let ABCD be a quadrilateral such that A, B, C, D are concyclic. Suppose AB = 11, BC = 12, CD = 6 and DA = 8. If the extensions of rays BA and CD meet at P, find the length of PA.
obtuse pebbleBOT
Let ABCD be a quadrilateral such that A, B, C, D are concyclic. Suppose AB = 11,...
restive jungle
i only know PAxPB = PDxPC
dont know hiw to make any more steps
just found out PAD and CPB are similar
crude ice
restive jungle
use another channel
$Let ABCD be a quadrilateral such that A, B, C, D are concyclic. Suppose AB = 11, BC = 12, CD = 6 and DA = 8. If the extensions of rays BA and CD meet at P, find the length of PA.$
warm shaleBOT
candies
restive jungle
what
obtuse pebbleBOT
@restive jungle Has your question been resolved?
restive jungle
yes i got it
civic bone
what's the best way to solve for a variable thats inside of a fraction in an exponent
obtuse pebbleBOT
what's the best way to solve for a variable thats inside of a fraction in an exp...
grizzled shore
Logarithms!
civic bone
like $2^{\frac{3 - 2x}{2 - x}} = 3$
warm shaleBOT
nchoosek
grizzled shore
Logarithms!
civic bone
right but then im stuck with a $\log_{2} 3$
warm shaleBOT
nchoosek
grizzled shore
That’s the same as being stuck with a +5
civic bone
what I'm really trying to solve is this
and I set x = log_4n... = log_3n...
then got x down to that form up there
but if I solve for x there im at a dead end
so did I just reach a dead end in general?
before I put myself there I also got $2^{3 - 2x} = 3^{2 - x} = n$
warm shaleBOT
nchoosek
grizzled shore
There’s no dead end
grizzled shore
**nchoosek**
From here you have $\frac{3 - 2x}{2 - x} = \log_2 3$
warm shaleBOT
Frosst
grizzled shore
Now it’s just a quadratic in x
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
main moth
How do you do this question it’s pretty confusing
obtuse pebbleBOT
How do you do this question it’s pretty confusing
@main moth Has your question been resolved?
obtuse pebbleBOT
@main moth Has your question been resolved?
slate hamlet
How do you do this question it’s pretty confusing
Have you tried plotting it?
main moth
How do you do that ?
slate hamlet
On the graph you have been given
Treat x axis as the East and west
and y axis as north and south
and origin as Point S
Can you plot it now?
obtuse pebbleBOT
@main moth Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
deft ginkgo
fellow nerds, i come to thee with a simple question
obtuse pebbleBOT
fellow nerds, i come to thee with a simple question
deft ginkgo
does i^2 and -i*i both equal -1?
-i*i comes out to -i^2 which in my brain comes out to 1
but it doesnt?
deft ginkgo
what about (-i)^2?
wet moss
That would be -1
hollow dove
for the latter, you can expand it out more as (-1 * i)^2
royal basin
-i * i = -(i * i) = -i^2 = -(-1) = 1
hollow dove
that might be more helpf:l
royal basin
@deft ginkgo are you getting somebody telling you that -i*i is not 1?
deft ginkgo
no no, but I just started dealing with i 2 days ago and its a bit confusing at times
your expanded equation for it makes sense though
so I can treat -ii as -1i*i
grizzled shore
put `` around your text
deft ginkgo
so I get -1*-1 in the end
grizzled shore
`like this`
deft ginkgo
-i*i=-1*i*i=-1*-1
grizzled shore
put one at the end as well
dreamy gale
obtuse pebbleBOT
dreamy gale
Subject: Multivariable Calculus
I honestly dont understand this, how is the flux of A have a value of 2Pi if the divergence is 0? I'd just assume that B is just 6Pi then
.close
silent spruce
obtuse pebbleBOT
silent spruce
frank monolith
start by $\log{a} - \log{b} = \log{\frac{a}{b}}$
warm shaleBOT
Dissrupt
obtuse pebbleBOT
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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
1
rich plume
**Dissrupt**
what do you not understand here?
silent spruce
i want the answer to this one
silent spruce
what do you not understand here?
