grand forum
yeah we arent allowed to use it here
#help-36
ivory vessel
does the fact that an augmented matrix is homogeneous directly make it easier to rref?
final saddleBOT
does the fact that an augmented matrix is homogeneous directly make it easier to...
formal trail
well you know that no matter what row operation you do, the augmented column will stay 0
<@&268886789983436800> ad
ivory vessel
hmmm
formal trail
well you know that no matter what row operation you do, the augmented column wil...
which means you don't even really need to include it in your calculations
sonic berry
can sm1 help me in 49
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slender sky
Hi! Can someone help me with this problem please
final saddleBOT
Hi! Can someone help me with this problem please
ivory vessel
find the tangent line of x^2 - 1 at (-1, 0)
ivory vessel
the normal line is the line perp to that tangent line
find the equation of that normal line, and do the rest
slender sky
find the tangent line of x^2 - 1 at (-1, 0)
so I found that its y=-2x-2
i'm confused on what to do afterwards
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slender sky
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sonic cairn
how can I solve this?
final saddleBOT
how can I solve this?
sonic cairn
substitution x^2 + y^2 = r?
mortal berry
Polar coordinates?
sonic cairn
I tried a straight substitution and I got
dr/dx = 1
Do you think this is the correct approach?
mortal berry
Can I see how you did it?
mortal berry
,w dy/dx=(-x+Sqrt[x^2+y^2])/y
soft zealotBOT
mortal berry
I don't understand what you have to do
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short harness
hi
short harness
How Could I make this porint I do onot know hwere to starht mayve solvuing in the right side?
warped rivet
How Could I make this porint I do onot know hwere to starht mayve solvuing in th...
use u-sub on the right side
the right integral*
white mantle
How Could I make this porint I do onot know hwere to starht mayve solvuing in th...
if you use u-sub on the rigjt integral you get $\int_{0}^{8} f(x)dx=40$ and than you only need to see the difference between the two integrals
soft zealotBOT
mercidy
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untold kiln
Is there a connective that yields the values of the column at the end?
white tiger
when in doubt, k maps
untold kiln
when in doubt, k maps
what is k maps?
white tiger
We should put this on a shirt.
id buy it
long flame
Me too
rocky tusk
Is there a connective that yields the values of the column at the end?
just not P and not Q since it would be true when both P and Q are false and false in all other instances
white tiger
what is k maps?
a method to get a minimized boolean expression from truth tables
untold kiln
just not P and not Q since it would be true when both P and Q are false and fals...
I know that it can be constructed using multiple operators. I'm asking if there is a single operator that performs that action.
static beacon
Is there a connective that yields the values of the column at the end?
What type of mathematics is this?
rocky tusk
I know that it can be constructed using multiple operators. I'm asking if there ...
so without using negations too, you mean just P Q and the standard 3 connectives
untold kiln
Is there a single binary operator which, when combined with P and Q, results in the truth values of the last column. Like P ? Q
rocky tusk
i mean not P and not Q is just not(P or Q) so maybe there’s something for that
yea
NOR
with a down arrow
$P \downarrow Q$
soft zealotBOT
knief
untold kiln
i mean not P and not Q is just not(P or Q) so maybe there’s something for that
nor it is. Thank you.
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weak silo
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weak silo
can someone check over my epsilon delta proof?
i know ive done something wrong but cant tell what
weak silo
OH
THATS WHY
because the person in the video did this
yeahh that makes sense
wait also
he got |u+2| < 2
but i got |u+2| < 4
did i do another thing wrong
ivory vessel
dusk laurel
i need a ton of help with related rates 
dusk laurel
rocky tusk
so maybe you should draw a sketch
you’ve only listed the givens
@dusk laurel
we know it’s a rectangular well so you should draw a rectangular prism
standing upright
like a nice rectangular box
with some water in it
dusk laurel
ohhhh i was drawing it laying flat LOL
rocky tusk
ohhhh i was drawing it laying flat LOL
lol i mean it also works it’s just the dimensions will be different because it’s supposed to be deeper than it is wide
rocky tusk
idk if this is right or not
this looks good as well
this is a good video btw
but if you have more questions you can ask here of course
best way to learn is by doing
arctic fiber
What is Area of triangle BCQ when AP:PD=2:3 and AC//PQ , □ABCD = 100
fast creek
What is Area of triangle BCQ when AP:PD=2:3 and AC//PQ , □ABCD = 100
is there a drawing for this
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arctic fiber
Yes
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frank folio
can anyone explain why the +1 exists?
final saddleBOT
can anyone explain why the +1 exists?
frank folio
in the last equation? This is simple recurstion
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trail mango
the 1 is from flipping once. then there is a 1/2 chance of being done and a 1/2 chance of expecting E[flips] more flips
i love expectation conditioning
frank folio
@trail mango why do we include it tho?
final saddleBOT
<@984319941417250907> why do we include it tho?
