kizzmybutt

give context
describe your thought process if you're stuck
try wolframalpha.com and stackexchange.com
How To Ask Questions The Smart Way catb.org/~esr/faqs/smart-questions.html

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girlDog

What are some applications of Bézout's theorem to basic projective geometry. Two degree one curves intersecting seems kind of trivial, since in a projective space anyways parallel lines will meet.

New_Cliche

Ayyyyyy
How many things are emergent properties of the interactions of fundamental particles? (Eg distance, gravity)
Is there a lower level of reality than fundamental particles, and what is it?

Illusionz

Basically, what things are the results of the interaction/nature of fundamental particles, and how do you get from them to the thing that is an emergent property of them?

Need_TLC

think I found a proposition: If $f(x,y)$ is a homogenous polynomial of degree $d$, then $Z(f)\subseteq \mathbb{F}P^{n}$ has at most $d$ points.
stinks of Bézout.

CodeBuns

how did we get pictures from voyager?

Ignoramus

Like this

Emberburn

I wanted to get in a CompSci college but barely failed the exam and got redistributed in a traffic and transport college in the same uni ,my aunt who is a teacher at that uni (and teached me math and phys for the exam because I was a brainlet in highschool) and my family all told me it would be better to stay in that college after a year has passed so the year I spent there would not be wasted ,so I did ,even though I felt it was a huge mistake now almost two years later I feel like this degree can't get me a good rewarding job and that thought took all my motivation away I can't sleep and I feel like I don't have anything to get up in the morning for
So I want to ask should I quit this college and go for the compsci degree and a job in programming?And what's a good job that I can aim for with a traffic and transportation degree?

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Emberfire

Why the fuck are trig Integrals so complicated

they aren't
just memorize reduction formulas

King_Martha

Me programmer irl
You should be programming on your spare time. Learn programming on your own, then develop applications for traffic and transportation. Make simulations: read papers on algorithms for effecient traffic signals, and implement them. Going to college for it is pointless because if you're going to be a successful programmer, you need to be able to do it on your own.

You might not be intelligent enough to pick up programming. Its too complex for people who don't have a knack for it to be able to pick up easily and use proficiently. Give it a shot. You may realize that it's not what you want to do. It happens to a lot of people who get into CS.

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Poker_Star

That's what people tell me but to get a job you have to have some sort of certification people trust in and that HR recognizes how would I get that without a college?

ZeroReborn

memorizing when Euler gave us $e^{ix} = \cos(x) + i\sin(x)$
Engineer detected.

Harmless_Venom

Theorically how much energy would it take to manipulate time?

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look faggot, if you cant pass entrance exams you cant program.

Crazy_Nice

to be fair it was a kind of math I haven't been teached and I almost got in

Supergrass

What are some good resources for learning R?

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takes2long

you _are_ manipulating time right now
you'll need a lot of energy to noticeably manipulate it though

RavySnake

suppose that $r\colon[a,b]\longrightarrow C$ is a parametrisation of a curve C and r* is the parametrisation of C in the opposite direction, that is r*(t)=r(a+b-t). Then what is [eqn]\int_C F\cdot dr^*\,?[/eqn]
I think it's $\displaystyle\int_a^b F(r(a+b-t))\cdot r'(a+b-t)dt$, which follows from the definition(?). but that's wrong and i cant see how it would be otherwise.

Boy_vs_Girl

I begin the dreaded Series portion of calc 2 next week. What do you guys recommend to get an upper edge

girlDog

Who "dreads" series? Series as a subject are way easier than antiderivatives.
There's like 4 or 5 convergence tests you have to know, just practice enough to develop a feeling for when each one works and the rest is autopilot.

hairygrape

be comfortable with simplifying factorials in different ways, learn each test, do a bunch of practice. it's really no worse than the rest of the course.

Skullbone

Think you're a minus sign off. See tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtI.aspx explanation above third green box

Raving_Cute

I need some form of strictly noncommutative algebra (i.e. a*b != b*a for all distinct (a,b)) for an proof attempt at representing combinatorical objects algebratically

The only noncommutative algebra I know of that can be calculated easily are matrices but they don't suit my needs. Are there any other such systems?

massdebater

due to representation theory you're going to find matrices if you want regardless

Poker_Star

Can anyone give me some links to me that prove IQ matters?

