/sci/ - Science & Math

After you take calculus 3, linear algebra, and differential equations, what comes next? Is there any point in going beyond these classes as an EE?

>>10019984
>what comes next?
Death
>Is there any point in going beyond?
72 virgins
Rent a truck of peace today my good brother

Is there a simpler way of doing this?

=(D3*40)/IF(COUNTIF(F3:3;TRUE)=1;2;IF(COUNTIF(F3:3;TRUE)=2;4;IF(COUNTIF(F3:3;TRUE)=3;8;IF(COUNTIF(F3:3;TRUE)=4;16;1))))

>>10019984
>as an EE
my condolences. real talk: pick up real analysis, and work your way up to functional analysis.

A bit of an off-topic question: what do people mean when they say X is a meme? What does the word meme mean in this context?

I have 15 ECTS points in the bachelor and 15 points in the master for electives.
Considering the current CS age I think I want it to be CS based. Btw I'm a biochemist.
What do you guys think is best to take?
Should I just take a bioinformatics course and some introduction to programming course in my bachelor's and then follow up with something like machine learning in my masters?
I'm not planning on majoring in bioinformatics in my master's but I'd like to get close by taking similar electives

>>10020087
Usually when something is referred to as a meme, it is not in the sense of a facebook-tier-tag-your-friends meme, but more in the sense of "virgin/chad" or "npc" memes. What I mean here is that it is much more than just a simple and relatable concept or idea, but instead one that digs deep into your insecurities, that scourges for the thing that troubles you most and displays it for everyone to see. You feel vulnerable because your fatal flaw has been found out. But that is not the worst part, no, it is the fact that you are simultaneously being forced to accept that it exists to begin with, a fact you have always had in the back of your mind, yet have always been reluctant to consider.

However, it just so happens that a concept so powerful goes both ways, in that both the reader and the maker of the meme are being impacted. The former as described; the latter in that he must divulge the meme so as to gauge if he is not the only one; if he is not alone, and his quest for his meme's popularity is a direct indicator.

How does this bring us to the topic of "X is a meme"? Firstly, you must consider that its writer has the direct intention of insulting those who like X by way of making them feel insecure about their taste. Secondly, they want to discourage any interested readers from doing X, usually and purposefully without any indication as to why it is so in an attempt to bait the reader into making a fool of themselves. Finally, it is most likely the case that the writer does not know why X is in fact a meme, they either speak from inexperience, are repeating what they've read, or simply cannot comprehend how or why there is value in X, and they save face by defacing X.

Is MATLAB the industry standard? It seems to be on a lot internship qualifications. I've have yet to come across Mathematica.

Is a Masters in Population Biology a good idea? Are there a decent amount of jobs in it?

This is a pretty open-ended question.
Past Calculus and Linear Algebra, what classes should a pure math undergrad be taking?
Number Theory and Intro to General Topology are obvious picks, but what past that? Just whatever's interesting?

>>10019775
Let's say I’m playing a card game with several friends. One aspect of the game is that roles are randomly assigned by dealing character cards to the players. Most character cards are similar but one is especially important. Let’s also say there’s one player I don’t want this card to go to (maybe it’s her first time playing) how can I assure that she doesn’t get this card and also prevent myself from knowing who gets it. How can I hide my slight of hand from the other players?

How do I optimize learning anatomy with anki? I've been copying the diagrams from lectures and cropping out the names of structures, but our lecturer is an incompetent bitch and puts in a lot of 'theory' questions as well, which you have to infer from the diagrams in the EXACT way she did. Like a question will be 'name 5 structures posterior to the pancreas' and if you don't put the exact five she thought of you get marked wrong.

I'm drinking and I had a math degree at one point but now I'm lousy but I'm here for you my dude

ask me anything. I've recently resigned myself to going back through a pre-calc text to solidify really what I know and don't know.

No question too stupid, idgaf. Let's do this.

>>10020661
Mathematica is for academic use and prototyping, matlab is for faster computation.
But better to just prototype in a common language like python, R, etc and drop to C for bottlenecks

>>10020902
what does P=NP mean and what is the relevance of the answer for my daily life?

>>10019984
>next
After early math, it opens into a complete graph on infinite nodes. People spending their life head down in diff eq won't know basic shit about analysis.
Pick your passion my dude. For EE, lin alg, diff eq, now go to control theory and fourier analysis which are both just applied earlier stuff

>>10020820
if you are dealing the cards, you can deal the second from the top instead of the top without anyone noticing, as long as you do it quickly and convincingly enough. If you can guarantee that the top card is not the specified card you just deal her that card

I gotta take either Thermodynamics or Heat Transfer next semester, looking for the little pussy way out. Which one is easier?

Can someone explain when to use * or .* when multiplying in matlab? I'm confused.

>>10020730
Does your school offer a course on formal logic and proofs? If so I'd suggest that.

I've taken linear algebra and differential equations, but haven't had occasion to use either and as a result am pretty rusty on both. Any suggestions for inexpensive refresher books on these topics?

Any advice on project management? I've got two large scale design projects. I'm a team leader on one and a senior advisor on the other.

>>10020493
>>10020087
Around here, it generally means X is bullshit. Otherwise, what I hope to god is copypasta said.

>>10020947
often the pussy way is the "harder" path
see: trying to solve physics problems by algebra rather than calc
see: getting random heat transfer problems rather than just understanding the underlying thermo

I just want to know, if CO2 is such a big deal then why aren't we doing anything to fill that huge gap of chlorophyll smack in the middle of the oceans, the part that gets the most sunlight? Can't we throw some floating (but submerged) structures anchored to the sea bed for algae and other shit to use as a substrate?

And if you're one of those that thinks global warming is a meme then cool, there's no problem, ignore this k?

>>10021054
algae doesn't need a substrate and all we'd have to do it toss some iron (or phosphorus) in the ocean to get the effect you're looking for
run-off from farms actually causes algal blooms in the ocean, but algal blooms kill everything by depleting the oxygen in the water
Ideally we seed algae growth out in the middle of the the ocean where there's fucking nothing, rather than on the coast.

>>10020087
Its value is inflated due to the cultural meaning or significance attributed to it.

>>10021060
That's interesting, I didn't know that, but I just used algae as an example so hopefully we could use more helpful organisms. And I also mentioned doing it "smack in the middle of the oceans" where, as you and the map I posted point out, "there's fucking nothing".

I'm curious now though, isn't photosynthesis supposed to produce oxygen? Why would algae deplete it? At night maybe?

>>10021072
Oh, it goes like this:
>algae grows really fast and uses up all the nutrients (hence "algal bloom")
>it all dies at once
>the rapid decomposition depletes all the oxygen in the water
Also, decaying algae smells bad enough to make you gag.

>>10021083
Ah so the blooming part is what fucks things up, I see.
>decaying algae smells bad enough to make you gag
You can say that again. What do you think is a better alternative to the example I proposed?

would be awesome if anyone could at least help me with even one of the three questions, i love you all

>>10019775
Do my homweerok

what's larger, 3rd root of 7 or 5th root of 5

Explain your reasoning because it's easy to wolfram :)

>>10021185
I honestly never failed to find an example of the solution to hw problems like that online by just googling the first scentence. I dont get how professors expect you to learn it without examples. Just make sure youre not one of the idiots only copying down the answer.

Any recommendations for basic chemistry books?

Looking to get the basic principles down before studying at college. Namely elements, compounds, the periodic table, atomic structure, covalent bonding, molecular shape and polarity.

Anything more is also appreciated.

>>10021199
cube both of them
cube root of 7 becomes 7
5th root of 5 becomes 5^(3/5) < 5

>>10021341
Brown or Raymond Chang. It's where my first semester teacher got his exams from, at least.

>>10020730
Number theory isn't really an obvious pick. Undergrad number theory is fucking stupid and grad number theory is only a "should" for aspiring number theorists.

In terms of courses taken you really _need_ three things by the time you graduate; 2-3 semesters of algebra (linear doesn't count), 2-3 semesters of real/complex analysis, and at least one topology course.
Anything else is flexible depending on your goals and what you find interesting.

>>10021199
A x A x A x = 7

1) A = 2
A3 = 8

2) A = 1,5 = 15 E-1
A3 = 3375 E-3 = 3,375

15 x 15 = 150 + 50 + 25 = 225
225 x 15 = 2250 + 1000 + 100 + 25 = 3375

=> 1,5 < A < 2
_____________________

B x B x B x B x B = 5

B = 2
B^5 = 32

B = 1,5
B x B = 2,25
B x B x B = 3,375 = 3375 E-3
x B = 3375 E-3 + 1500 E-3 + 150 E-3 + 375 E-4
= 4875 E-3 + 150 E-3 + 37,5 E-3 = 5062,5 E-3 = 5,0625
x B ≈ 7,5

=> 1 < B < 1,5

======> B < A

ayo hold up, so you be sayin that if i look at a plant hard enough that i'll see it's held together by electromagnetism?

>>10021054
>if CO2 is such a big deal

It's not. It's what makes plants grow.

>>10021383
Excellent.
I have known a lot of mathfriends who can't deal with very basic problems.
Good solution. I didn't check it because I have better things to do with my life lol nerd.

>>10021383
>using , as decimal point

Covers (either open or closed) need not be contained in the set itself, right?

>>10019775
yah

>>10021933
wat ?

Why don't you guys answer every question in here?

>>10021976
shit, I meant that the cover can exist outside the set, so long as there's a 'small' piece of it left inside the set.

>>10022016
I assume that your question is this: given a subset A of X and a covering U_i of A, does the definition of covering require that each U_i lies completely in A ? answer is no.

>>10022036
That's what I thought, thanks.

By convention, for an undirected graph G=(V,E), do we tend to write edges in both "directions" in E (i.e. do we repeat (x_i, x_j) and (x_j, x_i)), or only one of the two?

>>10019775
Are there any lists for different subjects so I can attempt to educate myself?