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frank folio
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frank folio
@trail mango sorry
if ur still here
im lost on the logic behind the 1
even if you can write it both ways
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prime halo
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prime halo
How do I do this one?
pliant shore
How do I do this one?
b or c?
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pliant shore
C
try drawing the graph of y = 2|x + 4| - 5
you should find that the limiting case is when l passes through the vertex, (-4, -5)
increase a any more so the line becomes steeper, or the part where x < 0 goes down even more
and there will be no solutions for x < 0
you can adjust the slider to change the value of a
ah and the other part is that two parallel lines never intersect, so your answer should look like
$p \le a$ and $a > q$ for some real $p, q$
soft zealotBOT
south
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prime halo
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prime halo
ah and the other part is that two parallel lines never intersect, so your answer...
So shouldn’t a just be not equal to 2?
I think the answer said a<2
Oh wait no
That would intersect twice if it was greater right?
pliant shore
I think the answer said a<2
a > 2 is one part of the answer
pliant shore
That would intersect twice if it was greater right?
exactly that is why
it will intersect if you make the slope larger than 2
prime halo
a > 2 is one part of the answer
Oh was this meant to be the other way round then?
pliant shore
Oh was this meant to be the other way round then?
yeah, you want the line to intersect at least once
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pliant shore
$p \le a$ and $a > q$ for some real $p, q$
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south
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tranquil pine
yo
deep jacinth
i can try helping
tranquil pine
bro
am i wrong in adding all history students/100 and x by geo students apart pyschology /99
deep jacinth
am i wrong in adding all history students/100 and x by geo students apart pyscho...
hmm that seems right to me
deep jacinth
ohh yeah makes sense
you need to exclude some cases from history students
what was your history students total?
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@tranquil pine Has your question been resolved?
tranquil pine
no
deep jacinth
help
see there was an overlapping student right?
you removed 1 from the 100
but pure history + geo case becomes repeated too
so remove another
which is why you have 25/100 * 36/99
and not 37
sturdy cypress
it's two cases probably
overlap or not
should be (25/100)(36/99) + (18/100)(37/99)
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crystal lion
I'm not sure why when I do simultaneous equations that both variables are eliminated???
final saddleBOT
I'm not sure why when I do simultaneous equations that both variables are elimin...
uncut tartan
Yea cuz where did your x go?
uncut tartan
you forgot ur x
tulip coyote
crystal lion
I did "let x = 1"
crystal lion
uhm idk my teacher taught me to do that and solve for y and z
crystal lion
kk I'll try that
crystal lion
Just solve normally but solve in terms of one of ur parameter
It's not looking too good atm 😬 idk how but somehow I got 6y = 6y
tho I did get x = 0 as one of the coordinates
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dapper quarry
can anyone explain why we dont need the -1 at the end?
final saddleBOT
can anyone explain why we dont need the -1 at the end?
urban burrow
seems like you're doing geometry, lengths can't be negative
glacial hill
Hey can i post smth here
urban burrow
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loud umbra
using bernoulli inequality i need help with this problem
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using bernoulli inequality i need help with this problem
lethal totem
you need more pixels
loud umbra
i actualy need help to solve lim(n----inf)q^n when |q|<1 using bernoulli inequality
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loud umbra
no 😦
candid aspen
yea
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knotty night
Was watching this video, wouldn't it be easier to use the 1 + tan^2theta trig identity instead of changing everything, or am I missing something?
final saddleBOT
Was watching this video, wouldn't it be easier to use the 1 + tan^2theta trig id...
rocky tusk
i mean yes but the point is that you’re showing why that identity holds
frank ginkgo
@rocky tusk Is cooking
rocky tusk
given that sin^2 + cos^2 = 1
frank ginkgo
Bye ...
rocky tusk
What do you mean?
like yes obviously tan^2 + 1 = sec^2 as it is a variation of the pythagorean identity but then you wouldn’t show any work
knotty night
True
rocky tusk
they want you to prove it using the assumption only that sin^2 + cos^2 = 1
that’s your toolbox so to speak
rocky tusk
So, I should always change it so it will be sin and cos
in more involved problems no but for something this simple yes
rocky tusk
it’s only because you’re trying to prove this property specifically that it would be meaningless to just state it’s true
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knotty night
How come he expanded everything? I thougt we would have to turn Sin^2x + cos^2x into 1. Can anyone explain to me the priorities or rules?
verbal pendant
How come he expanded everything? I thougt we would have to turn Sin^2x + cos^2x ...
bc the second term on the right side isn’t useful yet until you expand
left side*
knotty night
Over here, he left (1 + sinx) x cosx. I expanded it
How would I know all this
?
