/SQT/- STUPID QUESTIONS

I guess I'll start.
How do I figure out substitutions and all that stuff while doing integration?
Everytime I see a new problem I can't for the life of me figure out what to do to solve it but then I see the solution and it makes sense.But there is no way I could predict the substitutions and all the tweaking stuff.
Tl;Dr I suck at integration and need to learn it ASAP.
Pls tell me how to.

Best way to study math? How much repeating concepts until they stick properly?

What is even the meaning of these in the Laplace notation? Let's that I have

y'= -2x+4y

With initial y(0)=1

y'+2x-4y=0
L{y'}+2L{x}-4L{y}=L{0}

I understand everything up to here, but then

[sY(s)-y(0)]+2X(s)-4Y(s)=0

What is it with the Y(s) X(s) things? I know how to Laplace simple stuff and the results are always s's, but where are these giant X and Y even coming from?

>>8522287
X(s)=L{x(t)} etc

The frequency-domain version of the function uses the upper-case letter.

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Can someone explain to me why when Apostol used the first integral here, it became a negative value? Shouldn't it be?
[math]\int_{-1}^{0}(\frac{x^3}{4}-x)dx = \frac{1}{4}.\frac{(-1)^4}{4}.\frac{(-1)^2}{2}[/math]?
Also, why when I try to use absolute value as it says below, the answer is not the same?

>>8522412

I don't understand your buzzwords, but you seem to be implying that this is like the dx dy thing that's only there to let you know what are you integrating. Is this thing X(s) only there to remind me that I'm Laplacing x and not y?

>>8522444
I mean:
[math]\int_{-1}^{0}(\frac{x^3}{4}-x)dx = \frac{1}{4}.\frac{(-1)^4}{4}+\frac{(-1)^2}{2}[/math]*

>>8522447
Fuck, I hate latex.
[math]\int_{-1}^{0}(\frac{x^3}{4}-x)dx = \frac{1}{4}.\frac{(-1)^4}{4}-\frac{(-1)^2}{2}[/math]
Now I wrote it right.

>>8522445
It's often convenient to give the transformed functions names, i.e. X(s) rather than L{x(t)}, particularly if you're going to use something other than just X(s), e.g. X'(s) or X(s-s0), etc.

If you see x(t) and X(s), you can assume that these are the time-domain and frequency-domain representations of the same function, i.e. X(s)=L{x(t)}, x(t)=L^-1{X(s)}.

>>8522450
it should be
integral_{-1 to 0} x^3/4-x
= ((1/4)x^4/4-(1/2)x^2)_{-1 to 0}
=((1/4)0-(1/2)0)-((1/4)(1/4)-1/2(1))
=-1/16+1/2

>>8522234
Do you mind simplifying further please.

I'm pretty retarded.

>>8522450
It is negative. Looks like they omitted a '0' which might have made it more clear. The evaluated integral is negative since you're evaluating from 0 to -1. You subtract the value of the integral at -1 from the value at 0 -- which of course is zero. So the answer is: 0 - (what you said) = negative(what you said).

Does it make sense that the following must be co-prime for all k ?
[math] (2k+1, 2k+3, 2k+5, 2k+7) [/math]
If so, how could I prove it?

>>8522495
>>8522480
I see, thanks.

i assume you mean coprime as a 4-tuple, since otherwise 2k+1 and 2k+7 are not coprime when k is 1

if n divides all four then n divides (2k+7)-(2k+5)=2, so n is either 1 or 2

n can't be two since all of them are odd, so they're coprime

>>8522508
for >>8522496

>>8522512
Thanks for that!

>>8522109
Memorization, really. Memorizing associations between things for example powers of 2, etc. Working on textures for vidya gave me instant calculation for powers of 2 from constantly seeing things like 128x128, 256x256, 1024x1024, etc

>>8522096
Would being suspended in fluid help withstand more than 1g constant acceleration? Assume the force is perpendicular to the human.
I personally don't see how it will help keep blood circulating to the brain, although you'll feel buoyant?

Consider the language of math expressions, with an alphabet S = A ∪
{(,), [, ]}, where A denotes all legal math symbols except for the round and
square brackets. Design a pushdown automaton with a single state that
parses strings formed of symbols from S and accepts these strings if and
only if the opening and closing brackets match correctly.

How would I start this/What would be the first 3 states so I could work out the rest.