Not lists of suggested reading but this then this then this type of lists.

Which is better: Oxford or Cambridge for computer science? Will I miss out on more complex areas of computer science if I don't take a maths with computer science dual honours or is there a sufficient amount of maths in both of the computer science courses?

>>10022132
catalogue has suggested reading basically in order, its v silly and will waste your time. there is another just for math. if i get a chance i can post the math chart that /sci/niggers post on /lit/ to recruit people away from being humanitards. Im not going to most likely. Just go on the wiki and make a sensible chart yourself, its not difficult as its all ordered almost exactly how you’d progress through it if you were studying it in college.

>>10021996
I answer all pure math questions that arent brainlet arithmetic or graph theory

>>10021388
>It's what makes plants grow.
That's light. CO2 is just the material they use, there's a big difference in that we could be literary drowning in CO2 and plants wouldn't grow any differently.

what's the intuition behind this definition? The interval is finitely covered and the lengths are "small"? Would an associated picture be an interval with finitely many "small" intervals covering it?

In psycology, identification is the introjection of an object into your ego-ideal, right?

>>10022417
No, that interpretation is wrong because any non-degenerate interval can't ever be a zero content set. The interval isn't finitely covered, instead your set A is the one covered by FINITELY (as opposed to countably, which is important because that's the main difference with a null set) many intervals. As an easy example, just imagine a discrete set composed by {1,2,3}.

You can evidently cover the three elements of the set with three open intervals which are centered at each of the points respectively. Then you can pick any random ε, and you can state that the sum of the content of the three intervals can be smaller than that ε, no matter how small it may be. In this case it's easy because you can just say that each of your intervals' content will be ε/4, and thus the sum of the content of the whole cover will be smaller than ε.

I absolutely love these 1980s (?) mathematics videos from Open University/BBC:
https://www.youtube.com/watch?v=IkmFgNUWA4A

They seem to follow some kind of curriculum and there are numbered episodes etc. YouTube probably doesn't cover everything. Is there a place on the internet where you can watch all of them?

>>10022507

I've found a playlist here:
https://www.youtube.com/playlist?list=PLIHphqCitRAqXSEGCNeFNEpjmUdels0tO

But surely there must have been more videos produced

What's an example of a set which is closed and contains isolated points?

>>10022594
Pick any real number discrete set.

>>10021383
what is this garbage?

>>10021199
7^(1/3) > 5^(1/5) iff 7^5 > 5^3

5^3 = 125
7^5 = 49*7*49 = 343*49

So 7^(1/3) is bigger.

>>10022674
and actually you don't even need to calculate that: 7 > 5 and 5 > 3 so 7^5 > 5^3.

bup

how to be renaissance man in science?

I'm very interested in going through the following subject, just wondering if there are better options? This is purely out of interest.

https://ocw.mit.edu/courses/physics/8-01sc-classical-mechanics-fall-2016/

>>10019775
Studying the Fourir whatever they are called and got to this, I have no idea what I'm doing and there is no backtracking on the PDF. How do the limits here work (and I'm guessing they are limits because they look contained by that big thing on the left.

but why does [math]3[/math] divide [math](n + 1)! + 3[/math]
I hate burtons number theory so much

>>10023466
is this correct (assuming [math]n+1 \ge r[/math]) ?

[math]a = (n + 1)! + r[/math]

so [math]a = bq + r[/math], where [math]b=r[/math] and [math]q[/math] is all the terms in [math](n+1)![/math] except for [math]r[/math]

and thus [math]gcd((n+1)! + r, b) = gcd(b, r) = gcd(r, r) = r[/math]

>>10023486
this seems overly complicated
just note that since (n+1) >= r, (n+1)! is divisible by r. Obviously r is divisible by r so the sum of the two is also.

>>10023493
oh right that makes sense
thanks a ton

I'm trying to work out the algebraic equation of the strophoid, which is the curve given by [math]x=a\sin t, y=a\tan t(1+\sin t)[/math].

One can derive that [math]\tan t=\frac{x}{\sqrt{a^2-x^2}}[/math], giving [math]y\sqrt{a^2-x^2}=x(x+a)[/math]. Squaring both sides, you get [math]y^2(a^2-x^2)=x^2(a+x)^2[/math], but the book I'm using explicitly says that this answer is wrong because the case [math]x=-a[/math] is wonky.

So what is the correct equation?

probably gonna get shat on, probably deserving. But, where does the limits [0,4] come from on the solution?

>>10023574
The integral in question is in the y direction first, then in the x direction. How i like to think about it, is when integrating in the y direction, you go from bottom to top, that is in the case of this curve, you integrate starting from the curve y=x^2 all the way up to the curve y=2x. Then, in the x direction, it is from left to right, so in this case, from x=0 to x=2, while constrained to the area from the previous integration.

Hence, to switch up the order of integration, what you want to do is first: invert the formulas for y so that you get x= sqrt y and x=y/2. Now you want to integrate from left to right, but this time, you start from the curve x=y/2 and finish at x= sqrt y, so the integral has those boundaries. Now, you want to integrate from the bottom up while constrained to these boundaries. The boundaries are discovered by simply solving the equation 2x=x^2 which is x=0 and x=2, corresponding to y=0 and y=4. Hence, the integration in the y direction is from 0 to 4

>>10023592
dude thanks for the thought out response.

>>10023598
no worries, godspeed brainlet anon

>>10023547
never mind, the answer is that i just have to simplify the difference of squares, i just thought there was simply something fundamentally wrong with my derivation

>>10021345
Raymond Chang's book seems great, thank you. Do you think it would make much difference getting 11e over 13e?

Are you studying Chemistry at college?

>>10019775
need some help from biology anons specializing in plants. I'm not a biologist myself so idk why I was given this topic.
I have to write a project that explains why it is better to plant trees and shrubs on the western side of outdoor toilets.
what I have so far is basically:
>western facing gardens get direct sun later in the day, so the the plant's metabolic process is highest in the afternoon/evening
>this means they start using nutrients later in the day as opposed to earlier
why exactly would this be beneficial? I think that this may be because people generally use outdoor toilets around this time, but I don't see why this would make a difference.

some more info:
apparently toxic metabolites are removed by liquid crystal membranes, which need to be heated up by the sun before metabolism can occur. I don't exactly see how this connects to the topic at all.
thanks for any help anons

>>10023601
sorry to bother again, but for this similar problem. The book just gives the limits without really explaining the algebra behind it. I'm a brainlet at calculus and I am just one of those people who plugs shit in and grinds problems until I get them right and this shit isn't helping

>>10023624
>I have to write a project that explains why it is better to plant trees and shrubs on the western side of outdoor toilets.
This is a pretty obscure project, imo.

Possibly it's to break down the poop and urine (high in nitrogen) faster? I'm a biologist but not a plant biologist, sorry.

>>10023624
maybe its because of the wind direction

What’s the workload of a maths degree vs engineering?

>>10023676
I'm still at a loss for this report, but I wrote that the afternoon sun is warmest, so it heats up the soil near the plant, allowing the poop to break down faster and release nutrients right when the plant needs them most.
No idea if this is what's actually happening but it sounds plausible enough I guess

Can you represent a square with conic-like formulas? I mean, defining the points of the squares as a set of solution that fulfill a certain equation?

>>10023615
Not that anon but Chang is pretty good for entry-level/engineering-tier chemistry. I haven't read 13e (because I haven't found a PDF of it) but I guess it doesn't have any significant difference when compared to 11 or 12 apart from some new exercises.

>>10023794
You can think of it as the unit ball with respect to the infinity norm.

The mapping T : x(t) --> |x(t)| is continuous in both (a) and (b) (right?). So, X\A must be closed since it's the preimage of [1, infty) under T, and [1, infty) is closed in IR. Hence, A is open in both (a) and (b).

Is my reasoning (somewhat) correct? Would appreciate any help or hints.

>>10023799
Thanks anon. That's pretty good to hear, I'm trying to get the basics down. Majoring in Pharmacology but want to have good grounding in chemistry to the extent of doing it as a minor. I would like to think they go hand in hand.

>>10023874
(b) is false. You can show the complement of the "open" unit ball actually isn't closed (hint: find a sequence of functions that converges to 0 a.e.).

I think (a) is true, but your reasoning is insufficient. You're thinking of T as a map from [0,1] to R instead of from C[0,1]. The claim that T is continuous is essentially what they are asking you to prove.

>>10024048
Yeah, I figured my thinking was iffy, or else it would've been too simple. Really appreciate the input. Thanks!

How often do engineers, specifically bioengineers, use programming?

>>10022491
bumping my dumb question
>inb4 psychology is not a science

>>10022495
So how could I show the cantor set has zero content for a non-discrete example?

>>10022491
Yes... Though no one in psychology takes Freud very seriously (not even the few remaining dedicated psychoanalysts), so this is a lot like asking questions about Aether in a physics thread.

>>10022491
>>>/lit/

>>10021199
5rt(5)=b=(5/(b^2))^1/3 means b is smaller than a^3=7 becasuse 5/(b^2) is smaller than 7 cause b is positive

Many proofs in mathematics are proofs by contradiction. However, isn't there a huge problem with that sort of proof? Namely, that one assumes that the alternative to the contradicting case must be free of such a contradiction. But going by the Incompleteness Theorem, there is the possibility that the entire idea is built upon a flawed base and results in contradictions either way. Such proofs therefore don't seem to be rigorous at all, always allowing the possibility of perpetuating mistaken assumptions.

>>10024630
not if it is generic. and the teeorem is only negating something.
But come on i have even seen proofs by induction...

What the fuck does the dr even mean? only defining the variable you are integrating in (kinda like parcial derivatives)??