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Snarelure
ZeroReborn

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StrangeWizard

There is something about vector analysis that is just so damn sexy.

haveahappyday

sorry i just realised it's not clear from my question, but F is a vector field

cum2soon

What have you tried?

farquit

how do i prove that the intersection of two distinct planes is a straight line?
i've started by considering two distinct planes $n_1\cdot (x-x_0)=0\,,\quad n_2\cdot(x-x_0)=0$ where n1 x n2 is non-zero, but that's it, and idk how to proceed. I imagine the cross product will be involved since the direction of the line is ±n1 x n2, but can't think how to use it with where i am now

I think i could prove it by substituting n=(a,b,c), x0=(x0,y0,z0),... etc. and solving the two equations, but i'd prefer not to

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Re learning math from the ground up and stuck on a problem.

5/16 • 1/10 - 1/32

The answer is 0, but I got 40/5 for some retarded reason.

haveahappyday

how do i prove that the intersection of two distinct planes is a straight line?
You can not, since it is not true.

w8t4u

wtf
what's a counterexample?
post how you got 40/5

Carnalpleasure

wtf
what's a counterexample?
Draw some pictures of planes.

Emberburn

What is did I get the Least common denominator of 16, 10 and 32 and got 160.

5•10 50/160

1•10 10/160

1•5. 5/160

Then I multiplied 10/160 • 50/160
And got 500/160.

500/160 - 5/160 and got 495/160
Simplified that and now I’m at 99/32.

It’s a different answer but I’m still fucking up

Nojokur

I'm not sure where the 50/160, 10/160 and 5/160 came from, but the answer is
$\dfrac{5}{16}\cdot\dfrac{1}{10}=\dfrac{5\cdot 1}{10\cdot 16}=\color{blue}{\dfrac{5}{160}}$

and then

\begin{align}\dfrac{5}{16}\cdot\dfrac{1}{10}-\dfrac{1}{32}&=\color{blue}{\dfrac{5}{160}}-\dfrac{1}{32}\\ &=\dfrac{5}{160}-\dfrac{5}{32\cdot 5}\\&=\dfrac{5}{160}-\dfrac{5}{160}\end{align}

Stark_Naked

What I did is just find the least common factor of the three numbers and multiplied the numerator and denominator respectively then multiplied and subtracted.

I’m not sure why I did that.

When you did

5/160 - 1/32 why did you multiply by 5 ?
What’s the meaning behind that

w8t4u

Another thing I did is that instead of just multiplying I looked for a common denominator.

I didn’t know you only do that when you’re adding and subtracting fractions.

Snarelure

You can only subtract terms if they have the same denominator. So i multiplied the denominator by 5 since 32*5=160. The fraction is still the same though since i just multiplied by 1:
$\dfrac{1}{32}=\dfrac{1}{32}\cdot 1=\dfrac{1}{32}\cdot\dfrac{5}{5}=\dfrac{5}{32\cdot 5}$

Emberfire

nice. how do you make it blue?

Flameblow

Can somebody explain how to find a formula for (x^n-1)/(x-1)?

LuckyDusty

Can somebody explain how to find a formula for (x^n-1)/(x-1)?
en.wikipedia.org/wiki/Polynomial_long_division

kizzmybutt

Ok, I'm not trying to be a brainlet right now, but I still can't figure it out, how do I deal with the exponent being n?

Gigastrength

Ok, I'm not trying to be a brainlet right now, but I still can't figure it out, how do I deal with the exponent being n?
en.wikipedia.org/wiki/Mathematical_induction

Burnblaze

So induction would be used to do the division?

BunnyJinx

Try it with n = 1, 2, 3, etc until you notice a pattern, then assume it's true for a general n and show how that assumption leads to it being true for n + 1.

100lbs of potato if it is 98% water and 1% solid stuff and by mathematics removing 2% of the water would mean it would weigh 98 lbs right? No! If you try out the math by solving it it always ends up with 98.89898989...% and that is wrong. Instead the right answer is it would weigh 48lbs and that's because if you divide it the answer would be "exactly" 98%.

How does removing 2% of a mass decrease it's weight over 50%.

Pls explain

girlDog

[eqn]\sum\limits_{i=1}^n i^{n-1} = x^{n-1}+x^{n-2}+...+x^2+x+1[/eqn], would this be right for the formula?