How would I know when to expand, when not to
when to do anything?
knotty night
bc the second term on the right side isn’t useful yet until you expand
What do you mean by isn't useful? Do you mean cannot be converted into a different identity?
or
verbal pendant
yes
knotty night
I see
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halcyon reef
this may be a completely stupid question but im getting a bit confused:
halcyon reef
is there any specific way I have to solve this other than putting it into my calculator?
long fable
is there any specific way I have to solve this other than putting it into my cal...
do u want degress
or radians
thats the only option u got
(u just have to plug it in)
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warm python
I'm trying to determine if $\begin{bmatrix} 5&4\0&5\end{bmatrix}$ is diagonizable
final saddleBOT
I'm trying to determine if $\begin{bmatrix} 5&4\\0&5\end{bmatrix}$ is diagonizab...
soft zealotBOT
A dense set(Ping when reply)
warm python
There is one eigenvalue
with a multiplicity of 1
which is why it isn't diagonisable
is that right
formal trail
what kind of multiplicity?
formal trail
there are two types of multiplicity for eigenvalues, arithmetic (multiplicity as a root of the char poly) and geometric (dimension of eigenspace)
warm python
the geometric multiplicty
warm python
there are two types of multiplicity for eigenvalues, arithmetic (multiplicity as...
Is that right
dense coral
which is why it isn't diagonisable
this isn't sufficient
wait hold on
okay, you said geometric multiplicity
yeah so, if any of the geo multiplicities are less than their respective algebraic multiplicities, then the matrix isn't diagonalizable
warm python
cool
dense coral
I'm gonna take your word for it when you say that the geo mult. of 5 is 1
the alg mult. of 5 is 2 obviously
so not diagonalizable
warm python
cool
one more question
Same question but for $\begin{bmatrix} 0&-1 \ 1&0 \end{bmatrix}$
soft zealotBOT
A dense set(Ping when reply)
warm python
The eigenvalues are not real
vivid cypress
The eigenvalues are not real
chat is this real?
warm python
so it isn't diagonizable
warm python
chat is this real?
I believe so?
dense coral
so it isn't diagonizable
I mean, this depends on what field you're working over
vivid cypress
I mean, this depends on what field you're working over
$\mathbb{C}$?
dense coral
if you're taking the vsps to be real, then sure
soft zealotBOT
utop!a
dense coral
if you consider them complex, then the char poly always splits
and so you'd need to check the geo multiplicites again
warm python
I mean, this depends on what field you're working over
$\R$
soft zealotBOT
A dense set(Ping when reply)
warm python
No idea what a Jordan form is
dense coral
just the "next best thing" to diagonalizability 
warm python
Eh, will either do it over the winter break, or i9n year 3
dense coral
warm python
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warm python
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spring lava
I'm not understanding what my prof is doing, I don't get why z = -3 sin(t)
I dont get why the direction vector is that of:
( -3 sin(t), 0, -3cos(t))
final saddleBOT
I'm not understanding what my prof is doing, I don't get why z = -3 sin(t)
I do...
spring lava
Ah I see this doesn't give the givens:
Let S be the surface defined by y = 10 - x^2 - z^2 with y >= 1, oriented with rightward-pointing normal.
Let F = (2xyz + 5z) i +(e^(x)) * cos(yz) j + x^2*y k
Determine the double integral over S of the Divergence x F dot dS.
This problem utilizes stokes theorem and sets a curve C
@ancient wharf do you think you can help w/ this?
ancient wharf
<@590544263964524575> do you think you can help w/ this?