>>10024630
The program of finding out how much of mathematics can be done without LEM and some other non-effective things is called constructive mathematics. What you said about the incompleteness theorems is gibberish.

jhjhkh

>>10023655
>>10023655
I'm gonna post it in several different posts because for whatever reason I'm not passing some spam filter


I don't know exactly how to explain it otherwise since it seems pretty natural, but when I'm doing a volume integral, first thing I do is write the definition of volume of a region R: [math]V=\int_R 1\;dV=\iiint_R 1\;dzdxdy[/math]. Notice the particular order of integration: Í chose to integrate z first because you already have an equation for z in terms of x and y that is pretty easy to integrate, so the z integral will have limits 0 to 16-x^2-y^2:

[eqn]\iiint_R 1\;dzdxdy=\iint_{R'}\left(\int_0^{16-x^2-y^2} 1dz\right)dxdy=\iint_{R'}16-x^2-y^2\; dxdy[/eqn]

>>10024663

Now notice why you wanna do the x integral first: if you were to integrate y first, doing the bottom to top method i described, you have two different bottoms: one bottom is the x=0 line, then you have a second bottom which is the y=4x-2 line. So these corresponds to two different integrations and that is double the effort.

Hence, you wanna integrate left to right, corresponding to x integration: the left boundary is x=y^2/4 and the right boundary is x=(y-2)/4, hence you integrate with respect to those boundaries. Now you just have the y integral remaining, so you have to find out the intersection of 2sqrt(x)= 4x-2, which is (1,2), that is, the y integral is between 0 and 2. Hence:

[eqn]V=\int_0^2 \int_{y^2/4}^{(y-2)/4} 16-x^2-y^2 \;dxdy[/eqn]

Is this correct?

>>10024634
Differential of your "radius" dr. As a mathematician, one would use ds for differential arc length. Basically, it's a differential of the length of your path that you are integrating along, because it is a path integral.

How do i integrate f(x,y) to (x,y) being f=(xy^2,xy) in the curve
C:1. y=0 ; 0<=x<1
2. x=1; x=1
Between the points (0,0) and (1,1)
My doubth is specifically where to evaluate the antiderivatives

>>10023466
That doesn't work for n<2 tho.

>>10024733
we don't need it to

This is a simple question, I guess.

Imagine a perfect conductor inside a region of constant electric field, like in pic related. I can understand why the electric field should be zero inside the conductor because of the counteracting field created by the induced charges. But how about the field outside? Is it simply [\math] \mathbf{E_0} [\math]? What if the material isn't a perfect conductor, but a dielectric. How I'd go about calculating the field outside?

This is a simple question, I guess.

Imagine a perfect conductor inside a region of constant electric field, like in pic related. I can understand why the electric field should be zero inside the conductor because of the counteracting field created by the induced charges. But how about the field outside? Is it simply [math] \mathbf{E_0} [/math]? What if the material isn't a perfect conductor, but a dielectric. How I'd go about calculating the field outside?

>>10023391
I'm unsure what exactly you're confused about... The notation?

Keyword "Analytic": 'especially : separating something into component parts or constituent elements'

Look at f(t) in the interval 0 < t < 5....

You can see it's composed of two parts, a downward sloping line and a horizontal line. You can see in the answer they use an open brace containing the equations of two lines for separate intervals.

So for 0 < t < 3, the function is f(t) = 2-t, for 3 < t < 5 it's f(t) = -1

f(t + 5) = f(t) means it's periodic with a frequency of 1/5 = 0.2Hz (assuming t is time).

>>10024521
There's a theorem that states that compactness and measure zero implies zero content, so if you already know that (or want to prove it) you can just proof that the Cantor set has zero measure. The first theorem is rather trivial to proof if you use the Heine-Borel theorem.

To proof that the Cantor set has zero measure recall that for each step in the building process the set's measure decreases by a given amount. So for every ε you pick there will be a step in the process after which the total measure will be less than ε, and thus it can be covered by a countable number of intervals that have a measure less than ε after a step. Since the Cantor set's building process is countably infinite, no matter how small ε is, you will always find a cover that has a measure lower than that. You can make that more formal but that's kind of the intuition to build the proof.

>>10024634
To help you a bit more... dr literally points in the direction of your path as you integrate along it.

>>10024630
> Many proofs in mathematics are proofs by contradiction. However, isn't there a huge problem with that sort of proof?
No.

> Namely, that one assumes that the alternative to the contradicting case must be free of such a contradiction.
Predicate logic is incomplete, but it isn't unsound. It's impossible for both a statement and its negation to be provable.

> But going by the Incompleteness Theorem, there is the possibility that the entire idea is built upon a flawed base and results in contradictions either way.
A statement is either true or false, if it's true its negation is false and vice versa, and if it's false it can't be proven (the converse isn't true; it might be true but unprovable).

> Such proofs therefore don't seem to be rigorous at all, always allowing the possibility of perpetuating mistaken assumptions.
What does that even mean? If anything.

inside is zero, outside is E1 + E0

if it's dielectric you have to do matching at the boundary

>>10024634
> What the fuck does the dr even mean?
\int \overrightarrow{F}\cdot\overrightarrow{dr}
is equivalent to
\int \overrightarrow{F}\cdot\frac{d\overrightarrow{r}}{dt} dt
The result is independent of how r is parameterised.

IOW, at each point on the path you find the dot product between F and an infinitesimal section of the path, then sum all the infinitesimal products over the length of the path.

>>10024876
>>10025066
forgot to quote the post

>>10025070
(forgot the tags)
[eqn]\int \overrightarrow{F}\cdot\overrightarrow{dr}[/eqn]
is equivalent to
[eqn]\int \overrightarrow{F}\cdot\frac{d\overrightarrow{r}}{dt} dt[/eqn]

>>10025066
>>10025072

I'm confused as to why should the field be the sum of [math] \mathbf{E}_0 + \mathbf{E}_1 [/math]. Isn't the electric field produced by a charge distribution given by:

[math] \mathbf{E(r)} = \frac{1}{4 \pi \epsilon_0} \int_{\mathcal{V}} \frac{\rho (\mathbf{r'}}{|\mathbf{r - r'}|^3} \mathbf{r - r'} d \tau'[/math]

And, in this case, [math] \rho = 0 [/math], since the material is neutral. So shouldn't the field be null?

Hope I didn't fuck up the TeX.

>>10025100

>>10025066
>>10025072

I'm confused as to why should the field be the sum of [math] \mathbf{E}_0 + \mathbf{E}_1 [/math]. Isn't the electric field produced by a charge distribution given by:

[math] \mathbf{E(r)} = \frac{1}{4 \pi \epsilon_0} \int_{\mathcal{V}} \frac{\rho (\mathbf{r'})}{|\mathbf{r - r'}|^3} (\mathbf{r - r'}) d \tau'[/math]

And, in this case, [math] \rho = 0 [/math], since the material is neutral. So shouldn't the field be null?

>>10025104
Hmm it makes sense from the point of view of the field lines. However, I can't see it from Coulomb's law (>>10025105).

>>10025104
>>10025105
actually i'm not sure if what i said is quite right there may be a factor of 2 missing and i'm too lazy to care but you can think of it like two infinite charged planes

Does stevia cause insulin levels to spike?

Undergrad here. How can I stop making careless mistakes on exams?

>>10019775
Why is the second equality true?
Everything else is just by definition.

>>10025197
To be clear, why is the index equal to the absolute value of the determinant?

>>10025197
>>10025201
I worked it out, you just diagonalize M.

>>10025196
slow down
most exams are not written at such a tight pace that you have to blitz through the questions to finish, and even if they were it's better to answer 8/10 questions and get them all right than to answer 10/10 questions and get 4 wrong.

I need to find the time it takes to get v(t)=0.
since the blocks accelerate together I thought friction cancelled itself out so I could just do a(net)=F(applied)/m(total) and integrate but that's wrong.
I've been on this for like 4 hours and tried changing things like which mass i use or subtracting friction from F(applied) but they only give us the max static coefficient so that's fucked.
I feel like I'm missing something very obvious, help me out /sci/sters

why do people give up in quantum physics and say that we can only 'statistically' describe innards of the atoms locations, doesn't that just mean humans are too dumb to pinpoint the exact location of an electron at a certain point and that our statistics are probably very very very wrong?

>>10025274
Find the total impulse with an integral then divide by m

>>10025276
Bell's inequality implies any hidden variable theory must be nonlocal, which is equally unpalatable.

>>10025276
i'm not sure if you're trolling, but if you aren't imagine this: I drop a pebble in a pond, a wave emanates outward. Can you give me a precise location of the wave?

>>10025276
It's simply once you reach down to the level of fundamental forces, you're no longer dealing with objects in the traditional sense. They aren't like ping pong balls and planets. Instead they are waves that interact, and those interactions resolve into particles.

>>10025287
figured it out, turns out i actually just dont know how to integrate e^(-.11t) bounded from 0 to another number and i'm stupid

is this correct

>>10025541
What did he mean by this

>>10025792
if you stretch one side to infinity the other will approach 0 if you get to keep the same area

>>10025796
So y = A/x, where A is a constant. y approaches 0 as x aproaches infty. That's pretty trivial.

If I prepared a 2g sample of soil in a 200mL volumetric flask and run it through an AAS, do I need to account for molar mass if I am trying to find the g/g mass of a mineral?

I am analysing for Na, according to google converting from ppm to g/g doesn't require molar mass. If this is right, why is that?

Ppm Reading on AAS: 35 ppm (or mg/0.2L)

Converting to mg/L:
35ppm/0.2L= 175 mg/L

Converting to g/g:
175/1000 = 0.175 g Na per gram of soil
Keep second guessing myself.

>>10019984
go into electromagnetism or some shit. it's like calc 4 on meth. an ee came into our 4th year electromagnetism theory class because it and its ee counterpart were only offered every other year

Calculus 2 exam review after the fact. Did absolute shit on the test. I really have no clue what steps to take since I had a gap in college and my first semester back jumped me into part of my worst subject. I basically have to go through 8 chapters of Calculus within the next week and a half just to have a chance in this class.