i understood that
rich plume
Okay
so at what value of x do you think y is equal to zero
(talking abt the red line)
obtuse pebbleBOT
@silent spruce Has your question been resolved?
viral tree
obtuse pebbleBOT
warm shaleBOT
Dissrupt
viral tree
what?
warm shaleBOT
Dissrupt
viral tree
1 3/4 is 7/4
frank monolith
yeah
viral tree
can any one point for me where I was wrong here
frank monolith
its $x^2 - 2 = \frac{1}{4}$
viral tree
I forgot to turn the 2 to positive
warm shaleBOT
Dissrupt
viral tree
so final answer is x^2 = square root 9/4
x = 3/2 , or x = -3/2
I just forgot to turn minus 2 to plus 2 then
thanks
frank monolith
np
viral tree
.close
frozen quest
can somone explaiin this one?
obtuse pebbleBOT
can somone explaiin this one?
frozen quest
can't understand what the language is trying to say 😐
timid silo
Twelve men play basketball, five of them at a time. The game is one hour long and [if] each man plays the same amount of time, how many minutes does each of them play?
frozen quest
Twelve men play basketball, five of them at a time. The game is one hour long an...
uh 5 of them play at a time so in 1 game 5 plays so how many games will they play?
very confusing
😦
even if we go by options it is still very confusing for me
i can't really comprehend this question
like if we take the answer 25 min and multiply it by 12 we get 300 min but 1 game was just 60 minute
and also if 5 play in a game so it is possible to get repeated person as the persons are 12
i hope you getting what i am trying to say 
timid silo
Yeah, the wording of the question doesn't help as well.
frozen quest
in the explanation of answer they did like 5*60/12 = 25min
and i'm like so confused 
timid silo
Oh, I get it now. I was thinking way farther than I should.
frozen quest
Oh, I get it now. I was thinking way farther than I should.
can you explain
timid silo
can you explain
We have twelve men, and the game requires five people on the field at any given time, and the game lasts for 60 minutes. Let's say each one of the twelve will play for $x$ minutes. We have $5\cdot60$ as our total time. Imagine that at each minute in every match, five "player-minutes" are used. Then, we have $12x=300$ (which is $5\cdot60$). Dividing both sides by $12$ we get the time-per-man.
warm shaleBOT
3317
frozen quest
how is total time 5*60?
5 players play at a time in a team for 60 minutes
i know that but cant get how is 5*60 the total time available
timid silo
I guess that's the intended answer for simplicity's sake.
In the end, if we take into consideration that any man (or men) can be repeated in each of the five men teams, it's undecidable, as I see it.
The number of minutes in game wouldn't be the same for all of them.
After all, they could play the same group of five indefinitely.
frozen quest
In the end, if we take into consideration that any man (or men) can be repeated ...
oh i see
frozen quest
I guess that's the intended answer for simplicity's sake.
ok thanks bro
timid silo
Glad to help.
frozen quest
.close
wraith reef
this is right? can i do this? (inside the blue rectangle)
obtuse pebbleBOT
this is right? can i do this? (inside the blue rectangle)
tardy epoch
,w plot -log(1-x^2)/x^2 for -1 < x < 1
warm shaleBOT
tardy epoch
yes
obtuse pebbleBOT
Available help channel!
glacial wolf
obtuse pebbleBOT
hard igloo
Solve the quadratics in the exponents then write 6^a=2^a×3^a
glacial wolf
tried that
glacial wolf
2^2a+1 + 6^a >= 3^2a+1
hard igloo
=
glacial wolf
2y^2 + yz >= 3z^2?
hard igloo
Now substract and then factor
glacial wolf
u mean send 3z^s to other side?
hard igloo
Yeap
glacial wolf
2y^2 + yz - 3z^2 >= 0
hard igloo
Yes
glacial wolf
now how?