What is this class lol
Integrals ?
ancient wharf
Sorry I got no clue in this topic 😔
ancient wharf
but like the biggest problem right now for me is the orientation of C
You in college ?
spring lava
yea
ancient wharf
I am in hs lol
tranquil pine
Man I wanna help with ez math what is this😭
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astral moss
I'm not sure how I should go about the proof for Exercise 3.25. I found that the statement is equivalent to the converse of theorem 3.24.
final saddleBOT
I'm not sure how I should go about the proof for Exercise 3.25. I found that the...
desert mantle
think about the word cyclic
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@astral moss Has your question been resolved?
astral moss
Cyclic groups havent been covered yet. Like i kinda have intuition for why the statement is true but i don't know how to prove it
desert mantle
wdym havent been covered
Z_n is the "only" cyclic group
cyclic groups are the most simple groups, how can you not talk about them
have you talked about the order of elements
desert mantle
<@1082220832740347926> Has your question been resolved?
and dont ignore the bot
astral moss
yes
i know the order of elements
and we proved two groups can only be isomorphic if they have the same number of elements with given order
for example in Z_4, the element 1 has order 4, but in Z_2 x Z_2, the highest order of any element is 2
desert mantle
good
aka Z_2×Z_2 is not isomorphic to Z_4
now do it for p and q instead of 2 and 2
astral moss
So in Z_pq, the element 1 has order pq, but in Z_p x Z_q the highest order of any element is something which is not pq
idk why im struggling to generalise it
desert mantle
if you have an element (n,m) in the product, what is the order of n and m? so what is the order of (n,m)?
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quick axle
Is x 113?
quick axle
Can anyone else also approve?
atomic moon
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severe canyon
Yes it's correct
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fathom stratus
Is this integration correct?
final saddleBOT
Is this integration correct?
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@fathom stratus Has your question been resolved?
fathom stratus
2A-A = A, not 3A
Its -2A-A
=-3A
<@&286206848099549185>
<@&286206848099549185>
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pastel spindle
ok
pastel spindle
ok: used to express agreement or acceptance
pastel spindle
<@&286206848099549185>
!15mins
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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.
fathom stratus
I substitued -2c as 2A
After substituiting -2c as 2A
It becomes 2A-A+B+2A
fathom stratus
Is this integration correct?
In the previous steps i showed of A=-C
Hence 2(-c)=2A
U get?
solar glade
No
solar glade
That is not correct
fathom stratus
That is not correct
Which part
solar glade
Solve again
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fathom stratus
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fathom stratus
@solar glade
Please check this
I tried correcting my mistake
Thank u for telling me where i had gone wrong
solar glade
A, B and C now look correct
solar glade
No
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carmine crow
Can someone please help with the part ii of this question? thank you
final saddleBOT
Can someone please help with the part ii of this question? thank you
@carmine crow Has your question been resolved?
carmine crow
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worn wave
What have you tried?
carmine crow
Did you already do the first part showing convergence?
No but I can just use the alternating series criteria, right?
just gotta show that 1/sqrt(n) is decreasing null sequence
@worn wave I basically just multiplied it into a cauchy product
and idk if a made a mistake or not
but idk how to proceed
can it be, that I have to keep the index starting from 0 when doing the product?
worn wave
I'd try starting from 0. You have a small issue of the last term being undefined
carmine crow
yeah
and for n=1, all the terms are undefined
sorry
for each n, the nth term is undefined
worn wave
Controlling the inner sum is the hard part. I think you need to find a bound away from zero in terms of n.
carmine crow
lemme try with leaving the the inner sequece as a_n
worn wave
The partial sums all have the same sign regardless of the parity of n
carmine crow
and keeping the index starting from 0
do you think that will help?
keeping the sequence as a_n?
worn wave
I think reindexing is good so that all the terms are defined
carmine crow
I think reindexing is good so that all the terms are defined
ok lemme try
carmine crow
I think reindexing is good so that all the terms are defined
I failed
it didn't work
worn wave
Reindex the original sum
carmine crow
Reindex the original sum
wdym
worn wave
Oh, the original question had a_0=0
The term under k=n is zero
You need to get a bound for your c_k. I suggest writing out partial sums for n=8 and 9 to see what it looks like
carmine crow
You need to get a bound for your c_k. I suggest writing out partial sums for n=8...
what is c_k
do you mean the series in my series?
worn wave
carmine crow
ah ok
@worn wave actually, I think I'm gonna give up on this question for now. Do you think you might be able to help me w another question?
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young birch
what does tr mean?
final saddleBOT
what does tr mean?
young birch
is it some matlab command im so confused 💀
gleaming pendant
matrix trace
twilit arrow
(P ∧ ¬Q) ∨ P ≡ P
is this because p and not q = p, and p or p = p?
final saddleBOT
(P ∧ ¬Q) ∨ P ≡ P
is this because p and not q = p, and p or p = p?
twilit arrow
my thought process:
P and NOT Q = P (because q is excluded)
New fxn P v P
P or P = P (because whatever you choose it's P)
formal trail
p and not q ≠ p
twilit arrow
how come
formal trail
,w truth table p and not q
soft zealotBOT
formal trail
the truth table disagrees in one row
twilit arrow
im coming from the logical standpoint of saying for example, it is hot and not cold
therefore it is hot
but why is it that way
formal trail
you are assuming p and q are incompatible. but what if they are not, e.g. "it's hot and not raining"
twilit arrow
mm
that makes sense
so then what would be the proper simplification
i know there's a rule called absorption law but I kinda wanna know how you'd do it step by step
formal trail
and distributes over or, & or distributes over and
twilit arrow
bcz idt i'll memorize it 😭
twilit arrow
and distributes over or, & or distributes over and
wdym
P or P and not Q or P?