Regarding the questions though, what are my options? Like what rules should I know to approach the problem? Honestly I only remembered what some of these symbols represented today ffs.

Questions 1-7

Questions 8-14

Like I need some reference so I can backtrack step by step of what to do. I plan on creating a metric fuck ton of anki cards so I can make sure I know what's what.

What are some of the pros/cons of the ELO rating system for ranking players in a purely 1v1 setting?
Why are others used instead of it?

>>10025907
Its the same isn't it?
[eqn] 35 ppm= \frac{35}{1000000}=\frac{x grams}{1 gram}[/eqn]

>>10025942
>>10025945
Above posts are me

If anyone else if going through Calc, I found some useful stuff for reference

Integral Info
https://www.rapidtables.com/math/calculus/integral.html

Derivative Info
https://www.rapidtables.com/math/calculus/derivative.html

Identities
http://2.bp.blogspot.com/-rtNBO7WFkO8/UDMeY9N88cI/AAAAAAAAAb0/VCsF-0iD84g/s1600/table-of-indefinite-integrals.gif

Integration by Parts
https://www.youtube.com/watch?v=dqaDSlYdRcs&feature=youtu.be

Integration using U Substitution
https://www.youtube.com/watch?v=qclrs-1rpKI&feature=youtu.be

Map to solving an Integral
http://i.imgur.com/11hGmBW.png

Step by Step Calculator (free)
https://www.symbolab.com/solver/step-by-step/

Also if you use Anki, there is a pretty useful browser extension for Firefox and Chrome that changes your new tab window into cards
https://github.com/corollari/ankiTab

How can I calculate how many samples I need from a population in my simple random sample? Say, I have 55,000 population and I want to conduct a 95% CI for income based on predictors. Is there a formula?

>>10026089
what is CI?

>>10026094
95%. Also, let's say that I would like to estimate to within $500 of the population mean of average income.

>>10026098
>>10026094
Sorry, the mean is 9,000

>>10025216
Thanks anon. I took your advice during the exam and I aced it. Cheers

>>10021388
they need water too, retard
globally, co2 fucks that up in a big way

>>10021402
germany+north europe does that

Is there an easy way to know the awnser to for example the fith root of 32 without memorizing the awnser?

>>10026316
Just factorize the number in its primes

>>10026316
No.

That specific example is trivial for anyone with a systems programming or digital electronics background because they'll know the low powers of two off by heart and 32=2^5.

Otherwise, you use a calculator, or use log tables or a slide rule (log(x^n)=n*log(x) => x^(1/n)=e^(log(x)/n)=e^e^(log(log(x))-log(n)).

>>10026354
Can you explain this?
Im not a native speaker so im not sure what you mean by factorizing a number in its primes

Two particles of the same mass are attached to the ends of a spring of negligible mass, initially with its relaxed length l0. The spring is stretched to twice that length and is released after there are equal and opposite velocities (v0; v0) the particles, perpendicular to the direction of the spring, such that kl02 = 6mv02, where k is the constant of spring. Calculate the radial and transverse components (vr; v) of the particle velocity as the spring returns through its relaxed length.
No one in the class could solve it.

>>10026316
2,4,8,16,32

What's a good book to start learning Genetics with?

what's the unit vector here?

Why/how do impurities in crystalline solids stop dislocations from moving around in the crystal and therefore make the material stronger? As an example beryllium copper alloy works this way.

Can someone explain to me why is velocity described as s/t and not s*t for example? Sorry I know it's a retard question, but I'm trying to understand the math behind physics.

>>10026761
Because its how much position, s presumably, changes over a certain period of time. Like you say the object went 10 meters in 3 seconds. s*t implies that the object went s distance each t time.

>>10026515
that divided by its modulus

>>10026515
Let r(t) be the position vector of the particle so
r(t) = [x(t), y(t)] = [cos(2t), -sin(2t)]. We have T=T(x,y), so that means we can compose T(x,y)=T(x(t),y(t)). Figure out for what value t0 r(t0)=[1/2, sqrt3/2]. Then calculate the partial derivative of T composed with r with respect to t at t0.

>>10026761
because the definition of (average) velocity is change in position / change in time. It is defined this way because when we talk about "speed" we are interested in how much ground is covered PER unit time.

>>10026796
Does "per" imply division or is it simply because we're calculating the slope, therefore we calculate the change in one variable for the change in other, aka (s2-s1)/(t2-t1)?

How much do we know about the far side of the moon, relative to the near side? What about the great circle which divides the two?

Has anyone ever checked the back side to look for the missing Apollo LM ascent stages?

>>10026806
>Does "per" imply division
Yes it does. The specific weight of a gas is the weight of the gas PER unit volume, hence the convenient unit of "pound/cubic foot". Speed is distance traveled PER unit time. If I know the speed v of an object and how much time t0 has elapsed from the starting position, I can say in very general terms that distance traveled = v*t0. Hence velocity has unit "foot/second".
Formally speaking: for a particle in one dimension, position s can be described as a function of time. So s=s(t) The instantaneous velocity v(t) at any time is defined the limit [s(t+dt)-s(t)]/[dt] as dt goes to 0.

>>10026806
>Does "per" imply division or is it simply because we're calculating the slope
It's quite literally both. The "per" you are probably thinking about is the discrete case. Say (as the most basic exmaple) if I have 10 apples, and I want to distribute it between 5 people equitably, I'd have two hand each person 2 apples - 2 apples PER person. Which happens to be 10 (apples)/5 (people). Saying 10 meters / 5 seconds (under the assumption that the function that describes the movement is linear) also simply means that we are moving 2 meters / 1 second (on average, at the very least), or 2 meters PER 1 second.

Naturally, when we talk about the discrete case, it isn't too intuitive to talk about slopes/tangents/rates of change. But the idea of slope is just a generalization of the concept I just mentioned above, the concept of 'x per y amount, or x distributed into y objects/units'. The difference is that now instead of dividing by a natural number of units, you divide it by a real number. Of course it means that now we can apply certain properties of the real numbers and real numbered functions, which is why it makes sense to find derivatives for example, which in this case gives us the instantaneous velocity.

>>10026829
>>10026834
Thanks!

Anyone here good with recursive definitions?
I'm stuck on one of my homework questions.

It says to "Give a recursive definition for the addition of two nonnegative integers"
I'm not really sure where to start. It feels like it would be super simple. Is it literally just saying (x+y)?

We've only just started on recursion and my teacher is a very quiet, elderly, chinese man. So I've had to try and teach myself most of this stuff on my own.
I can't seem to find any similar examples anywhere to help get a better grasp.
I learn best by watching people work through problems. So when that's not an option I struggle.

Not asking for just the answer, I really need to understand how to get to the answer more than the answer itself.

How much force *is* actually needed to pulverize a skull?

>>10026845
> Give a recursive definition for the addition of two nonnegative integers
x+0=x
x+(y+1)=(x+y)+1
You keep applying the second case with successively smaller values for the second integer until you hit the first case.

>>10026849
Average force on skull=mass*change in velocity/time of impact. It's more useful to talk about the pressure imparted at different parts of the skulls surface.

>>10026845
>Is it literally just saying (x+y)?
That doesn't fit the requirements for a recursive definition to begin with. A recursive definition consists of at least a base case (which is what you are missing first and foremost) and the general case. Since in your example the smallest valid value is 0 (the smallest non-negative integer) your base case needs to show what happens when you add a non-negative integer with 0.

The next step is showing how the addition of 2 such integers works if you know a previous result in the recursion, because the idea is that you apply the function over and over again with itself until you arrive back at your base case, after which the answer will finally be apparent. Using >>10026861 as an example (because he beat me to it), the first equation shows the addition with zero, which is the base case in the recursive definition for addition. After that, it shows how to add 2 non-negative integers, assuming you already know how to add one of them with the antecedent of the other. If you use two integer values for x and y, and apply the general case over and over again, you will eventually go back to the base case, after which you'll find the final answer.

I'm taking a quiz online and I'm pretty fucked.
I tried using the method for an exact ODE and got

(y^2)sin(x) - (x^3)y + 1/y + (x^2) = 1/e

But then realized that the eq is not exact. What should be used in this case?

>>10026861
>>10026890
So, the next iteration would be
>x+(y+2) = (x+y)+2
?
Just trying to see if I understand where you're going with it.

>>10026902
*what method

it's not linear, the argument of ln, cos and sin are non-zero so it's not homogeneous.

Am I retarded or does the partial test coming out with a 2ycos(x) and a -2ycos(x) actually make it not an exact DE?

>>10026905
The process goes like this:
>x+(y+1) = (x+y)+1
If y =/= 0 then we don't know the value of (x+y). Hence, we need to evaluate it in our general case yet again. Notice that:
>(x+y) = x+[(y-1)+1]
In this form, we can apply the general case again, so:
>x+[(y-1)+1] = [x+(y-1)]+1
I know that the notation may seem needlessly unsimplified, but formally speaking that's what we are doing. So, back at our first equation, we have, using what we got from the previous step:
>x+(y+1) = (x+y)+1 = [[x+(y-1)]+1]+1
Next you need to find [x+(y-1)], assuming (y-1) isn't 0, because in that case you would be already at the base case. And so on. If you want to see for yourself how this gets us to the final result, try substituting small values for x and y, like 2 and 3 respectively.

>So, the next iteration would be
>x+(y+2) = (x+y)+2
The reason this doesn't work is because the idea of the RECURSIVE iteration is to "advance" backwards, so the next iteration must necessarily have (y-1) in it, the next one (y-2) and so on, until the last case becomes 0 (which is just y-y), and here is where you apply the base case. Again, try the process I described above with small numbers to get a feel for what you are actually doing with the function.