hard igloo
(y-z)×(2y+3z)>=0
glacial wolf
now should i substitute
hard igloo
You have to find when its equals zero
glacial wolf
either y = z
hard igloo
Yes
hard igloo
2^a-3^a= 0 when a =0
hard igloo
Hmm you can do that but i dont know what to do with y=3z/2
2^a+1+3^a+1 is always bigger then 0
So we can ignore it
glacial wolf
i am getting x = (5+-(21^1/2))/2 when a=0
glacial wolf
2^a+1+3^a+1 is always bigger then 0
maybe we can equate a = -1
hard igloo
No
glacial wolf
then it will become 1-1
hard igloo
i am getting x = (5+-(21^1/2))/2 when a=0
At this value the equation =0 we should see what happens at a bigger and a smaller value
glacial wolf
oh sry a = 1
hard igloo
Or sory thaths 2 value
At both of them the equation is 0
Test a value between them to see wheter its positive or negative
Thats working because every quadratic equation has an upwards parabola
And at those two value does its function intercepts the x axis
glacial wolf
when z = -2y/3??
hard igloo
Yes
glacial wolf
so we need to equate right part to 0
glacial wolf
so a = 0
hard igloo
Yes
glacial wolf
then we found values of x for that
glacial wolf
i am getting x = (5+-(21^1/2))/2 when a=0
.
hard igloo
Yes two values, do you understand until here?
glacial wolf
ya
hard igloo
Do you know what graph a quadratic equation has?
hard igloo
Yes it can be upwords or downwords right?
hard igloo
What is ve
hard igloo
Do you mean it has an x-axis intercept at a negative x value? Or that there is an x solution with + and - sqrt21?
glacial wolf
so we have to take the negative root and positive root and all the values above positive root?
glacial wolf
Do you mean it has an x-axis intercept at a negative x value? Or that there is a...
there are two x solutions
glacial wolf
is that the solution?
hard igloo
No
glacial wolf
so we have to take the negative root and positive root and all the values above ...
..
hard igloo
But roots are positive
glacial wolf
ohya
hard igloo
So now we dont know wheter we have an upwords or downwords parabila
Thats why we test a value between x1 and x2 (these are the 2 roots
If thats positive now we know that x1=<x=<2
And if negative
x=<x1 and x>=x2
glacial wolf
they are in between
glacial wolf
(1/10)^2 - 5*1/10 + 1 = 51/100
hard igloo
To check an outside value
glacial wolf
so inside it is -ve?
hard igloo
(1/10)^2 - 5*1/10 + 1 = 51/100
What is this
glacial wolf
because we are getting outside +ve
hard igloo
But not in this
glacial wolf
in which one?
glacial wolf
so answer is everything excluding the numbers in b/w roots
glacial wolf
k
hard igloo
Wich is between them and it will be negative
glacial wolf
-5
hard igloo
Yes so between them is not part of the solution as you figured it out before
glacial wolf
so we got the solution?
hard igloo
Yes
hard igloo
Hungary
hard igloo
Nice
glacial wolf
kbyee then
hard igloo
Bye
hard igloo
.close
timid silo
Need help in Linear Equations with one variable
obtuse pebbleBOT
Need help in Linear Equations with one variable
timid silo
I don’t know what to do after this
Usually when something is added/subtracted to the number with the variable
I’d know what to do
versed stratus
Hint:- Bring the Ts to one side and the constnats to the other
nocturne minnow
Move terms with variables to one side, everything else to the other
timid silo
So 18t-16t = 18-12?
nocturne minnow
Yeah
nocturne minnow
Then maybe it wants you to do one term at a time (really stupid)
So don't move both 18t and 12 at the same time, do one then the next in separate steps
timid silo
I do not understand
nocturne minnow
12 - 16t = 18 - 18t to 18t-16t = 18-12
You moved both 18t and 12 at the same time, correct?
timid silo
Yes
nocturne minnow
I'm saying do one at a time, either move 18t first then 12 or vice versa but don't move both at the same time
nocturne minnow
So 18t-16t and keep the other side same?