P and not Q or P
^ I'm fairly new to logic so forgive me if I'm mistaken
formal trail
that's valid. although i was thinking of first writing P = (P or 1) then "factoring out P" from the expression (basically distributive law in reverse)
twilit arrow
that's valid. although i was thinking of first writing P = (P or 1) then "factor...
ok so then after we have P and not Q or P how do we simplify
to get only P
ok according to chatgpt
this is how you do it
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tranquil pine
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abstract bramble
are you supposed to use stokes theorem
vital crag
$n = \nabla (z - \sqrt{x^2 + y^2})$
soft zealotBOT
riemann
abstract bramble
oop wait nvm
abstract bramble
i think its stokes
tranquil pine
is stokes theorem curl (F) ?
vital crag
my bad i always confuse green's with divergence
similar problem here https://math.stackexchange.com/questions/3026712/divergence-theorem-cone
tranquil pine
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old hornet
What is the l1 norm of f(x)=x1^2+x2^2+3x3^2 at (1,1,1)?
final saddleBOT
What is the l1 norm of f(x)=x1^2+x2^2+3x3^2 at (1,1,1)?
keen skiff
What is the l1 norm of f(x)=x1^2+x2^2+3x3^2 at (1,1,1)?
I will find you.
vital crag
what's the definition of a norm at a single point
old hornet
what's the definition of a norm at a single point
What is it?
vital crag
yes
old hornet
L1 norm
vital crag
you gave a point
vital crag
what's the definition of the L1 norm of a function at a point
old hornet
idk
vital crag
......
old hornet
||x||_1
sum of absolute value of xi
for 1 case
so how do you find it if is a function
vital crag
the L1 norm is defined for functions yes
vital crag
old hornet
the L1 norm is defined for functions yes
so you evaluate it at (1,1,1) first?
old hornet
ah yes so sum up the absolute vlaues
vital crag
because this definition does not incorprorate a function
old hornet
i dont see any functions in that def
vital crag
What is the l1 norm of f(x)=x1^2+x2^2+3x3^2 at (1,1,1)?
but you have a function
vital crag
show the original question
vital crag
picture or screenshot
old hornet
vital crag
no idea what it's asking then ¯_(ツ)_/¯
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tranquil pine
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vital crag
what formulas were you given
just this one?
and you're just algebraically solving for $x_0$ ?
soft zealotBOT
riemann
tranquil pine
Yes
tranquil pine
what formulas were you given
yes
vital crag
you messed up your signs in this step.
Exf - Ex0 = delta v
add Ex0 to both sides
Exf = delta v + Ex0
subtract delta v both sides
Exf - delta v = Ex0
now you can divide by E
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dusky dock
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wicked flare
Would you not just use trig functions?
dusky dock
we didnt even learn that though
wicked flare
let width = w
tan 50 = w/35
tan50 * 35 = w
dusky dock
im not at that stage 💀 this is about hinge theorems
winter axle
try using your intuition for the first question in terms of how the side lengths & angles given will relate with the width
wicked flare
ohhh.
dusky dock
try using your intuition for the first question in terms of how the side lengths...
tried that
wicked flare
idk why you get 50 either
so the triangle is a right triangle right?
dusky dock
yea
wait
nvm lol
i just dk the 50 part
nor how u get the more than 35 less than 50
dusky dock
180
dusky dock
ye
winter axle
tried that
think of the opposing angles for each triangle, remember that the side length adjacent to the larger angle will always be shorter than the other
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@dusky dock Has your question been resolved?
snow ore
how to do qeastion one \
final saddleBOT
how to do qeastion one \
snow ore
,rotate
soft zealotBOT
jagged flare
you already did this
jagged flare
was it?
snow ore
idk
i am not sure
the answer was like 2/7
but that is the ratio for the lines i think
jagged flare
.-.
snow ore
.-. how you do that
jagged flare
what do that
jagged flare
dot hyphen dot?