Uh so far I am really liking Hefferon's intro to linear algebra book. I kind of want to buy the paperback just to have to carry around. It's only $20, I am assuming not much changes between the editions. Should I buy it?

honestly have no intuition when it comes to Lebesgue outer measure and solving problems. I don't understand how covers work in the definition. I need to see some pictures but I can't find ANY anywhere. any anons want to draw me some intuition?

So apparently this is not a context free language

a^n b^n c^n | n >= 1

But why can't I just use this grammar?

S -> aS | bS | cS | E

Pls halp

>>10027007
>I don't understand how covers work in the definition
Just look at the picture as an easy example. The union of red lines/points is the set you are trying to find the Lebesgue outer measure from. First of all, the intervals that make up your open cover are, well, open. The sequence of intervals that are supposed to cover your set CAN contain the empty set (under most definitions at least), and even when it can't the result doesn't change, it's just that the building process is more annoying. We do know beforehand that the length of any open interval of the cover that has the form (a,b) is |b-a|.

Now, note that with this cover, we are overshooting. We are covering more than what is included in the original set, so of course the outer measure is going to be inaccurate by excess.

Ignore that set for a little while. Imagine we are covering a single point x; one of our covers may include inaccurately the interval (x-1/2,x+1/2). But then another of our covers can include (x-1/4,x+1/4). Maybe the next one (x-1/8,x+1/8). And you can probably see where we are going with this.

The reason we take the infimum of all the covers is precisely this, to make the excess as small as possible. So if you have an infinite amount of covers of said point, it would be the same as just having a sequence of intervals (x-(1/2)^n,x+(1/2)^n). The length of these intervals is |x+(1/2)^n-(x-(1/2)^n)|=|(1/2)^(n-1)|, and the infimum of all these lengths will be 0. Keep in mind that not all the covers of the point will have the form I mentioned earlier, but since the ones that do are actually included in our set of all possible covers, we can build this sequence of intervals.

The same process goes for all the subsets you see in the picture, no matter whether they are open/closed/neither, you can find countable open covers because of Lindelöf's Covering Theorem. You just want to make the covers as "narrow" as possible, while still strictly containing the set.

>>10026944
The way my book shows them it does recursion like S(n) =...
Also, I think the variables the book uses are m and n.

Would I write it like "S(0) = m+0"?
Then we need to find S(n+1)? which, if I'm following this right would be S(n+1) = m + (n+1)?
If that is the case, it may be easier to solve it like this. I found a video about it (https://www.youtube.com/watch?v=LS2lFEzlZXA))
Could I follow what he does after establishing a base and the S(n+1) = m + (n+1)?
As in, subtracting n -1 "S(n+1-1) = m + (n-1+1)"?

Like I said this is all really new to me. I'm trying to get it done a little early so I can start studying for my lin algerbra test this week.

Let's say you took a bunch a rocky planets and started smashing them together to make a larger and more massive planet. Could this giant rocky planet become a star, or can a star not form from rocky material?

>>10027122
Yes, with that notation you'd have:
>S(0) = m+0 = m
Also:
>S(n+1) = m+(n+1) = (m+n)+1
However, from what you just mentioned:
>S(n+1-1) = m+(n-1+1)
Which is basically:
>S(n) = m+n
And to get a general recursive formula, you just substitute:
>S(n+1) = (m+n)+1 = S(n)+1
Keep in mind that the constant m doesn't show up in this formula. That's just because it is already accounted for in the base case, so when you eventually hit n=0, that's when it gets added into the mix.

>>10027139
So, all I'd need to write down is that? Or is there another step I'd need to include?

Just to make sure, one last time, the full solution would go
S(0) = m+0 = 0.
S(n+1) = m+(n+1) = (m+n)+1
s(n+1-1) = m+(n-1+1) = S(n) = m + n which the general would be S(n+1) = S(n)+1?

>>10027152
Yes, most of what I wrote was just an additional explanation to get the idea across, but in practice it's not all that long. Just remember that you need:
1. Base case
2. General case in terms of a previous term in the recursion. Any value you use as an input must arrive eventually at the base case, so no infinite loops
Also I don't know if your prof is more strict about how you explain the process, but as far as the raw answer goes that's correct.

How do i decide what field of science to go into?

>>10027174
Thanks, I appreciate it.
My teacher obviously knows this stuff like the back of his hand, but he is not very good at articulating this stuff. We'll go through whole problems and he doesn't really explain why he does certain things. Or, I guess I should say he assumes we already have some idea what we're doing (it's the second discreet structures class you have to take, 90% of this stuff is new). I haven't missed a class since the start of the semester, but my notes are difficult to cipher since he writes it down but doesn't tell us what it is. A lot of times it feels like I'm doing this alone.

>>10027115
Thanks, this cleared up some problems. Is there such a picture for zero content as defined >>10022417 here.

>>10027214
If you want a graphic intuition honestly https://youtu.be/cyW5z-M2yzw?t=328 is as good as it gets. Keep in mind that the concept that is shown here is for zero measure instead of zero content. The idea for zero content is the same but instead of a countable cover you are looking for a finite cover, which is immediate for compact sets w/ zero measure.

Can someone help brainlet with his calc

>>10027315
You're taking the derivative of a constant function.

>>10027345
Yeah but how can I write it so that it's not dividing by 0. The answer is 1/2

>>10027353
could maybe start with the simplification of [math]\ln(2+h)-\ln(2) = \ln(2(1+\frac{1}{2}h))-\ln(2) = \ln(2)+\ln(1+\frac{1}{2}h)-\ln(2) = \ln(1+\frac{1}{2}h) [/math] then probably dig up some version of a limit identity involving [math]e[/math].

Is there a faster way to write the Y and Z boolean equations for this circuit? It’s going to take waaaay too long to find each output, then make a K-map.

>>10027372
Start with a rotation of [math]\frac{\pi}{2}[/math] clockwise.

>>10027379
>>10027372

>>10019775
Anyone here used Poisson Superfish from the Los Alamos website? Using it because free and can't afford better software. Any online resources I should check out?

>You divide a deck of 52 cards evenly between 2 players. What is the chance that each of the players gets at least one ace?

I've been stuck on this for hours. It really seems to me that it would be 1-((48C22)/(52C26)) because that should give you the compliment of the probability of one player getting all the aces.

>>10027129
Gather enough of any matter less dense than iron in one place (about 0.05 solar masses), and its weight will start a fusion process in the center, so yes, you'll get a star.

Unless you get more than about three solar masses squeezed into a space about the size of the moon, in which case, you'll get a black hole. (Which the outward force of the fusion process normally prevents, until it runs out of shit lighter than iron to fuse.)

>>10027384
Nevermind, I found a simple way by just finding the minterms without making a truth table

>>10026816
Fairly detailed maps from about 30 miles out. The LRO managed to find the Soviet Union's old Lunokhod 1 rover, which had been lost for almost 40 years, but SFAIK they still have not found 16's LM ascent stages - though all the others are accounted for (or re-entered, in the case of 11).

Question about multiplexers.
4:1 would have 4 inputs and 1 output
and
8:1 would have 8 inputs and 1 output.
So what are select inputs? When do I add one? I was watching an example on 8:1 multiplexers and she added 8 inputs, 1 output, and then 3 select inputs

Integral of x((sqrt(x-4))

Is there an easy way to do this through distribution, or do I gotta do some gay ass u-substitution? Calc II btw.

>>10027423
The odds one player has all the aces is (1/2)^3

I'm having trouble evaluating this integral
[eqn]\int_0^{\infty}\frac{\sqrt{x}}{x^2+1}\mathrm{d}x[/eqn]
The result is supposed to be [math]\frac{\pi}{\sqrt{2}}[/math]. I tried using Residue Theorem, but my result is off by a factor of a half. I can't tell where I went wrong with calculating the residues. I keep thinking it must have been some brainlet-level mistake with adding fractions.

>>10027726
how do you tex the nice lower case differential d

>>10027736
I used \mathrm{d} for that one, though \text{d} works as well.
It helps my brainlet eyes keep track of what's going on in the integral since it's a slightly different font from the rest.

What do you put in a lab report of a practice that was supposed to give you clean results but the actual data was all over the place?

>>10027726
Well, I'm sure you just fucked up the algebra but I could transform this integral into a well known integral. Pretty easy.

First, take [math] u = \sqrt{x} [/math]. That turns the integral into [eqn] \int_0^{\infty} \frac{2x^2}{x^4 + 1} dx [/eqn]. Now take [math] u = \frac{1}{x} [/math]. That turns the integral into [eqn] \int_0^{\infty} \frac{2}{1+x^4} dx [/eqn]

Which is pretty common. And if you don't know it then at least it is easier to apply the methods of complex analysis to this integral.

>>10027835
Shit, turns out I was way off base. I considered tackling the [math]2(1+x^4)^{-1}[/math], but thought I could circumvent it by going for the original function.
Anyway I think I got it from here. Thanks.

>>10019775
have we isolated which genes help determine intelligence? Is there a difference between the races of man?

>>10027862
>>>/pol/

>>10027862
No. Yes, and likely not in a way you think.

Thank you to the anon that suggested Raymond Chang's Chemistry book. It's perfect & Chemistry is amazing.

>>10027590
This is wrong it's 1 - (25/51 * 24/50 * 23/49)

how do I find the molar enthalpy of neutralization of NaOH when mixed with H2SO4 if both solutions have different initial temperatures. DO i take the average initial temperature

>>10027014
> But why can't I just use this grammar?
Because that produces [abc]^n, i.e. any string of those 3 symbols.

Note that a^n b^n is CF, because you can just use S -> aSb | E

>>10026951
if you prefer reading out of a book, yes

if it's "only" 20$, yes

>>10027315
[eqn]\lim_{h\to 0}\frac{\log(2+h)-\log(2)}{h}=(\log(x))' (2)=\left(\frac{1}x\right)(2)=\frac12[/eqn]

>>10027588
integration by parts to get rid of the x, then u-sub to get rid of the complicated expression under the square root

>>10027726
>my result of is off by a factor of a half
are you sure you didnt forget to take off the factor of 2 coming from taking the even part of the integral? (ie in the step [math]\int_{(-\infty,\infty)}=\frac12\int_{[0,\infty)}[/math]

>>10028095
Doesn't that only work with even functions? The square root in the numerator would mess it up I think.