No
12 - 16t = 18 - 18t
Moving just 18t you get 12 - 16t + 18t = 18
timid silo
Why do we move 18t
nocturne minnow
I'm saying do one at a time, either move 18t first then 12 or vice versa but don't move both at the same time
12 - 16t = 18 - 18t
do one at a time, either move 18t first then 12 or vice versa but don't move both at the same time
timid silo
Yeah but what does that do
nocturne minnow
Because
Move terms with variables to one side, everything else to the other
You're doing that
But that site might be requiring you to do one step at a time
timid silo
I’m saying what does moving 18t to the left side do to help us get the answer
nocturne minnow
The goal is this
Move terms with variables to one side, everything else to the other
You did
12 - 16t = 18 - 18t to 18t-16t = 18-12
Moving both 18t and 12 at the same time
So instead of moving both at the same time, I'm saying do one at a time
Meaning move the 18t first
nocturne minnow
12 - 16t = 18 - 18t
Moving just 18t you get 12 - 16t + 18t = 18
Yes that's what I showed here
timid silo
So now what?
nocturne minnow
Try it
timid silo
nocturne minnow
That's -16t + 18t
nocturne minnow
Try it
timid silo
I did it works
nocturne minnow
12 + 2t = 18
Move the 12 now
timid silo
2t = 18 + 12?
timid silo
How do you get the 12 on the left the right?
When we change sides we change the symbol too?
So it becomes minus?
2t = 18 - 12?
nocturne minnow
Yes
timid silo
Oh
nocturne minnow
I didn’t really understand why we move but hold up can we do one more problem
It's how you solve the equation
Move terms with variables to one side, everything else to the other
Recall that statement
timid silo
It's how you solve the equation
nocturne minnow
That's all you're doing
nocturne minnow
But you are literally doing one step at a time
nocturne minnow
Solve for x, what's the next step?
timid silo
Get the 6x and the 12x on one side?
At least that’s what we want to do
But the system might not let us
nocturne minnow
The 6x is already on one of the sides
timid silo
So bring the 12x there
nocturne minnow
Yes
nocturne minnow
Yes
Then you simplify
Just repeat what you did with the last problem
That's all you're doing
timid silo
Wait I don’t understand one thing
What about the other side?
6x + 12x
On the other side though
Does it become 15 + 12 or 15 - 12
timid silo
https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:solve-equations-inequ...
Okay thanks
.close
green sinew
matrix eigenvaslue question
obtuse pebbleBOT
matrix eigenvaslue question
green sinew
can we use matrix A= [1,-1,-2;0,m,0;0,-1,p] instead
and lambda1=1, lambda2=2, lambda3=2
fathom flicker
the sum of the eigenvalues equals the trace
the product of the eigenvalues equals the determinant
try that
sage geode
That matrix is different from the one presented, why do you think replacing A with that will help?
green sinew
reason is i did this question already with the A in the picture, i wanted to try it with a diff matrix
and diff lambda
sage geode
Ah, I misunderstood then, yeah sure you can try
green sinew
could i use characteristic polynomial to solve for m and p
sage geode
Yes, and what Austin suggested is also note-worthy
green sinew
ok for characteritic plnml i got this
P(λ)=−(−λ+m)(−λ+p)(λ−1)
then for p and m i got 2
m=2,p=2
@sage geode
@fathom flicker
is this right
<@&286206848099549185>
obtuse pebbleBOT
@green sinew Has your question been resolved?
woven wharf
i can do Sp-Sq and Sq-Sp it doesnt matter right?
obtuse pebbleBOT
i can do Sp-Sq and Sq-Sp it doesnt matter right?
frosty river
p+q=q+p
woven wharf
yea
but if i am doing p+q
then would i have to do p-q
or anything
because my answer is different
woven wharf
p+q=q+p
to find the difference i would definitely have to do Sp-Sq
the question is is vice versa applicable
no?
frosty river
Bro, the exercise could be stated as:
If the sun of q terms is p, and the sum of p terms is q, then the sum of q+p is...