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snow ore
.close
normal jasper
Pls help asap how do i do this
final saddleBOT
Pls help asap how do i do this
normal jasper
wait i will get a more clear pic
How do i do nukber 3
How do i start it
i think i can like put all the points down but is there a way to like solve it by just subtracting or adding and flipping the signs and switching x and y?
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normal jasper
.reopen
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normal jasper
<@&286206848099549185>
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finite delta
When I'm calculating part a, I would have to take the curl of F, right?
tranquil pine
can anyone help me?
final saddleBOT
can anyone help me?
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
tranquil pine
put it in a different channel, this one is being used
kk
tranquil pine
No
finite delta
oh ok
tranquil pine
I could have in no time opened another one
finite delta
I could have in no time opened another one
thats fair, i just feel bad when i do it because i find it annoying when it happens to me
tranquil pine
ok
upbeat harbor
When I'm calculating part a, I would have to take the curl of F, right?
Hey just you know asking
Is the answer 0?
finite delta
Hey just you know asking
Is the answer 0?
im sorry?
im not certain i quite understand what youre asking
upbeat harbor
The Flow rate of hot air
Is it 0?
finite delta
when i used stoke's theorem i got 0, and so i should also get 0 when i do part a
but im trying to figure out why i didnt get 0 for part a
upbeat harbor
Hmmm wait...
finite delta
because when i solved this, i didnt take the curl of F. and i think thats why i didnt get 0
upbeat harbor
Okay so here we need the surface integral
Which is flux
Flux is integral of F•n dS
For sphere we take $ndS = r^{2}sin{\phi}d\phi d\theta$
soft zealotBOT
@upbeat harbor
upbeat harbor
Here curl(W) = F
upbeat harbor
Okay
finite delta
are you telling me that i solved part a and b backwards?
upbeat harbor
are you telling me that i solved part a and b backwards?
No no
finite delta
because i used F*n dS for part b
upbeat harbor
because i used F*n dS for part b
See
The formula for flux is
$\int_{S}F.ndS$
soft zealotBOT
@upbeat harbor
finite delta
right
upbeat harbor
Since here the surface is a sphere
We can say $ndS = R^{2}sin(\phi)d\phi d\theta$
soft zealotBOT
@upbeat harbor
upbeat harbor
And since we just shifted to spherical coordinates
We need to rewrite $W = <-y,x> = <-R^{2}sin(\phi)sin(\theta),R^{2}sin(\phi)cos(\theta)$
soft zealotBOT
@upbeat harbor
upbeat harbor
Now if we do $\int_{S}W.ndS \$
It is
$\int_{0}^{2\pi}\int_{0}^{\pi}W.R^{2}sin(\phi)d\phi d\theta$
soft zealotBOT
@upbeat harbor
finite delta
wait whyre we using W instead of F?
upbeat harbor
This can be also written as
Flux = $-\int_{0}^{2\pi}\int_{0}^{\pi}R^{4}sin^{2}(\phi)sin(\theta)d\phi d\theta + \int_{0}^{2\pi}\int_{0}^{\pi}R^{4}sin^{2}(\phi)cos(\theta)d\phi$
Of u first integrate w.r.t. $\theta$ here , you'll get 0 ig
soft zealotBOT
@upbeat harbor
upbeat harbor
wait whyre we using W instead of F?
Because in the formula of flux, F is the vector field, which in your question is W , and F in your question is curl(W)
Which is used in stokes' theorem
finite delta
hm
upbeat harbor
<@947094726513942578>
This
upbeat harbor
is that not relevant here?
Okay wait yeah it is useful...
Did you use that information while solving part B?
finite delta
Okay wait yeah it is useful...
Did you use that information while solving part ...
well i think i solved both parts incorrectly now so i cant really answer that.
what i can say though is that im asking if its relevant to part a?
upbeat harbor
Yeah it is
Yk the answer has to be 0
No matter what
Because have you ever seen the vector field <-y,x,0>
finite delta
no
upbeat harbor
Look for it on Google
It's a stirring like motion
I.e. concentric circles
So that means all particles will just revolve around in a circle of fixed radius
So nothing goes in or out
finite delta
is it like the vortex field?
upbeat harbor
<@947094726513942578>
One thing we can do is
We found the flux of whole surface with this
Just subtract the flux of the surface below z = -R/4 from this
And I think it should still be 0
upbeat harbor
is it like the vortex field?
Kind of
But it's like concentric circles
Or or we just change the bounds!
finite delta
Ok but it’s at least similar then
upbeat harbor
Ok but it’s at least similar then
What the vector field?
finite delta
Yeah, i was just asking so I could get like a general idea of what it looked like
finite delta
One thing we can do is
We found the flux of whole surface with this
Just subtr...