>>10028230
yeah, i didnt know what you did exactly with respect to integrating over what contour, i was just guessing where a factor of 2 might have come from

Where can i get matlab for free? Asking for me because trial ended and im a poor college stuent

Hey, it's me again. Normally I wouldn't double-dip in the same /sqt/ but I'm desperate because I need to get this done so I can study for my other class' exam.
I've come on another problem, the last one I need to get done. But I'm really not sure what it is even asking.

It says to "give a recursive definition for the set of all well-formed formulas of integer arithmetic, involving integers together with the arithmetic operators +,-,*,/"
I'm not sure what exactly it is even asking me.
Does it just mean the recursive definition of each one? Or all of them together?

I have these 3 linear equations
30x - 10y - 15z = 660
-10x + 60y - 50z = 20a
15x + 50y - 90z = 0
a = y-z

How do i solve this using a matrix if possible? Trying to solve a circuit using current-mesh

It has occurred to me that I must first master the intuition behind rates for most of calculus to feel intuitive to me as well.
Whats strange is the standard definition (using speed as an example)
speed = distance / time
was immediately very intuitive; I can imagine some object traveling a particular distance over a particular time, and it makes sense that that is the object's speed. It is also clear that
distance = rate * time (I could verbally explain why this is intuitive too but I won't waste your time).
What isn't so obvious to me (intuitively; algebraically it makes sense) is why
time) = distance / rate.
We take the total distance traveled and divide it in into the quantity rate. If someone could help me visualize this I would be more than grateful.

How do I evaluate
[math]\Gamma(1/2)\Gamma(4)/Gamma(9/2)[/math]
It's supposed to equal 32/35, but I don't know how.

>>10029047
Oh shit
[math]\Gamma(1/2)\Gamma(4)/\Gamma(9/2)[/math]
You get the point

>>10028900
I would first substitute a = y - z into equation 2 abd then collect like-terms, then solve the system using gaussian elimination

>>10028863
I haven't brushed up on my propositional logic in a while, but I'm pretty sure that it's not simply just asking to give a formula for the 4 operations. Do you understand what a well-formed formula is? A set of well formed formula (for the arbitrary set of logic operations at least) is naturally defined recursively. It usually goes that you have a finite set of base propositions that are each considered a well-formed formula, and from that you can build additional compound propositions using the elementary logic operations, such as negation, implication, disjunction, etc. As long as the resulting statements are also propositions and thus can be true/false, they are WFFs too.

In this case I assume that your base assumptions are that the basic operations (a+b, a*b, etc, when a,b are integers), and you need to define all operations using the composition of those. However, I'm not really sure on the division part, since a/b isn't an integer for every combination of a,b. Maybe you need to find restrictions for that. But the idea is that you don't need to proof that +,X,-,/ are WFFs. The compositions must also be (e.g. a+(c*b)*(a-c) must also be a WFF, and so on).

>>10029147
>(for the arbitrary set of logic operations at least)
Also I don't know how the fuck I ended up typing this shit, I meant 'for an arbitrary set of logic propositions under the natural logic operations'.

>>10028900
> -10x + 60y - 50z = 20a
> a = y-z
=> -10x + 60y - 50z = 20y - 20z
=> -10x + 40y - 30z = 0

So:
30 -10 -15 | 660
-10 40 -30 | 0
15 50 -90 | 0

> How do i solve this using a matrix if possible?
A.x=b => x=A^-1.b
You'd invert the 3x3 matrix and multiply by the right-hand column. By hand, it's quicker to just reduce the system to an identity matrix and the solution:

6 -2 -3 | 132 (r1/5)
-1 4 -3 | 0 (r2/10)
3 10 -18 | 0 (r3/5)

6 -2 -3 | 132
-1 4 -3 | 0
0 22 -27 | 0 (3*r2+r3)

6 -2 -3 | 132
0 22 -21 | 132 (r1+6*r2)
0 22 -27 | 0

6 -2 -3 | 132
0 22 -21 | 132
0 0 -6 | -132 (r3-r2)

6 -2 -3 | 132
0 22 -21 | 132
0 0 1 | 22 (-r3/6)

6 -2 -3 | 132
0 22 0 | 594 (r2+r3*21)
0 0 1 | 22

6 -2 -3 | 132
0 1 0 | 27 (r2/22)
0 0 1 | 22

6 0 -3 | 186 (r1+r2*2)
0 1 0 | 27
0 0 1 | 22

6 0 0 | 252 (r1+r3*3)
0 1 0 | 27
0 0 1 | 22

1 0 0 | 42 (r1/6)
0 1 0 | 27
0 0 1 | 22

=> x=42, y=27, z=22

>>10027467

Thanks for the reply. Meaningful traces of 11 should still exist though, I would think, due to the lack of an atmosphere. Given enough time, I would think that the bits and pieces could be found.

Something fun, and related: Snoopy is the only surviving flown LM ascent stage, still out there somewhere unless it collided with some other object. At least someone is trying to look for it.

https://www.space.com/13010-snoopy-nasa-lost-apollo-10-lunar-module-search.html

>>10029223

This reminds me: 10's descent stage (IIRC) is also unaccounted for, jettisoned towards the surface as the ascent stage climbed back up. 10 did not land of course, but got very close-to, to prove out the basic down-and-up sequence.

so I have in my notes:
[math]\displaystyle \sum_{n=1}^{\infty} \frac{\epsilon}{2^{n}} = \epsilon [/math] how the fuck does that work

>>10029321
[math]\displaystyle \sum_{n=1}^{\infty} \frac{1}{2^{n}} = 1[/math]

How do I go about taking the complex conjugate of the time-dependent Schrondinger equation? Pic related. Thanks.

Have I done this right?
Claim: [math]m^{*}(\emptyset)=0[/math]
Let [math]\epsilon >0 [/math] and let [math]I_{k} = \left[ -\frac{\epsilon}{2^{k+1}} , \frac{\epsilon}{2^{k+1}} \right] [/math]. Obviously, [math]\emptyset \subseteq \bigcup_{k=1}^{\infty} I_{k}[/math]. Now, [math]|I_{k}| = \epsilon[/math]. Since the outer measure is the infinum of all sums of lengths, [math]m^{*}(\emptyset) \leq 0[/math]. But also, [math]0\leq m^{*}(\emptyset)[/math], so the result follows.
Likewise for the set of reals, I can just pick a cover like
[math] [a_{k}-\frac{\epsilon}{2}, a_{k}+\frac{\epsilon}{2}] [/math] and since the sum of the lengths is divergent for all [math]\epsilon>0[/math], so outer measure is infinite. Is this reasoning sound?

>>10029357
or, for [math]\mathbb{R}[/math] should I not write the interval in that way? Would it be alright to just say for any arbitrary covering of the reals the sum of the lengths is infinite?

>>10021199
7^(1/3) ~= 2
5^(1/5) ~= (5^(1/2))^(1/2) ~= 2^(1/2)

You basically just play engineer. Didn't even need to do much desu.

>>10029147
Would you care to go through the parts you do understand? Because I'm still lost.

>>10029380
Also, prof elaborated a bit. Said that it was "One recursive definition for the set of all valid arithmetic expressions"
And gave an example(?) 123, 12+34, 45*7, 12-3*45/6+23

Still not sure what it means

>>10029386
This is done two steps, terms and formulas. First recursively define the terms, since your language has no constants, then you want to start with the variables. Say all variables are terms, this is the base case of the terms def. Now you want to close under the function symbols, like if t and s are terms then t+s is a term and so on for all your function symbols. Now you want to recursively define formulas, the base case of this is the atomic formulas. Since your language has no relation symbols there is only one kind of atomic formula, that is which is equality =. Say if t and s are terms, then s=t is an atomic formula. Now, finally, for the recursive clause you have to close formulas under the logical connectives, for instance, if A and B are formulas then A 'and' B is a formula. That's it, you should fill in the details. A nice reference for this is the open logic project pdf.

>>10019775
How do I multiply binary decimal numbers:

10110010.1101 x 1.11

I have no idea for some reason in class the decimal was pushed to the right one place when on-top of eachother.

Chem question here
What's a ligma bond exactly?

What kinds of industry jobs are available for graduates with a degree in fields like molecular biology or biochemistry?

>>10029357
You got the right idea for the first proof, there are just a few things I'd change though. First of all, the length of the intervals isn't ε, but ε/2^k. I think you mean that the sum of the length of the sequence of intervals I_k is ε, but keep in mind that you can't just sum the length of the union of all those intervals, because the union of those intervals isn't disjointed. Instead you have a sequence of different, decreasing covers, and the limit inferior of that sequence is 0 for every ε>0. So the infimum of the length of all covers will be zero, but that's because you found a sequence that prevents the outer measure to be any higher than zero.

Look at the cover you defined as the union of all L_k. The whole union is the same as L_k for k=1, because they are all contained within the first interval in the sequence.

For the second proof I don't fully understand what you mean. >>10029363
is kind of the gist of it, but of course more formal. The proof usually is that first you need to proof that if a set Y is strictly contained in a set X, then the outer measure of X must be higher than the outer measure of Y. Then Y can be any interval (0,y) where y is a positive real number (so the length of Y is y of course) and since Y is strictly contained within the real numbers (which will be the set X) then the outer measure of X must be higher than y for every possible real value. Doing it with the cover definition can get messy, since covering the whole real line with bounded open intervals isn't all that easy to work with. But please elaborate a bit more, I think I'm missing something.

Let p be prime, and let G = GL_2(Z/pZ).