The exercise is completely symetric
frosty river
No problem, p works the same as q, they're interchangable
frank monolith
show what you did
woven wharf
ok wait
its on my notebook, gimme 1 min
Here in my common difference there is no negative sign
warm shaleBOT
woven wharf
But the one they have done, does have negative sign
Ans the answer should not be different
woven wharf
Yes
frosty river
And the answer must be symmetric for p and q so I guess it's a sum, I'm thinking aloud, didn't do any cacluations
woven wharf
its obvious that q-p is not equal to p-q
frosty river
Exactly
woven wharf
but to find the difference i can do the vice versa
the only difference would be of negative sign
no i mean
frosty river
For the difference, you have to suppose p>q or q>p, innit?
woven wharf
according to the question the difference would be same but with negative sign
woven wharf
For the difference, you have to suppose p>q or q>p, innit?
its not given
frosty river
Do you mean the differences p-q and q-p are opposite?
woven wharf
yes pls
frank monolith
it was q^2 - p^2
frosty river
The thing is, there must be a p+q, not a p-q
frank monolith
which is (q-p)(q+p)
woven wharf
where
woven wharf
ohhh
frosty river
That creates a negative sign
frank monolith
yup
frank monolith
btw be aware of the site you're using
frank monolith
its known to have wrong answers
woven wharf
its known to have wrong answers
the topper one?
frank monolith
yup
frosty river
so i can do vice versa in this question
Obviously, because the exercise is symmetric in p and q
woven wharf
how did u know its topper.com
frank monolith
toppr*
frosty river
true but irrelevant + misleading + confusing
It was the core of his question.
frank monolith
i just know
obtuse pebbleBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
warm shaleBOT
thin hare
use both of these in tandem
warm shaleBOT
thin hare
✅
wild dock
hidden compass
Read #❓how-to-get-help @wild dock
And go on another channel among the available ones
thin hare
try showing that if f(2) < 2, then f(x+5) = f(x) + 5 doesnt hold
so we know this is a fact right
so we can use this to show f(6) = f(1) + 5 = 6
now i'll assume for the sake of contradiction that f(2) != 2 and use that to show f(6) != 6
and clearly that means f(2) must be 2
with me so far?
oh thats "not equal"
warm shaleBOT
thin hare
just use the f(x+1) <= f(x) + 1
over and over
clearly f(2) is not greater than 2 since f(2) <= f(1) + 1 = 2
so assume f(2) < 2
that means f(3) <= f(2) + 1 < 3
and keep going
yea
✅
npnp
clear oar
obtuse pebbleBOT
clear oar
So i ahd someone explain this yesterday
but im still a little confused
as to how we got d ≅ 0 mod n
i know now
that d is the same between each p value
and that I should try and prove the contrdiction
obtuse pebbleBOT
@clear oar Has your question been resolved?
zenith raft
as to how we got d ≅ 0 mod n
isn’t that just what it’s asking you to prove lol
where was it explained to you before?
clear oar
uh in this channel actually
clear oar
isn’t that just what it’s asking you to prove lol
yeah but we got to k*d mod n gives remainders
where k is between 0 and n-1
and that it spans all possible reminders
cycles through them whatever
but then we got to something about -p1
and i got hellla confused
zenith raft
i was expecting you to link the previous conversation
zenith raft
that’s too much reading for me to catch up on sorry
clear oar
.close
vital pawn
I am stuck on another discrete math problem. The problem represents F(x,y): x can fool y. We need to translate the statement "There is exactly one person whom everybody can fool"
obtuse pebbleBOT
I am stuck on another discrete math problem. The problem represents F(x,y): x ca...
forest sinew
using quantifiers, yea?
vital pawn
Yep. With the domain of all people
I think the answer to the following statement is: Ey ∀x F(x,y)
But, I don't think that's correct based on the answer key
coarse ibex
That says that there exists a person, but not that there's exactly one person (meaning, it doesn't rule out the possibility of multiple such people)
forest sinew
That says that there exists a person, but not that there's exactly one person (m...
yea
vital pawn
I found this online, so I don't know of the accuracy (self-studying)
forest sinew
im curious how they intend for you to go about
wait you said theres an answer key?
is that this?
I should say I found discussion here https://math.stackexchange.com/questions/394609/write-there-is-exactly-1-person-without-the-uniqueness-quantifier
vital pawn
I should say I found discussion here https://math.stackexchange.com/questions/39...