Playboyize

The right hand side looks good but does not correspond to what you've written on the left.

Sharpcharm

Thanks, I see the issue

Evilember

HAVN'T SLEPT FOR 30 HOURS
WENT THROUGH 400 PAGES OF LINEAR ALGEBRA BOOK
2 LITERS OF ENERGY DRINKS CONSUMED

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CodeBuns

You’ll forget it all for cramming

Nude_Bikergirl

Why do we reduce to 9? Is this essentially just cross division to reduce to its lowest terms?

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girlDog

You can rewrite the top as (9)*1*11/(9)*4*44, the 9s cancel each other out.

PackManBrainlure

Not quite understanding you

VisualMaster

$\left(\frac{9\cdot11}{44\cdot36}\right)=\left(\frac{9}{36}\cdot\frac{11}{44}\right)$

New_Cliche

But 9•11 is 99? I’m definitely missing something here sorry

5mileys

$\frac{9}{36}=\frac{9}{9\left(4\right)}$

I’m too retarded, I’ll figure it out eventually thanks though

Techpill

Wait are you just flipping the denominators and then just dividing the numerator and denominator?

Methnerd

So I am finishing up Real Analysis and thinking about being a teacher's assistant next semester for calc, but I have never learned integration by parts and my teacher keeps asking questions that need it/ What is the single best way to learn it?

Emberburn

you can do all other types of integration??

VisualMaster

Pretty much, I self-studied them, but I hated doing integration by parts so I didn't practice it enough to learn it well.

FastChef

When it says the subsapce given by the equation, does it mean the orthogonal complement of the vector (1,1,1)? That's what I assumed and when I worked it out I ended up with nice numbers typical of a contrived example question but I have a feeling i've been going in the complete wrong direction. I'll post my working in a sec in case it helps anyone

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I'm a messy cunt sorry

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CouchChiller

/a/ was able to make a cripple fucking simulator. What have you done with your life?

Would Veeky Forums be interested in working with some autists from /g/ to develop a FOSS android? We're talking from the ground up - motor skills, intelligence, human interaction, and human "interaction". We were thinking Veeky Forums could especially help with materials science, but volunteers with any knowledge/skills/ambition are welcome.

Stark_Naked

Does anyone know of any YouTube videos to help a brainlet learn this stuff? I really want to understand it

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idontknow

What does it mean when a problem asks me to find the thermodynamic identity for a certain variable? I thought the thermodynamic identity was just dU = TdS - PdV.

CouchChiller

About to finish my first two semesters for my 4 semesters MS (EE). I have the choice of a few internships over my subject at good companies or doing research at my university over the summer, should I just do the internship if I am not 100% on whether I want to pursue a PhD after my MS yet? I plan to go into industry whether I get an MS or PhD anyways.

MPmaster

What are some good freeware circuit drawing and/or simulation software for Mac OS X?
I tried Spice, which works fine for windows, but is absolute ass on Mac. No fucking icons, everything's done through keyboard shortcuts, and running a simulation is painful as all hell.
Any and all alternatives are welcome

Spamalot

Spazyfool

Consider this:

$\frac{2}{1} * \frac{1}{2} = 1$
essentially 2*1/2
so that's why we can simplify by reducing 2s

haveahappyday

Shouldn't the directions on the middle spring be flipped? As in I push m to the right, middle spring is compressed; meanwhile push m2 to the right, resulting in a restoring force to the left in the middle spring

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JunkTop

If b is a power of a you have a problem.

Need_TLC

You are correct. You are basically subtracting out the mean of x y and z, giving you three new numbers having zero mean

Raving_Cute

Shouldn't the directions on the middle spring be flipped?
no, you could flip the direction of the arrows if you wanted to, but then you would also have to change the sign of the equation
As in I push m to the right, middle spring is compressed; meanwhile push m2 to the right, resulting in a restoring force to the left in the middle spring
the spring is only ever going to push the masses apart or pull the masses together, it doesn't make any sense for the spring to be pushing one and pulling the other

SomethingNew

how do i prove that the intersection of two distinct planes is a straight line?
You don't. Consider the case in which the planes are the same, or the case in which they are parallel.

Without thinking about it, my guess is that you consider a system of linear equations which describe where the planes are.
From that you will conclude that, either they intersect everywhere, have a one dimensional solution set, or there is no solution.