Thanks

Does specialization in EE undergrad really matter for finding a job? Want to get into defense or aerospace company and not sure which electives would be best

>>8522481
If f is continuous, then x being "close" to x0 means that f(x) will be "close" to f(x0). More precisely, for any given positive "distance" ε, there's always some non-empty neighbourhood around x0 where all f(x) are within that distance of f(x0), i.e. f(x0)-ε<f(x)<f(x0)+ε.

A series y[n] converges to L if for any given positive "distance" ε, all of the y[n] which are farther from L than that distance are confined to a finite portion at the start of the sequence, with all remaining elements closer.

The sequence x[n]=x0+1/n converges to x0. To prove this, note that |x[n]-x0| = x[n]-x0 = 1/n. So n>N => |x[n]-x0|<1/N. Or conversely: given ε, |x[n]-x0|<ε => 1/n<ε => n>1/ε.

You're being asked to prove that
> yn = f(x0+1/n) converges to f(x0)
I.e. (from the definition of convergence) that
∀ε>0, ∃N∈N, ∀n∈N, n>N => |f(x0+1/n)−f(x0)|<ε

From the definition of continuity,
∀ε>0, ∃δ>0, ∀x∈R, |x−x0|<δ => |f(x)−f(x0)|<ε
Restrict x:
|- ∀ε>0, ∃δ>0, ∀x∈{x0+1/n : n∈N}, |x−x0|<δ => |f(x0+1/n)−f(x0)|<ε
|- ∀ε>0, ∃δ>0, ∀n∈N, |(x0+1/n)−x0|<δ => |f(x0+1/n)−f(x0)|<ε
Simplify |(x0+1/n)−x0| to |1/n|:
|- ∀ε>0, ∃δ>0, ∀n∈N, |1/n|<δ => |f(x0+1/n)−f(x0)|<ε
|- ∀ε>0, ∃δ>0, ∀n∈N, n>1/δ => |f(x0+1/n)−f(x0)|<ε
Existential generalisation:
|- ∀ε>0, ∃δ>0, ∃N∈N, ∀n∈N, n>N => |f(x0+1/n)−f(x0)|<ε
Existential Instantiation:
|- ∀ε>0, ∃N∈N, ∀n∈N, n>N => |f(x0+1/n)−f(x0)|<ε
QED

The thing is, you're asking for a "formal" proof, but I don't know the exact level of formalism which is required, so the above may be excessive or it may have omitted some "required" steps. It's been 20 years since I last had to do a predicate logic proof step-by-step.

>>8522557
If you were in equilibrium (i.e. same average density as the fluid), then acceleration wouldn't change that. As the acceleration increases, the fluid pressure would increase.

The reason why you lose blood flow to the brain under upward acceleration is that the forces accelerating you upward tend to be applied more to your skeleton than to your blood. So the forces are basically lifting your brain upward, out of your blood supply

If the forces were applied via a fluid in which you were (barely) floating, that wouldn't be the case; they'd affect all of your body equally.

>>8522096

Is it possible to make a box such that all 3 side lengths and all 4 diagonals (one on each face, and one from bottom left front corner to top right back corner) are integers?

>>8522660
So it would "sorta help"?

>>8522102
Practice more. You'll see patterns eventually.

>>8522666
http://mathworld.wolfram.com/PerfectCuboid.html

very very open and hard problem

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>>8522666
>>8522698
http://www.mast.queensu.ca/~kani/lectures/cuboids-b.pdf
http://www.mast.queensu.ca/~kani/lectures/cuboids2-b.pdf
^ some slides from a talk I saw once that went through some of the history of the problem, gets into hardcore arithmetic geometry very quickly

I'm struggling with the fundamentals of sets.
For instance, the line:
({7, 14, 21, 28, ....} ∩ {5, 10, 15, 20, 25, ....})
Is this the one set where the sets {7,14,21...} and {5,10,15...} intersect?
And also, how would you specify this set?
"The set of the multiples of 7 intersecting with the multiples of 5"?

How do I stay motivated to study for my finals.

1 week left...

>>8522820
yup, you got it

>>8522824
if you don't study you'll be more of a failure than you are now.

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I'll be blunt; how the fuck do I into linear programming? Pic related is an example problem that I got the answers to only because myomlab is retarded

can someone tell me what the 3x3 matrix does geometrically.

cos(a) -sin(a) 0
sin(a) cos(a) 0
0 0 1

There is this exercise that while I know the answer, I don't know how to get to it:
Find a linear function f(x) = ax+b that the graphic intercept the quadratic function g(x) = x^2 in only one point (1,1).
The answer is f(x) = 2x - 1. I can find it using the derivative, but this is from a pre-calculus book and the book didn't even teach derivative. So how do I go in finding it?
There is also this one:
Find a linear function f(x) = ax+b that the graphic intercept the quadratic function g(x) = x^2 in only one point (p, p2).
And this one I don't know how to do even using derivative.