Show that the set of upper triangular matrices with 1’s on the diagonal form
a p-Sylow subgroup of G. Show that the order of its normalizer is at least
p(p − 1)^2.

Ok so |G|=(p-2)^2(p+1)p
Let H be the subgroup above. |H| = p.

I've shown its a slyow p-subgroup ezpz. But the order of the normalizer part is tricky for me. I've thought about using the orbit stabilizer theorem since the stabilizer under conjugation is the normalizer.

>>10029440
special case of a sugandese bond

>>10029454
Correction

|G|=(p-1)^2(p+1)p

>>10029462
I googled everywhere and checked my book and I'm not finding anything about this "sugandese bond"
What is this special case exactly?

>>10029439
Multiply them as integers then put the point in the right place. In your example, there should be 4+2=6 digits to the right of it. (a/2^4)*(b/2^2)=a*b/2^6.

>>10029465
Ok progress. So the subgroup of upper triangular matrices has order p(p-1)^2. So if I can show that this is contained in the normalizer, I'm done.

>>10019775
I've been shitted by a bird recently... And I want to know how lucky I was. Don't ask why, it is probably the only thing I won, besides being born...
But I have some problems calculating probability. (Because I completely forgotten how to use it)
Typical bird shits once in 15 minutes. The problem is that shitting process takes infinitely small amount of time (compared to 15 minutes). Let's assume shitting takes a second. It looks like that probability is 1/900 then. Seems too small.
Okay. Now I will calculate number of birds per area of my body (bird view). It looks like I have 10-ish birds per 100 square meters. My body area is aprox. 0,25 square meters. So I will have 0,025 birds over my head and shoulders.
So, it looks like overall probability is 1/900 * 0,025 ≈ 0,0000278... To be shitted in 15 minutes?
And now I don't know what to do in order to calculate it over time. If I put this in power, it will get even smaller, and it is definitely not true. If I will multiply - It will eventually get a number that is greater than one, and it is wrong too.
What I'm doing wrong?

>>10029538
Don't know if I'm already late, but isn't that fairly simple? The product of any element of the subgroup of upper triangular matrix and the subgroup of upper triangular matrices with 1's produces the same cosets, right? If you find the product using variables in your matrices (with the required restrictions of course) you end up obtaining that both the left and right cosets contain the same elements on their diagonals, and the upper right value is different but it depends entirely on which element of the subgroup of upper triangular matrices with 1's on the main diagonal you pick, so even if the elements themselves are different, the cosets are still the same.

>>10019775

>>10029641
Yeah I was able to get it. It is pretty simple, but I guess the stress is getting to me. Thank you.

>>10029623
It looks like probability should be 1-(1-P)^t, where P is that tiny number, and t is time.
Am I right?

>>10019999

I'm doing Maths at a Uni in the UK in the 3rd year, this year we get to choose the modules we do. How important are my choices for modules? I prefer the pure and statistics modules but for the first semester I chose a differential equations module and advanced fluid dynamics. I kinda regret this now and I still have time to change, would changing to ones I find easier be worth it or is there anything important in these modules I should do them for?

>>10029758
Shit, when you mentioned modules I thought you meant the ones in algebra.
Kek, maybe check out modules instead.

I don't get why [math]\tau_{xy} EC sin \theta[/math] is the transformation of [math]\tau_{xy}[\math] originlly acting on EC. Shouldn't it be cos? As in, if [math]\theta[/math] is 0, ED is the same face as EC, so it should be the same exact stress, not 0.

>>10030250
Also what's more annoying is that for theta = 90, I think it would again be the exact stress, not 0 (complementary shear), hence sin instead of cos would be valid

I've got the following problem (pic). Basically it's a system where u is the initial signal and y is the system's answer and I need to find y/u. It should be done with Fourier transformation if that helps. I also have the answer if you need it.

flip a coin twice
X is #heads
Y is #tails
Expected values are EX=EY=1
and E(XY) = 1/2

Cov(X,Y) = E(XY)-EX*EY = -1/2

so how in the fuck is the covariance just -1/2?
If I flip 2 tails, then Y=2 & X=0
2 heads: Y=0 & X=2
1 ea: Y=X=1
where is the -1/2 variance? -1 variance would make much more sense to me. How am I supposed to interpret -1/2 here?

what is happening to me bros
i think i am legitimately developing some form of dementia or something
i keep writing phrases wrong, switching one word for another, straight omitting a word from a phrase
im only 20

>>10030419
Based on your post I can only conclude that you disregard capital letter and full stops. Apart from that everything looks in order.

>>10030419
Pot or vape?

>>10030437
i initially wrote "i think writing phrases wrong" but i noticed and corrected before posting

>>10030419
>only 20
>only
yikes.
you can't say "only x(years old)" after 12

>>10030452
i was 19 a week ago

>>10030455
happy late birthday. good for you, son.
*unsubs*

>>10030460
sorry i shouldnt be bothering the thread with my retardism

>>10020910

This is a computing and computer science question, not a maths question (kind of... everything is connected to some extend when we speak of math)

Why does i, or j (the imaginary number), equal to phase angle 90 degrees?

>>10029339
the map [math]f:i\mapsto -i[/math] has the property [math]f(z_1z_2)=f(z_1)f(z_2)[/math]

>>10030491
multiplication by i is a 90º rotation in the complex plane: pick any number [math]z\in\mathbb C[/math]. Then plot it on a graph. Then plot on the graph [math]iz[/math].

>>10020730
Real analysis, abstract algebra, differential equations, complex analysis, topology.
Number theory is fucking worthless, no clue what makes you think you should bother.

>>10030445
nothing

>heat transfer
>not an element of thermodynamics
Son, what school you go to ?

What hearing range and vocal range would a Lilliputian (1/12th the size of a human) have? What about a Brobdingnagian (12x the size of a human)?

>>10030321
Dunno about solving it via Fourier, but using Laplace you have
s.x = A.x + B.u
=> (A-s.I)x = -Bu
=> x = -(A-s.I)^-1.Bu
A-sI = [[-s,1],[-11,6-s]]
(A-sI)^-1 = [[6-s,-1],[11,-s]]/(s^2-6s+11)
=> x = [3u/(s^2-6s+11), 3su/(s^2-6s+11)]
y = [2,-5].x
= (6-15s)u/(s^2-6s+11)
=> y/u = (6-15s)/(s^2-6s+11)
The inverse transform gives
e^3t*(-(39/√2)*sin(√2t)-15*cos(√2t))
i.e. y(t) is the convolution of this with u(t). Note that the eigenvalues of A are 3±√2i.

The Fourier transform should just be s=i*w
=> y/u = (-6+15iw)/(w^2+6iw-11)

(made my own thread but trying to delete because it belongs here)

According to my Biochemistry textbook, osmolarity is determined only by the number of solutes and the size has nothing to do with it.

While I accept this as true because I'm just an undergraduate, no explanation is made and no references are made in the text as to why this happens. Intuitively, one would think that 100 Na+ ions dissolved in a solution with a semipermeable membrane would have less of an effect than 100 large sugars, just in the sense that there would be less water and more solute inside.

Is there an intuitive explanation why molecules being separate is the defining factor of osmolarity? Thanks

>>10030805
This is interesting. It could be that only the concentration of solute matters because only a specific number of water molecules interact with each solute. This way size wouldn’t matter, just the number of interactions with water would

>>10030823
That was my first thought, but how would you hydrate molecules of differing sizes? The lipid molecules aren't dissolved, sure, but it seems like there's no limit on molecule size there. The Na+ and Cl- ions (and other ions of differing size) definitely require different numbers of H2O molecules to dissolve, and one would assume that Br- would require more, K+ more, and dissolved alcohols would require drastically more than the ions.

>>10030729
Also, is this a question for this thread? Its own thread? Or /tg/?

I sleep on my side because I worry if I sleep on my back I'll swallow my tongue and die, is this fear irrational? is it possible? would I wake up?

>>10030996
while it's theoretically possible that your tongue could do this (I imagine this is not something that happens very often unless you are taking muscle relaxers or something) you would wake up coughing as soon as you started to choke on it.
There's no possibility you would die or even have problems with oxgen deprivation unless you were doped up heavily enough that you couldn't wake up to normal stimuli.

>>10030999
ok thanks this put my mind at ease a lot, going to sleep on my back tonight since side sleeping on my terrible mattress is giving me a lot of back pain, thanks again

What's the difference between A=B and A⊆B?

QM/statmech

or

biochemistry

>>10030870
Very >>>/tg/. Not enough of an audiophile to answer it myself, but some of the folks there are actually autistic enough to do the research and find out. (Though there's also a lot of audiophiles in certain threads over at >>>/g/.)

>>10031006
[math] A = B [/math] "Equality" means that the two sets share all the same elements, neither has any element that is not in the other one.
[math] A \subseteq B [/math] "Improper Subset" means that all the elements in A are in B, but it is possible for there to be elements in B that are not in A.
[math] A \subset B [/math] "Proper Subset" means that all the elements in A are in B and there must also be elements in B that are not in A

>>10030797
I guess I'm a bit late to reply but anyway. Your solution is right and the answer as well and I know how fourier and laplace function in principle. However I'm not sure how exactly to do it with the matrices. I guess I'll try again tomorrow. Thanks for the answer though.

>>10031062
Use ⊊ over ⊂ for proper subsets

>>10031088
Why?

>>10020910
found a proof for it, but it won't fit this textbox. your daily life sucks anyway, so what do you care? opfer

>>10031088
Use ⊆ for "improper" subsets and ⊂ else

Alright Einsteins, help me with this.

You have 12 different gifts. You will give some gifts to five of your friends, but you do not need to give out all 12 gifts.

If each friend can receive any number of gifts (including zero), how many different ways do you have to distribute these gifts (you don't need to give out all the gifts)?