Let me read this. I haven't seen this
vital pawn
Hm... Can you explain it. I have read the explanation in the forum about 3-4 times and I am still getting tripped up on the second part that explains "exactly one"
vital pawn
All good. No rush!
forest sinew
I think the answer to the following statement is: Ey ∀x F(x,y)
we definitely agree on this part
for every person, we can find a person, who they
uhh
vital pawn
Yep
forest sinew
in yours was fool
now the other part that we need is this
well, maybe you can finish this implication we want to be true
eh
If, for some person, we can find a person whom they can fool, it is the same person
trying to think of the best way to say it 
its like, we scan over everybody
everybody who can do the fooling
and everybody who can be fooled
say everybody who can do the fooling is w
everybody who can be fooled is z
as we scan through everyone, and put the pairs one after the other into F, if w ~ z, then z better be that same person
does that help at all 
morrow is typing 
think like
"everybody gave money to dan, and no one else got any money"
coarse ibex
Might help to write $P(y) = \forall x\ F(x,y)$ to simplify things
warm shaleBOT
Morrow
forest sinew
one way to do this is to make sure that everyone gave money to someone
then go around and ask every pair of people if they exchanged money
if they said yes, the reciever better be dan
coarse ibex
P(y) says that everyone can fool y. We need to express the idea that there is a unique y such that P(y) is true.
vital pawn
Oh... I think that makes sense.
coarse ibex
What you wrote above is $\exists y P(y)$.
warm shaleBOT
Morrow
coarse ibex
But we need to add more to express that there is no other person that satisfies P.
vital pawn
So, we would add ∀w (P(w) -> x=w)?
coarse ibex
y=w, but yeah
vital pawn
Yeah. That's what I meant
So, essentially first we are checking to see one person who everyone can fool and then we are saying if there is anyone else, they are actually just the first person that everyone can fool
coarse ibex
Yep
there is a person y who satisfies P, and anyone else who satisfies P is actually just y
vital pawn
Okay. That makes sense now.
And just to confirm that's the same thing as $\exists !y P(y)$
warm shaleBOT
Salaga
vital pawn
when we use the uniqueness quantifier...
coarse ibex
Yeah
I saw in that SE thread there's another nice way of doing it
Which is $\exists y \forall x(P(x) \iff x = y)$
warm shaleBOT
Morrow
coarse ibex
Which says that the people who satisfy P are exactly just y.
vital pawn
yeah. I think this is a little more intuitive to understand.
This makes a lot more sense. Thanks a lot for the help morrow and jan Niku. I really appreciate the time and help!
.close
frosty fractal
Quick question, if I have a function f(x,y,z) = 4xy+z
obtuse pebbleBOT
Quick question, if I have a function f(x,y,z) = 4xy+z
frosty fractal
Would the derivative vector of this function i.e. gradient be <4y, 4x, 1>?
random rose
Yes it would
obtuse pebbleBOT
urban patrol
hallow hatch
oh im sorry i thought it was open
obtuse pebbleBOT
Available help channel!
hazy vine
obtuse pebbleBOT
urban patrol
how are there two variables without two integrals
slim cove
Can you post the full screenshot from the book?
urban patrol
nvm thats prob just a me being bad thing
slim cove
It's probably a typo
hazy vine
ok so then the answer must be wrong too right?
slim cove
Most likely yes you'll have to change it a bit
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
viral tree
obtuse pebbleBOT
viral tree
Can some one point me where I'm wrong here
barren yew
i have to use the numbers 3, 5, 9, 11, 23, and 25 to get a total of 351, there are no other restrictions besides that but im sure that itll be simple taking its only extra credit
viral tree
what?
tardy epoch
i have to use the numbers 3, 5, 9, 11, 23, and 25 to get a total of 351, there a...
!help
obtuse pebbleBOT
i have to use the numbers 3, 5, 9, 11, 23, and 25 to get a total of 351, there a...