>>8522820
the intersection is 'the set of multiples of 35'

>>8522848
picture the x-y-z 3d space

the matrix rotates the entire space counterclockwise around the z-axis by 'a' radians

>>8522855
graph* and intersect*

>>8522826
What about this kind of sequent?
{x ∈ N : 7x = x2} ∪ {x ∈ N :6+ x = x2} =
I'm somewhat confused as to what to make of the 'such that' symbol when there's more than just a single symbol to the left of it. I also don't see what I'm meant to deduce from this. The question wants me to specify something from this set but I don't know what to make of it.

>>8522826
>>8522856
The answer was:
{35, 70, 105, 140, ....} = {35n : n ∈ N}
I have no fucking clue how this works.

>>8518262
Someone mind telling me what this means & what the transition rules would actually be + how it would achieve binary addition?

Thank you.

So I needed to build an energy extraction device for a system of baking soda and vinegar and i need to be able to calculate the work output. I used a U-tube filled with water to measure the work. One end of the tube is connected to a reaction chamber/valve system, the other is open to the air. How do i calculate the work if I measure the change in height of the water?

>>8522847
Talk to a tutor

Taking notes.

Rewrite notes you made in class - yes or no?
What's your opinion on it? Seems like a huge time waste to me

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How do I find/describe the area over which we are integrating?

Sets make no sense to me.
Specifically, A ∩ B ⊆ A ⊆ A ∪ B
So if AuB is {1,2,3,4,5,6}, and A is {1,2,3}, how the fuck can A∩B contain A? Where is it ever explicitly stated that A∩B contains every element that's within A?
This kind of math just seems like the most illogical concept ever. It feels less like it makes sense and more like I'm learning some ancient rules that someone came up with off the top of their head.

>>8523077
rewriting them reinforces them

also you might come across something you took for granted and learn something new while filling in that gap

>>8523134
>how the fuck can A∩B contain A? Where is it ever explicitly stated that A∩B contains every element that's within A?
it doesn't, that's why you wrote A ∩ B ⊆ A, which means A contains A∩B

you can think of ∩ as 'and' and ∪ as 'or'

>>8523114
Consider the regions described by each integral starting from the inner most one. Ignoring the first two integrands for the moment, the first one's limits indicate that the region is half an infinite cylinder of radius 3 of which the x-axis goes straight through, the half of which constitutes the region is of course the half where y is positive, since y is only positive in the integral. Now it shows that given a fixed z, x varies from 0 to z. Consider the plane defined by x=z. To visualize this, look at the xz plane and the like x=z, basically the identity line. That line is the edge of the plane. This plane will intersect the infinite half cylinder. Clearly we only consider the plane that is in the first octant, because only in this octant will x, y, and z all be positive. The intersection will split the cylinder into two pieces, only one will be finite. Lastly we only vary z from 0 to 2. This part is easy, just "erase" the figure past z=2 and below z=0 (The figure would terminate after z=3 anyways)

>>8523140
I'm confused as to the fundamentals of sets.
So is A∩B the set that contains EVERY element in A and EVERY element in B? Or is it the set that contains the elements that are ONLY shared by A and B?
If it's the forner then I can't really see a difference between A∩B and AUB, since they both seem to encompass both sets in Venn diagrams.

>>8523134
>Where is it ever explicitly stated that A∩B contains every element that's within A?
You have it backwards. The statement A ∩ B ⊆ A means that every element in A ∩ B is an element of A. This is, of course, true, because the all of the elements in the intersection of two sets are members of both sets.

>This kind of math just seems like the most illogical concept ever. It feels less like it makes sense and more like I'm learning some ancient rules that someone came up with off the top of their head.
I suggest you read the first few chapters of How to Prove It, if you aren't reading that already. I'm sure it will clear up a lot of things. Sets are very logical.

>>8523145
>So is A∩B the set that contains EVERY element in A and EVERY element in B?
no, thats AUB

>Or is it the set that contains the elements that are ONLY shared by A and B?
yes

>>8523146
Right, thanks.
Is it safe to ignore the sizes of the Venn diagrams then? I feel like I'm approaching this wrong.