Lunatick

what's the proper definition of a linear ODE?
i've seen a lot of definitions like it's an ODE where "all y terms appear in a linear manner", "all y terms appear to a power no higher than 1", etc., but they confuse me and don't seem particularly rigorous

So I've got a particle theory module at uni that frequently uses Einstein summation, the spacetime metric, spacetime vectors and things like that. But despite mentioning lowering and raising indices like
[eqn]V^\mu=\eta^{\mu\nu}V_\nu[/eqn][eqn]V_\mu=\eta_{\mu\nu}V^\nu[/eqn]
It is never explained what the difference is between the index being above or below. It's not really explained what its even called so I have no idea what to google/youtube to learn more. Apparently I might want to raise or lower the indices to help simplify equations but I have zero intuition as to when or why I might do this

5mileys

I think I'm retarded, max is supposed to be a non-linear function, but using the usual proof of linearity, it keeps coming up as linear. Is max() not what I think it is?(the highest value of a matrix)

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likme

surely the best way to prove this is to find a counter example?

Techpill

It's just that I'm trying with different numbers and I'm getting the same result. Using max() on wolfram alpha also wields the same results, yet everywhere I see, they say that max() is nonlinear. As usual, I'm probably doing something wrong, and I don't know what.

TreeEater

max is supposed to be a non-linear function
Why?

TurtleCat

A subscript index is a covariant vector and a superscript is a contravariant vector.
They are different ways to discribe the same vector in non euclidean space.
You can read about it in every book about differential geometry and tensors, they will do a way better job explaining it than I could.

VisualMaster

It isn't?

StrangeWizard

I got it, as usual, I'm dumb. the problem is that I kept using the same place to hold the highest number.

King_Martha

Do you anons find any value in it? If yes do any of you know some literature that I could read to have an understanding of how the human brain really functions? Can be a thesis or a book, anything to know more if theoretically you'd want to create a true artificial intelligence for example.
I know we are far from knowing enough to achieve that but I'd like to know were science currently is on the subject

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w8t4u

So what you’re describing is cross cancellation?

massdebater

$x_0=\gamma(0), x_\alpha^+=\gamma(\alpha), x_\alpha^-=\gamma(-\alpha)$, these three points define a circle, so you can write some equation and the center $c_\alpha$, then differentiate this equation and you get two zeroes, $y_1,y_2$, then differentiate again and you get a zero $z_0$, if $\alpha\rightarrow 0$ then all those points tend to $x_0$ and the equations tell you they satisfy exactly what you wanted in the limit, or something.
Been a while since I took differential geometry, often you just have to find the right equation and you get the result you want right away.

likme

Well, the cross-cancellation is just a handy term but i dont think it is an actual thing.

Basically, if you multiple a number by X and then divide by X there is no point doing so, thus you can remove X's

Firespawn

Ahh thank you

TechHater

hey a dumb physicist told me once that subzero temperatures (Celsius) are 'hot'. What did he mean by that??

Spamalot

What should I do if I am interested in CS but do not want to study baseline engineering?

cum2soon

What do you call this letter?

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Inmate

xi
(pronounced "zee")

idontknow

Study CS without studying baseline engineering

Emberfire

Does this not belong in this thread ? Should I make another one just for this ?

BinaryMan

well it would certainly be better than the constant iq thread spam

Spamalot

Self teach yourself CS.

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cum2soon

Now I have become true Pajet

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BlogWobbles

Should I put MATLAB as a language or software on my resume? I've never really "coded" with it in my eyes, just used it for matrix calculations, graphs, laser things, and as a big calculator basically

takes2long

Your parametrization is a joke: try the parametrisation is r*(t)= r(b-t), where t goes from 0 to a.

Notice that the minus sign comes out of the derivative r'.

Evilember

Notice that the positive curvature at 0 implies that it is not a point of inflexion, in particular, in a small neighbourhood of beta there exists a circle. You could show this by, say, doing a Taylor expansion of Beta around 0 up to second order.
Uniqueness is obvious. The rest comes from the first part.

ZeroReborn

can someone explain to me what the fuck a pullback of multilinear form represents

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Raving_Cute

How many of you guys have taken the physics GRE? How is it overall? How much did you study for it?

I know it's mostly stuff from intro classes, so I'm gonna go back this summer (between sophomore and junior) and try to basically be able to recite Halliday, Resnick, and Walker in my sleep. But what did you guys do about the special topics that you might not have covered by the end of you're junior year.