Please read #❓how-to-get-help
river finch
Can some one point me where I'm wrong here
i think its w/ your foiling from step 1
oh wait
sry i didn't see the 2
warm shaleBOT
viral tree
can you particularly check if I got this part right?
oh
it should have been 4* 2* 8 and I did 4 (2+8)
Solution x1 = +- Square root 10 is correct
Solution x2 = 8 2/3 is incorrect as 50/3 == 40/9
.close
gleaming rose
obtuse pebbleBOT
gleaming rose
i know what u_k is
kinetic friction
i know what N is normal force
what is F_k?
hushed pebble
F_k is kinetic frictional force
gleaming rose
so the force of kinection friction basically?
gleaming rose
F_k is kinetic frictional force
so this formula says its like the entire frictional force?
hushed pebble
it is specifically the force of kinetic friction
in sliding motion, it is entire frictional force opposing that motion
obtuse pebbleBOT
@gleaming rose Has your question been resolved?
gleaming rose
in sliding motion, it is entire frictional force opposing that motion
thanks alot broski
fervent scaffold
how do i know when to
- parametrize the surface when doing a surface integral
- parametrize the curve when doing a line integral
, and how do i know what to parametrize them as?
obtuse pebbleBOT
how do i know when to
1. parametrize the surface when doing a surface integral ...
thin pawn
Do you mean parameterize to then differentiate using Leibniz' rule?
fervent scaffold
wait what?
thin pawn
I'm not sure what you mean by parameterize?
fervent scaffold
sample question
parametrize the surface S
like put it in terms of two variables
thin pawn
well, sorry, that's not quite what I had in mind, and I can't help you with that. I'm sure someone else will though :)
slim cove
how do i know when to
1. parametrize the surface when doing a surface integral ...
whenever you want to do a line integral explicitly, you have to parameterize the curve
fervent scaffold
oh ok
so just basically whenever i do a line integral instead of using stokes theorem or greens theorem
slim cove
yeah exactly
so like, you can either do it from the definition by parameterizing
or if you're integrating the gradient of something, you can use the fundamental theorem of calculus for line integrals
fervent scaffold
and the definition is usually harder?
slim cove
or if you're integrating along a closed curve, you can use like green's theorem or stokes's theorem
slim cove
and the definition is usually harder?
Yeah, if something simpler applies like the other things you mentioned
It's the same thing with surface integrals
fervent scaffold
how should i know how to parametrize this surface?
slim cove
Either parameterize or use something like gauss's theorem or stokes's theorem to simplify the problem if possible
slim cove
Right
fervent scaffold
but idrk know what to do since its - z^2
slim cove
There's multiple different ways you could do this, but might I recommend cylindrical coordinates?
SInce there's an x^2 + y^2 and then a constraint in terms of z
fervent scaffold
so like
ucosv i + usinv j + h?
how do i know what z becomes?
or is z just sqrt( u^2 - 1)
slim cove
can you post the entire problem?
fervent scaffold
slim cove
ah okay
slim cove
or is z just sqrt( u^2 - 1)
yes that's correct
and now the only question is: what values are u and v allowed to take?
fervent scaffold
oh ok
slim cove
Let's do it in terms of z and theta since we already have the constraint 0 ≤ z ≤ h in terms of z
So what would our parameterization be?
fervent scaffold
oh + 1 mb
fervent scaffold
oh ok could i try another one?
slim cove
sure!
fervent scaffold
after the parametrization i can solve it i just
always struggle on setting it up
slim cove
So this is basically the same process right?
slim cove
Hm
Try doing cylindrical coordinates
Because you have a constraint on z
but it's symmetric about x and y
slim cove
perfect! and remember when you do these parameterizations you always have to write what the domain of the variables is
fervent scaffold
oh so for theta 0 to 2pi
slim cove
yup
fervent scaffold
and z 0 to h
slim cove
exactly!
fervent scaffold
also, does it matter which variable is first for parametrizing?
fervent scaffold
oh ok
fervent scaffold
oh
slim cove
do you know how to figure out the orientation of a parameterization?
fervent scaffold
uh not really sorry
slim cove
that's okay!
so basically
you take a look at dr/dtheta and dr/dz
and then you use the right-hand rule
do you know the right-hand rule?
slim cove
awesome
fervent scaffold
how do i parametrize for a line integral
slim cove
so dr/dtheta is just the direction of increasing theta, so just counter-clockwise
fervent scaffold
like if i wanted to verify stokes’ theorem for those
slim cove
and then dr/dz is just the direction of increasing z, so just up
slim cove
so if you use the right-hand rule you should be able to verify that the normal points outwards
if you do theta and then z
fervent scaffold
ohh
slim cove
just check that you understand that first before we move on :)
slim cove
yup!