>>10031196
are the gifts unique? and are the people unique?
if I just give out gift "a" to person "B" is that a different way of distributing the gifts then just giving gift "b" to person "A"?

>>10031219
>different gifts
The gifts are definitely unique, I'm guessing the people are too

Bruhs. I'm an amateur at this shit. This is a paint representation of a reaction going off in a gun chamber. I read that during the ejection the bullets acts on the gun and the gun acts on the bullet, which is perfectly logical. However, the rapid expanding gas pushes on both and throughout the gas the forces mediate. I'm just not sure how. Anyway, I get confused when I'm supposed to add action/reaction forces, if I wanna do an FBD of a bullet I'm supposed to write only the reaction on the bullet without the bullet's reaction on the gas, right?

Please excuse this disturbingly retarded post, hopefully you understood what I want to understand.

I'm taking a pretty basic calculus 3 (or real analysis) course, and i can't fucking find a book that talks about lim inf and lim sup for multivariable (usually just 2 tho) functions. they all seem to skip the limits and go directly to more advanced topics. Could you point me to a good book please?

>>10031351
The motion of the bullet is dictated by the forces acting upon the bullet. The motion of the gun is dictated by the forces acting upon the gun.

In the simplest case, you'd probably assume that those are equal and opposite (i.e. assuming that the gas has negligible mass).

>>10019775
i have some data on altitude and a map of my city, is there any software i can use to make contour lines or heat map from it?

>>10031404
why would you care about that? Usually you can just transform the problem into a single variable case anyways (limits are taken along lines, ie 1-dimensional vector spaces, ie the real line)

>>10031196
label your gifts 1 to 12, and choose | as a separation between your friends, gifts that aren't given away, so that for example the configuration 1 2 3 | 4 5 | | 6 7 8 9 10 | 11 | 12 means that you gave gifts 123 to friend 1, gifts 45 to frind 2, no gifts to friend 3, gifts 678910 to friend 4, gift 11 to friend 5 and gift 12 to noone. then the answer should be the amount of permutations of these 12 different gifts and those 5 |'s. I'll let you figure out what that is.

>>10031531
I don't know, i just have a bunch of excercises that i have no idea how to solve, for example how would i calculate lim inf and lim sup of x^2/(x^2+y^2) for (x,y)->(0,0) transforming it into a 1 variable case?

>>10020910
Research on wikipedia for more specific info. It's a general term referring to the problem of finding a faster way to compute the solution to a set of particularly slow-to-solve problems. These problems can be deceptively simple; for instance, checking if there is a way to divide a set of numbers into two sets whose sums are equal.

Relevance to your life... depends on your life. But many computer applications would become faster if all problems could be solved in polynomial time (P).

>>10031580
consider any line through (0,0), ie: take y=mx for any m in the real numbers (note that you will have to take the case x=0 separately). Substitute it in to your expression. Now take the limsup as x tends to 0. If you do it for arbitrary m (and x=0), then you've shown that any limsup (if it exists), is what you got.

>>10021185
They are all equilibrium problems. For each force acting on the object, break it into X and Y components (using cos and sin). You wont know all of the forces, so just sub in some variable. You know the forces in both the x and y direction have to sum to 0, so solve for the unknown variables.

For instance, in #3: the weight of the block is 2.4 N, so the y-component of the tension in the string must also be 2.4 N, or it wouldn't be sitting still. The y component of the tension is T*sin(53.1). So T must be 2.4 N / sin(53.1). Bam, now you know the net tension force.

Likewise the x-component of the tension force is T*cos(53.1), so now you can find it since you know T. The forces in the x-direction have to cancel out, so P must be equal and opposite to T*cos(53.1).

>>10031610
not the original question asker, but I'm wondering if it is always good enough to only consider lines of the form y=mx, or do you actually need to compute the liminf/limsup by using a more general line?

>>10031580
>>10031622
It can often help to consider a change of coordinate system. Here, for example, to polar coordinates: [math]\frac{x^2}{x^2+y^2} = \frac{r^2 \cos^2 \phi}{r^2} = \cos^2\phi[/math]

>>10031566
This makes the problem akin to one like "How many ways can 12 books be arranged on a shelf" the answer to which is 12! Because here we basically have 17 unique "objects" to arrange differently, thus the answer is 17!

17! may very well be correct but it seems wrong to think that this question has the same answer to "How many ways can you arrange 17 books on a shelf."

Hi, I want to learn the basics of Fluid Mechanics. I'm planning to use the cheap cchaum book and some lecture videos on youtube. Is this a good way to learn it?
The recommended textbooks on the wiki are too damn big,

>>10031699
>schaum
Fix

>>10019775
why does
[math] rw^2(cos(wt)i - sin(wt)j = -w^2r [/math] ?

Is it correct to say that an atom is a molecule (of 1 atom)?

>>10019775
Is the earth a hypersphere?

>>10031697
It should be 17!/5! since the |'s are not unique, but yes, it's basically just a mathematically more intuitive way of writing OP's problem down.

>>10032034
No, it's a hyposphere.

>>10031622
well, if you're considering the limsup at a point (a,b), then surely you have to consider lines through (a,b). In this case it's through (0,0), so there is no intercept term which makes life really easy

>>10032220
I was more wondering about lines of the form y=x^m or the likes.

>>10032296
or maybe even one should take the infimum and supremum values on the circle x^2+y^2=n and then let n go to zero.
What I'm trying to say is that I don't know what the rules are for calculating the liminf or limsup, I was wondering if there are functions that when looking at all curves y=mx not one of them will approach the actual liminf or limsup, but always a value between them, but maybe that's dumb, since close to (x,y)=(0,0) the function y=x^2 can be approached by y=0, in that these functions approach (0,0) from the same direction, but then I'm wondering about function that approach (0,0), but in no particular direction, like y=x*sin(1/x), but maybe these kind of functions alone can be used to calculate the liminf as well as the limsup, while if one uses lines of the form y=mx then all m's must be taken into account.

>>10019775
I'm new to the notation here so can someone help clarify?

I'm guessing the numbers right next to the container thingy that looks like a cleft note correspond to the limits of the y-axis while those left of them are the limits of the x-axis though its muffling me more than it should.

>>10032317
I'm pretty sure it just means that delta_n = 1 if n = 0 and that delta_n = 0 if n = +-1, +-2, ...

>>10032322
Oh, thank you so much for sorting that out. *hug*, hope the rest of your day/ night goes well!

>>10031610
I'm pretty sure i've heard the professor say that we would have to try *every* possible curve and not just straight lines to be sure the limit exists, as this guy >>10031622 says. Is that not the case?

>>10031681
I know about that trick but in this case how does it help? What does it even mean to calculate lim inf(cos^2(phi)) for (x,y)->0? r goes to 0 but phi is undefined i think?
I guess my problem is that i don't even know the definition of lim sup and lim inf for anything other than a X_n sequence for n->+infinity

>>10019775
absolutely stuck

i just can't figure out a way to introduce a contradiction- like, at all

Why does {Λ}* = O*?

>>10032506
With O meaning empty set of course. (sorry for newfaggotry)

>>10032296
>lines
>of the form y=x^m
cmon bro

I know what you mean though. Why lines? Why not some arbitrary curve? Well, the point is, does it matter? When you consider the limit along any line y=mx as I explained, you're essentially looking at the composition f(x,mx) and see how it behaves close to 0 (ie: for small values of x). So now imagine you have a continuous curve passing through (0,0) along which you want to take the limit (why does it need to be continuous? hint: if not, how do you take the limit?). What will the limit be like close to 0? Does it matter? Every one of those points still lies on a line passing through 0, and the value is not going to be any different.

If the curve is differentiable, then you could take a linear approximation using Taylor's theorem easily, and then use the line method.

>>10032625
ye, that seems to make a whole lot of sense. Though, I'm still wondering if there is some formal proof that there isn't some space alien function for which the line method doesn't work

>>10032625
I'm the original asker, that makes sense but i'm still not convinced. For example how would that method work with the function in pic related?
Sorry if it's retarted

I'm >>10032735
>>10032754
I guess here it wouldn't even matter which direction you come from, since for every m, and an arbitrarily small epsilon there is an x1,x2 < epsilon such that mx1 = sin(1/x1) and mx2 != sin(1/x2) so that the liminf=0 and limsup=1 over any of these lines, but I could be misunderstanding the definition of limsup and liminf.

>>10032754
well, that's not continuous at 0 so the limit doesn't exist there

>>10029449
So what would be a better choice of I_k in the first proof? a_k-ε/2, a_k+ε/2 ?

>>10031978
Bump. I really wouldn't ask this if it wasn't such an embarrassing thing to ask elsewhere.

>>10033114
The family of covers you decided to analyze is fine. Or just I_k=[-(1/2^k),1/2^k]. It's just the argument that needs to be improved. What is the exact definition your textbook/class uses anyways? Trying to bound the outer measure by an arbitrary ε>0 is not wrong, but since you already have a sequence of intervals that decreases into a set with measure zero it's also not necessary, unless that's what the definition you are using requires. If anything you can just say the for any ε>0 there will be an interval in the sequence with a smaller length, since the lengths of those covers converges to 0. Then by transitivity the empty set must also have a measure smaller than ε, and you are done. No need to find the length of the union of those intervals, since each of them is an independent cover of the empty set.

>>10033166
The definition we use for outer measure is as follows: Let [math]A\subseteq \mathbb{R}[/math]. The Lebesgue outer masure of [math]A[/math] is [math]m^{*}(A) = \inf \{ \sum |I_{n}| : \hspace{0.1cm}\text{where}\hspace{0.1cm} \{ I_{n} \} \hspace{0.1cm} \text{is a closed covering of A}\} [/math]

>>10033176
>closed covering of A
Consider the measure of any singleton {x}, where x is a real number of course. That set contains the empty set, is closed and has measure 0 (assuming you proved that, well). That way you also avoid having to work with infinite covers.

>>10033203
new
>>10033203