>>8523148
Thanks

>>8523145
A∩B (intersection) is the set of all elements that are shared by both sets. In logical terms, if an element is in A∩B, then that element is in A AND B.

AUB (union) is the set of all elements in both sets. In logical terms, if an element is in AUB, then it is either in A OR B (or both).

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>>8523142
Ey buddy, appreciated. Thank you very much.

>>8522875
They're the same set
[math] \{35n: n\in N\}=\{7,14,21,...\}\cap \{5,10,15,...\} [/math]

Saying two sets are equal is the same as saying that each is contained in the other. So to see they're equal, you have to show that the left side is contained in the right side, and vice versa.

Let's show the left is contained in the right. To do this, we need to show that every element of the left set is also an element of the right set.

Take any element x of the left side. By definition, it is of the form x=35n for some integer n. But also x=7(5n), and since 5n is an integer, we conclude [math] x\in \{7n: n\in N\} [/math]. Similarly, x=5(7n), and since 7n is an integer, we have [math] x\in \{5n: n\in N\} [/math]. Therefore, [math] x\in\{7,14,21,...\}\cap \{5,10,15,...\} [/math] since the intersection is by definition all elements in both sets.

We've shown that every element of the left side is also an element of the right side. i.e.
[math] \{35n: n\in N\}\subset\{7,14,21,...\}\cap \{5,10,15,...\} [/math]

To finish up, you also have to show that every element of the right side is also an element of the left, which it would behoove you to try yourself.

Not a question but a request to believers in the almighty God. I currently have a 79.8% percent in my College Trigonometry class. I just took the final today. Pray that I at least got a 40%. I will at least pass the class then. I'm also taking a Pre-cal final Friday. I'm not too worried about that one as I find Pre-cal a lot simpler even though it's considered slightly higher on the totem poll.

Is there any difference between something that can't be known and something that can't be proven?

>>8523317
define 'know'

>>8523335
Like, say I look up at the sky, I know in my mind that it is blue. But I might not have the means on hand to prove it in any exhaustive way.

So I'm taking a first topology class, and so far it's gone pretty good. Covers from metric spaces to fundamental groups. Everything makes perfect sense, except last test there was a question on a topic we spent no time on, only went over the definition. It asked if there could be an open map from [math]S^n\to\mathbb{R}^n[/math]. I would assume, "no", and this had something to do with compactness, but I couldn't manage to figure it out within test time, and I'm still having trouble figuring it out after the fact, because 99% of the work we had done involved continuous functions. Anyone able to tell me what I'm not seeing? The class was not well versed in properties of open maps, and even after consulting the text further, it's not clear.

>>8523386
Define "prove"

>>8523015
You need to calculate the change in potential energy, which is equal to the weight (i.e. mass multiplied by gravitational acceleration) of the water multiplied by the change in mean height.

For a U-tube, you can ignore the water in the U portion and consider only the water which moved. I.e. if the water level dropped by H on one side and increased by H on the other (so the difference changed from zero to 2*H), a cylindrical section of water of height H moved up by H.

The mass of that section is H*area*density, its weight is H*area*density*g, and the change in potential energy is H^2*area*density*g. The density of water is ~1e3 kg/m^3, g is 9.81 m/s^2. H and area should be in m and m^2 to get energy in Joules.

>>8522096
Would it be difficult to get my doctor to prescribe me HGH? I'm fairly sure poor nutrition stunted my growth
(mentally ill parents, i was never allowed to eat much, weighed 52 kilos or 114.64 pounds at 18, and i'm around 4 inches shorter than all of my cousins on both sides)
Would HGH even help? I'm 19 now and my economic situation is vastly different, I have plenty of money for food but i'm fairly sure my growth spurts have ended.

Without much context, is this plot irredeemably shitty? I was thinking of differentiating some of them with different styles(dots, dashes, etc.), but it seems to just make them busy. It's supposed to represent 5 different sets of data - for each one, an upper (red) curve, a lower (blue) curve and an average(green) of those two's asymptotes

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>>8523578
silly me

>>8523535
i don't know your doctor, but if you're exceptionally short for your height, then there is a case to be made. my brother was very short and skinny for his age, and his doctor suggested putting him on hgh (although he didn't take it and now is over 6 feet lol)

>>8523000
Can anyone help please?