fervent scaffold
oh
slim cove
you look at the direction you go when you increase the first coord
and then the direction you go when you increase the second coord
fervent scaffold
ohh ok
slim cove
and then you take the cross product (right hand rule) to find the normal
fervent scaffold
ohh got it thank you so much
slim cove
no problem!
slim cove
how do i parametrize for a line integral
can you post a specific problem that you're thinking about
if you have one
fervent scaffold
slim cove
so what does the boundary of the surface look like?
slim cove
well first answer, what does the surface look like
and what does its boundary look like
fervent scaffold
it looks like a sphere with a bit of the top cropped out
slim cove
yup so what is its boundary
fervent scaffold
wait sorry what do you mean by boundary
slim cove
the boundary of the surface
like
idk how to describe it 😭 like for example the boundary of a filled-in rectangle is its edges
fervent scaffold
oh so like the circle?
slim cove
oh so like the circle?
can you send a drawing of the surface and what you think its boundary is if possible
it'll make this easier
slim cove
the part at z=0 AND what other part?
fervent scaffold
at z=h
fervent scaffold
oh
slim cove
So to get from the orientation of the surface to the orientation of the boundary
You can use the right-hand rule
Your thumb should point outwards since the surface is oriented outwards
And then your fingers curl in the direction that the boundary is oriented
fervent scaffold
counter clockwise
slim cove
Does that make sense?
fervent scaffold
ohh yeah
fervent scaffold
the bottom
fervent scaffold
clockwise?
slim cove
yup!
perfect
so
now we can figure out how to parameterize them
since it's a line integral, how many parameters should there be?
fervent scaffold
1
fervent scaffold
fervent scaffold
slim cove
very close: remember, which way does it have to be oriented?
fervent scaffold
oh
slim cove
let's start a little simpler: how do you parameterize a circle clockwise in like, 2D?
with radius 1 centered at the origin?
fervent scaffold
you make the dr/dtheta negative?
fervent scaffold
oh
slim cove
it's a velocity vector
but can you give me a clockwise parameterization of a circle in 2D with radius 1 centered at the origin?
slim cove
it's okay, so the clockwise one is just the counterclockwise one but the y-coordinate is flipped right?
like
instead of going up to start, you go down
fervent scaffold
ohh
slim cove
so here you were close except you shouldn't have made the x-coordinate negative
fervent scaffold
so the only negative component is y
slim cove
yup
fervent scaffold
slim cove
yessss
:)
and now once you have that, verifying stokes's theorem should be easy but kinda tedious
fervent scaffold
yeah
i set up the line integrals for eCh
then i do the bottoms line integral - the top’s
how do i know that the top is supposed to be clockwise?
slim cove
how do i know that the top is supposed to be clockwise?
because of what we said earlier
fervent scaffold
wait im confused at how
because in both ways it should be going outward if its counterclockwise right?
slim cove
because in both ways it should be going outward if its counterclockwise right?
does this picture help?
fervent scaffold
does this picture help?
wait but im still confused at how the orientation changes as you go higher
oh wait
slim cove
wait but im still confused at how the orientation changes as you go higher
The orientation of the surface is always outwards, no matter your height
fervent scaffold
ohh
slim cove
The orientation of the top circle is clockwise
The orientation of the bottom circle is counterclockwise
This is because the orientation of the surface induces and orientation on its boundary (the two circles)
fervent scaffold
so they have to be opposites?
slim cove
so they have to be opposites?
the orientation of the boundary is whatever aligns with the orientation of the surface
you see how the arrows all align together?
fervent scaffold
yeah
slim cove
another visual
slim cove
yup
fervent scaffold
its like the
only half of a full loop of wire inside a field
ohh thank you so much
could i try doing the prove stokes’ theorem for the other question?
slim cove
could i try doing the prove stokes’ theorem for the other question?
sure go ahead
fervent scaffold
since i account here for the top being clockwise by it being negative
does that mean i should add the line integrals now?
or is it still bottom - top?
slim cove
does that mean i should add the line integrals now?
always add them
fervent scaffold
oh ok
slim cove
yes this is correct