I mean all we’d need to do is change the bounds of phi to make it be above -R/4 right?
That shouldn’t be too hard
upbeat harbor
Yeah, i was just asking so I could get like a general idea of what it looked lik...
This is it
upbeat harbor
I mean all we’d need to do is change the bounds of phi to make it be above -R/4 ...
Exactly
Wow you got it!
finite delta
Exactly
Wow you got it!
It’s just changing the bounds kinda sucks lmao
upbeat harbor
I mean all we’d need to do is change the bounds of phi to make it be above -R/4 ...
Now notice
We get 0 at the very moment we first integrate with d(theta)
So like the bounds for phi don't even matter
finite delta
Fair, but I’d still like it to be correct just so I don’t risk having points knocked off
upbeat harbor
Fair, but I’d still like it to be correct just so I don’t risk having points kno...
Yeah its a good habit
Hey btw
What bounds did you get for phi?
finite delta
<@947094726513942578>
would the -R/4 change this at all?
finite delta
Hey btw
What bounds did you get for phi?
working on that right now
I got 0<phi<arccos(-1/4)
but im not certain that thats correct
upbeat harbor
I got 0<phi<arccos(-1/4)
Hmmm
I get arccos(3/4)
finite delta
how so?
upbeat harbor
Oh mb
But it should be arccos(1/4) then
upbeat harbor
Oh yeah I'm so sorry
I calculated for the other one
You're correct
finite delta
ok, so it is arccos(-1/4)?
upbeat harbor
Yeah surely
finite delta
maybe. let me try and fix the rest of part a, and then ill look over part b again
upbeat harbor
Alright
finite delta
ok so for part b, i parametrized the curve as <Rcos(theta),Rsin(theta),-R/4>
then I found the normal vector using that, and dotted it with F to get 0
upbeat harbor
Dotted...?
Curl is gradient cross vectorField
?
finite delta
wait hold on, i forgot i solved these backwards
for b, id use F because thats the curl of W, right?
upbeat harbor
Yeah
finite delta
ok
upbeat harbor
What value did you get of F?
finite delta
<0,0,2>
upbeat harbor
How........
finite delta
W is <-y,x,0>
this means that when you take the curl of it, you get:
<0-0, 0-0, 1--1>
upbeat harbor
,w curl of vector field <-y,x>
upbeat harbor
this means that when you take the curl of it, you get:
<0-0, 0-0, 1--1>
Oh ok yeah we can parameterize
Fine
Ok
finite delta
did i do something wrong? or something that doesnt make sense?
finite delta
Oh ok yeah we can parameterize
Fine
Ok
well we know what the curl is then. so now we'd just need to dot it with n, right?
upbeat harbor
Okay yes
finite delta
ok, now we just need n then
could I use the surface <Rcos(theta),Rsin(theta),-R/4> for this?
upbeat harbor
Umm yes
But
Is the fluid or smoke whatever it was
It isn't flowing through this surface....
upbeat harbor
Okay see here our normal vector will be a vector which yk like goes out the sphere and is normal to the surface of the sphere
finite delta
mhm
finite delta
ok
upbeat harbor
With components dependent on phi and theta
finite delta
ok so wed end up with <Rcos(t)sin(p),Rsin(t)sin(p),Rcos(p)>?
(t is theta and p is phi, i just wrote it like that to try and make it look a little neater)
oh wait no thats the curve
not the normal
hm
ok yeah i really am not sure now how to do that. would i take the derivatives with respect to theta and phi and then cross them?
upbeat harbor
Yeah there's a formula to find the unit normal vector for this
finite delta
taking it with respect to theta gives
<-Rsin(t)sin(p),Rcos(t)sin(p),0>
taking it with respect to phi gives: <Rcos(t)cos(p),Rsin(t)cos(p),-Rsin(p)>
upbeat harbor
Cross them
finite delta
working on it
this is disgusting
<-R^2sin^2(p)cos(t),-R^2sin^2(p)sin(t),...>
the z component is gonna take me a good minute
upbeat harbor
Huh, why do I think something's getting wrong here
Will you give me some time please
upbeat harbor
Normal vector will be <smthng,smthng,cos(phi)> ig
finite delta
-Rsin(p)<Rcos(t)sin(p), Rsin(t)sin(p), Rcos(p)>?
I found something in my notes that helped me get this
Oh wait that points inward doesn’t it
I mean we could just multiply by -1 and fix it
finite delta
I think I messed up everything
What do you mean?