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Trig identities question:
Simplify: tanx csc^2x - tanx

What does the matrix geometrically:
-1 0 0
0 -1 0
0 0 -1

>>8523646
youre a fucking retard

>>8523654
its the diagonal of a unit cube in the octant of negative x,y,z

>>8523646
> tanx csc^2x - tanx
tan = sin/cos
csc = 1/sin

tanx csc^2x - tanx
= ( sinx / cosx ) ( 1/sin^2x - 1)
= ( sinx / cosx ) ( (1 - sin^2x) / sin^2x)
= ( cosx / sinx)
= cotx

>>8523654
Scale by -1, i.e. reflection in the origin.

>>8522096
number theory fags, help me out here

suppose g is a primitive root mod p( a prime) and we want to solve the discrete log problem, g^x = a (mod p). Critique this approach, ie what are the pros and cons(if any) of this method.

First, find a LEGENDRE p (a/p) and therefore, the parity of x.

If x is even (a LEGENDRE p) == 1, rewrite x as 2x and take a square root on both sides and go back to step 1.

If x is odd, (a LEGENDRE p) == -1, multiply both sides by g, rewrite x+1 as 2x, take a square root on both sides, and go back to step 1.

What about if p = 1(mod 4)?, p=3(mod4)?

>>8523837
why do you still not have anything that actually outputs the x you want?

>>8523846
this is theoretical, if i wanted to solve a discrete log i'd use an index calculus, but just for this question, how does it help anyone find a discrete log? is there anything about this that would aid in someones quest to find x?

>>8523582
>>8523578
I just need a random person's opinion on this, if it's readable by an outsider

>>8523871
maybe pick a less offensive shade of green, mark each trio of data lines by a letter

What jobs can I get with a Bachelor's in Mathematics?

>>8524002
your employability will be unequivocally related to the brand of knee pads you invest in

>>8524007
Is this a blowjob joke?

>>8524008
it means if you happen to have great connections, you can expect literally 200k+
if you don't, then you won't come close

>>8524011
I know, but is it a reference to sleeping your way to the top?

>>8523389
If you further assume the map to be continuous, then f(S^n) will be compact and open in R^n, so by Heine Borel open and closed. Since R^n is connected you get a contradiction

>>8524019
Yes, that would be the case under a continuous map, and we've covered many such similar cases, however the map was not assumed to be continuous, which is why I'm having difficulty moving forward.

>>8524017
He's talking about IT. 'Cause they climb ladders and are generally datacenter monkeys and shit like that.

>>8523389
do it by induction

I know how to write in LaTeX.

How do I input LaTeX in a post on /sci/?

>>8524169
put it between a [math]
and a
[/math]

>>8524173
[math] testing [/math]

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Please help, in telling me exactly what the transition rules would be from this example of a Turing machine.

Thanks

>>8522096
What's the necessary condition on [math] f [/math] for a solution [math] x: \mathbb{R} \to \mathbb{R}^n [/math] to exist for a given autonomous differential equation:
[eqn] \dot{x} = f(x) [/eqn]

Also why do you have to specify the standard basis for [math] \mathbb{R}^n [/math]. Why isn't it just that [math] \mathbb{R}^n [/math] is defined by induction on the Cartesian product?
[math] \mathbb{R}^1 = \mathbb{R} [/math]
[math] \mathbb{R}^{n+1} = \mathbb{R}^n\times \mathbb{R} [/math]

Why is thermodynamics so confusing? Why are the Kelvin and Clausius statement of the second law equivalent? How to make sense out of the Clausius theorem? Why does S always increase? Why does the maximization of S imply the minimization of the internal energy?

What is the geometrical interpretation of complex integrals?

What exactly are branch points and why there is a discontinuity between two "sides" of a cut line but not discontinuity between two "sides" of a countour line?

>>8522686
Yes, it would sorta help, dependant on the density of the fluid and the severity of acceleration

>>8522096

why does pussy feel so good?

>>8524196
Please someone answer this is an emergency.

Why is w perpendicular to [math]w^Tx + b = 1[/math]?

Factorize x^4+x+1

Help

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>>8524282
>implying you've felt pussy before

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Why does pic related not work? I understand that that my logic must be flawed but I don't understand where.

>>8523149
What do you mean by "size of Venn diagrams"? Venn diagrams just illustrate the relationships between sets. For the general rules you're talking about, you don't even need elements; just draw each set as a circle. That's size-invariant, so I don't see how that plays into it. Can you MSPaint an example?

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>>8523149
>>8524384
Here's the rules graphically illustrated.