Thanks boys and good luck.

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Inmate

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kizzmybutt

Why you people don't just ask Alexa?

TurtleCat

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AwesomeTucker

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Spamalot

Does anyone know why when i measured the attenuation coefficient of optical fibre across varying wavelengths of light, my findings show shorter wavelengths (green LED 550nm) outperform longer wavelengths (IR LED 750nm) by an order of magnitude or more?

The reason i ask is that a little research has shown the most common wavelengths of light used in long distance fibre optic transmission are all in the IR range (850 nm, 1300 nm, and 1550 nm).

Any help would be greatly appreciated as all google wants to tell me is what i already know.

takes2long

you should combine programming languages and software into 'IT skills'

Bidwell

it's 'ksee', friend

likme

I took it last year. It's pretty gnarly. I studied for about two months prior. There's five practice tests available online. You gotta take all of those. Make flashcards based off of the questions you miss. There are also official flashcards online somewhere. Learn elimination strategies. This was my regimen and I got a 910. Good luck user.

TurtleCat

1/infinity
Any amount is enough.

RumChicken

Thanks man.

StonedTime

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Spazyfool

nice work user

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Booteefool

doesnt a matrix of orthogonal projection have to be its own square

Harmless_Venom

Find the Σ0 → Λ0 transition matrix element < Λ0 ↑ | sumi=1 µi(σz)i|Σ0 ↑>. Express the results in terms of µu, µd, µs.

Our professor never went over how to transition between two baryons, and only used a similar technique for finding the protons magnetic moment. So I am very lost.

haveahappyday

Is there something like the Luxel Body Badges but digital?
I need something that records cumulative dosage in mrem(dosimeter).
Rads and Gy are fine too as I can covert it manually if needed.

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hairygrape

Trying to find good resources on $\sigma -$compact sets that are not compact. I know that $\mathbb{R}^{n}$ satisfies the above, but I can't seem to find any other examples. How can I construct such metric spaces?

Need_TLC

Your parametrization is a joke: try the parametrisation is r*(t)= r(b-t), where t goes from 0 to a.
don't think it really matters. they're both the same and that's the one that was in the text. it also feels more natural to define the reverse parametrisation that way since r and r* have the same domain.
Notice that the minus sign comes out of the derivative r'.
why though? i imagine it's the chain rule, but how exactly do i apply it there?

Flameblow

you are right sorry, i was in a hurry when i made that one

Need_TLC

This time again, but correct, was in a hurry so didn't think right

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Booteefool

usually you have to divide by dx where x is some variable, usually time, and then you use the definitions and maybe integrals to find the variable

kizzmybutt

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Playboyize

thanks bud, does the method youre using have a name, we got the same answer but it looks a lot cleaner than my working

Evilember

Well idk just logic i guess? Also note I wrote the map P the wrong way round in the pic lol.

How I thought of approaching the problem:

I need to describe a linear map between R^3 and the subspace W
the map has to project onto the subspace W
any linear map can be described by its effect on the basis (of R^3)
what does projection mean in this setting? Well, it means taking off the component of the vector that goes out of the plane
The normal vector to the plane is the "purest" element that goes out of the plane (by definition) so I take that one
Take a basis vector, and subtract the "normalness" out of it
ok great, now I know how the projection acts on e_1, now i should calculate e_2 and e_3
Every linear map has an associated matrix, call it A. Then Ae_1 = (what i calculated for e_1). Repeat for the other two and you can discern A.
In this case, since I could use the standard basis (and not just an arbitrary one), it is pretty easy to see what A is.

TurtleCat

interesting, thanks for your help dood

StonedTime

ask more math question stupid anons

t. bored phd student

StonedTime

How do I go into evaluating $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\rho{e}^{-\rho}sin\phi d\rho d\theta d\phi$ with all the ugly indeterminate forms?

Spazyfool

this integral does not converge

what do you think it means to integrate dphi or dtheta from -infinity to infinity? Generally they are limited to 0 to 2pi and 0 to pi.
see pic

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VisualMaster

-inf to inf
uh what are you doing

iluvmen

Plugging my /wsr/ thread. It's about identifying the name of a [thing] related to math and programming.

cum2soon

Can someone explain why the dy/dx in partial notation is 0? This is for implicit differentiation btw

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likme

y doesn't depend on x

hairygrape

Optics or Materials science? Currently just doing BS in Physics, but plan to do PhD in the chosen field.