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finite delta
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@finite delta Has your question been resolved?
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finite delta
<@&286206848099549185> can I have a hand with this please?
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@finite delta Has your question been resolved?
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@finite delta Has your question been resolved?
lament vale
When I'm calculating part a, I would have to take the curl of F, right?
First let's calculate F = curl(W)
I think it's (0, 0, 2) (verify that)
Now let's compute the flux
Luckily the normal vector to the sphere is parallel to the radius vector
In spherical coordinates, n = (sin(theta)cos(phi), sin(theta)sin(phi), cos(theta))
The surface element in spherical coordinates is R^2 sin(theta) dtheta dphi
So the flux is
$$\int_{-\pi}^\pi \int_0^{\arccos(-1/4)} 2 \cos(\theta)R^2\sin(\theta)d\theta d\varphi$$
soft zealotBOT
EQUENOS
lament vale
In order to solve (b) just add a flat circular "cap" on the bottom of the sphere and note that div(F)=0
finite delta
So the flux is
$$\int_{-\pi}^\pi \int_0^{\arccos(-1/4)} 2 \cos(\theta)R^2\sin(\t...
Why is it from -pi to pi?
celest crane
Why is it from -pi to pi?
From pi/2 to pi, 2cos(theta) is negative and will zero out the integral.
finite delta
From pi/2 to pi, 2cos(theta) is negative and will zero out the integral.
Wouldn’t it be 0 to 2pi though? Because it’s a sphere?
celest crane
I'm basing that on the 2*cos(theta) that Equenos wrote.
finite delta
From pi/2 to pi, 2cos(theta) is negative and will zero out the integral.
Mm. I’ll try and calculate it using this, and 0 to 2pi, and I’ll see if my result changes
celest crane
Where did that cos() come from in Equenos equation?
finite delta
Where did that cos() come from in Equenos equation?
The cos(theta)?
celest crane
Yeah.
finite delta
That’s from the normal vector I think
lament vale
Why is it from -pi to pi?
Because each horizontal section of the balloon is a full circle
By horizontal section I mean a section parallel to the XY plane
finite delta
Because each horizontal section of the balloon is a full circle
Right, so wouldn’t 0 to 2pi give the same result?
lament vale
Right, so wouldn’t 0 to 2pi give the same result?
Of course it would
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@finite delta Has your question been resolved?
stray hare
- Determine the real zeros and state the multiplicity of any repeated zeros
- list the x intercepts where the graph crosses the x axis and those where it does not cross the x axis
- Sketch the graph
final saddleBOT
1. Determine the real zeros and state the multiplicity of any repeated zeros
2. ...
vital crag
All the terms have an x so you can factor
stray hare
@vital crag am i on the right path? I use these as my zeros right
vital crag
,w factor x^4+x^3-4x^2-4x
soft zealotBOT
stray hare
ok cool um
how would I sketch the graph out
I know there are concepts to find when it increases and decreases do u know abt that
@vital crag ?
ashen shoal
They might be referring to differentiation
stray hare
.close
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agile thunder
why does triple integrating a 1 (constant) give us the volume?
final saddleBOT
why does triple integrating a 1 (constant) give us the volume?
agile thunder
desert mantle
think about how integrating $\int_a^b 1 dx$ gives us $b-a$ which is the length of the interval $[a,b]$ over which we are integrating
soft zealotBOT
Denascite
desert mantle
yours is the same thing, just 3d
if K is for example a cube you can easily compute the integral and see that you indeed get the product of the side lengths
formal trail
if you do a single integral of 1, you are multiplying base * a height of 1, which means you get an area which is numerically the same as the length.
similarly if you double integrate 1, you are multiplying area of base * a height of 1, so you get a volume which is numerically the same as area.
so if you triple integrate 1 specifically, you will get the volume. if you were to triple integrate 2, you would instead get twice the volume
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small wing
is there supposed to be a "t" when forming the integral next to -r?
wicked robin
Likely yes
silent meadow
Idk how to do this
final saddleBOT
Idk how to do this
silent meadow
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silent meadow
Please only use the <@&286206848099549185> ping **once** if your question has no...
Ahhh
I keep forgetting
Mb
violet echo
hi
silent meadow
hi
violet echo
how ya doin
silent meadow
good u?
silent meadow
Take ur time
violet echo
where u from?
violet echo
Idk how to do this
is that h = 21 - 20 cos(pi t) or - 20 cos(ht)
silent meadow
born in brazil live in switzerland
I wanna live in Switizerland, I live in Canada tho
This country is shit