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what does that mean desu? im dumb af can u explain

Snarelure

en.wikipedia.org/wiki/Partial_derivative#Geometry

See their example for a cone. In your case y(x,z)=y. i.e. it doesn't depend on x. Say for the unit circle y=+-sqrt(1-x^2), so y depends on x and its partial derivative is -x/sqrt(1-x^2)

PurpleCharger

so if it doesnt depend on x does that mean the partial change of y is 0?

Boy_vs_Girl

Ok yeah I'm a retard, that makes sense. I also meant for the left half of the integrand to be rho^3 instead of just rho.

But when integrating by parts with respect to rho, I get
$\int_{0}^{\infty}{\rho}^{3}{e}^{-\rho}d\rho \rightarrow \left ( 0 \right ) + 3\int_{0}^{\infty}{\rho}^{2}{e}^{-\rho}d\rho \rightarrow (0) + 6\int_{0}^{\infty}\rho{e}^{-\rho}d\rho \rightarrow (0) + 6\int_{0}^{\infty}{e}^{-\rho}d\rho = 6$

And then solving for the other two integrals, I get
$6\int_{0}^{\pi}\int_{0}^{2\pi}sin\phi d\theta d\phi \rightarrow 12\pi\int_{0}^{\pi}sin\phi d\phi = 24\pi$.

But the ((((((((textbook)))))))) says the answer is only $2\pi$. So are the coefficients not actually part of the problem?

CodeBuns

how much harder does calc 3 get after partial derivatives

and if possible, how much harder does phys 2 get after induction for magnetic fields

Illusionz

Is quora just a /sqt/ website?

SomethingNew

Algebrists are useless monkeys.

King_Martha

Can anyone explain pic related to me?
I guess I understand a, b, c and d, but I'm lost at e.
is e just some illustration that does not have to be correct and you are actually required to multiply whatever functions are behind $g(\lambda)$ and $h(t-\lambda)$ and plot the result, or is there a graphical way to multiply to curves?
It looks like they are added, but if I measure it, it does not really fit, but multiplying the singe values is off way more.
I would think that there should be some graphical way (book's from the 60s where they did integrals by drawing them, cutting them out and wheighting the paper), but if there is, I can't find it.
Any help?

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TalkBomber

Not technically a science question but does anyone have the pic with Cauchy making an interjection about calculus?

VisualMaster

Its just a schematic drawing showing the general principle of convolution. The Wikipedia page on it is actually quite good.
Convolution is almost always solved by Fourier transforms. Convolution in real space is multiplication in Fourier space.

TechHater

You can approximately multiply curves if you know where y = 0 and y = 1 are, and know about the derivatives of the curves.
The graph in e isn't supposed to be exactly correct, it's just a qualitative thing. You can see that gh is about right, it has the right max and min, correct sign of the derivative, etc.

PurpleCharger

im not american so i did most of that in highschool /first year, but not very - unless you refrain from thinking: you most likely will need to set up 3D integrals and there's no general procedure to do it, although it's relatively easy for HW/exam problems to see the problem geometrically and understand that there's a canonical choice of parametrisation. This obviously will not make sense to you now, so I'll just say: think geometrically and not just think everything is algorithms.

As to phys 2 i have no clue what comes after but if its intro to modern physics (ie relativity and baby QM) then it's basically plug and chug integrals for the latter (easy shit), or just play around for the former.

Nojokur

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haveahappyday

The graph in e isn't supposed to be exactly correct, it's just a qualitative thing.
Thought as much, but I thought (hoped…) that you could simply multiply curves graphically, and the shown graph is quite close to a graphical addition, which seemed off to me.

Dreamworx

It's not possible in my country i.e. India

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Methshot

I'm hypoglycemic. Why do I feel drunk when I have low blood sugar? I make terrible decisions and easily get angry. What's going on in my brain?

5mileys

Is there any easy way to prove addition is commutative in PA without using induction?

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I have difficulty when it comes to figuring problems like these. I don't feel confident in my ability to convert the given information to a quadratic equation. Is that indicative of poor critical thinking skills? If so, is that something which can be improved? Any recommendations on how?

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