The mnemonic for keeping the union and intersection symbols straight is that "U" stands for "union".

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>>8524296

>>8523871
>>8523582
>>8523578
The highest and second-highest red lines are too close for all the data sets to be distinguishable like that. OTOH, within each set, the three lines are very different, so maybe try making all three lines the same color in each set (i.e. if your 320 mm line is green, change the highest red and blue lines to green; then change the 370 mm line to blue along with its natural and forced curves, etc.)

Also, primary colors at the same value and saturation clash like a mofo. Desaturate them and/or use hues that are closer together, but still easily distinguishable. But that suggestion is even more /gd/ than the original question, so...

>>8524369
C(∞,k) * (1/∞) =/= 0

In fact:

C(n,k)/(k^n) -> 1/k! as n -> ∞

And sum[0<=k<=n](1/k!) -> e as n -> ∞

Quite rapidly, in fact. The first 5 terms are
2.0, 2.5, 2.666..., 2.708333..., 2.71666...; the 5th term being within 0.16%. At n=16, the difference is less than the accuracy of double-precision floating-point.

Scientifically speaking, why do we like sweet drinks but not savory ones?

>>8524406
I want to be all
>black coffee
>unsweetened tea
>soup
but I find all but tea gross, and love soda.

>>8524410
>futa anime girl jizz

>>8524410
>I find all tea gross
Sweetened tea tastes good. If you're not a brainlet.

>>black coffee
Literally makes me gag

>>unsweetened tea
Oh god, why

>>soup
Ok, but that's not a drink.

Why are you doing this to yourself, fag? Don't tell me you're falling for some meme diet.

>>8524403
thanks, couldn't find this explained anywhere

Can I ask for java help here?

I'm trying to set up a class that receives a date and if it's not a valid date sets it to 1/1/2000 (among other things it does)

Let's say this is how I check if a date is valid
https://codetidy.com/9783/

And these are my constructors for the Date object

public Date(int day, int month, int year)
{
_day = day;
_month = month;
_year = year;
}

// Additional constructor
public Date(Date date)
{
_day = date._day
_month = date._month
_year = date._year
}


How do I 'fuse' them so that every date I receive is checked to be valid and reset to 1/1/2000 if not, as a private method within the same class?

How can I demonstrate that all attractive fixpoints in the Mandelbrot set are attractive in all the complex plane?

>>8522820
Just remember U for Union and flipped U for the opposite, intersection. Another method is that U is a smile, so everyone gets in, and flipped U is a sulk/frown, which means only the popular kids get in. This is probably a good way to summarize my life in a para. Lol

>>8524428
>I find all BUT tea gross
Also,
>[soup]'s not a drink
>my tastebuds magically know how I define different aqueous solutions of nutrients
I know expectation plays into subjective experience a lot, but come on.

Can you get jobs in Tech and Engineering if you have a BSc in Mathematics and a Master's in Statistics?

What website should i visit to read interesting science stuff but accessible to someone with no science background

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Help with (c) please. No idea how to factor this.

What does the dot represent between two statements/characters?
I.e. (a + b) · (a − b)

What makes an argument invalid in logic?
For instance, the argument:
(~S^L)vS
S
------
~L

There is an outcome where both (~S^L) hold, and where S also holds, but the argument is still invalid. Why? Can an argument only be valid if both S and Q are true AND the outcome is true also?

>>8524674
Dot product aka multiplication

2 · 3 = 6

>>8524688
So it just means multiplication? Or is that different to product?

>>8524656
[math]
a + 1 + a^2(1 + a) = 1(a + 1) + a^2(a + 1) = (1 + a^2)(a + 1)
[/math]

>>8524641
leddit

>>8524750
I gotta figure you do the exact same thing.
1+a+...+a^14=(1+a+a^2+a^3+a^4)(1+a^5+a^10)

>>8524794
nope:
[math]
\left(x^2+x+1\right)\,\left(x^4+x^3+x^2+x+1\right)\,\left(x^8-x^7+x
^5-x^4+x^3-x+1\right)
[/math]

Why are rockets so primitive? why do we still need to detach parts in order to reach space costing millions of dollars, can't we reuse them? can't we make a ship that can go on orbit and back and then be ready to be used again?

>>8524910
do you not know what spacex has successfully done a few times now?

>>8524745
For numbers, it's just multiplication. Other structures (e.g. vectors) can have different types of product.

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Is the earth round or flat

>>8524410
>black coffee
Quit being a bitch, that's how

>>8525143
The earth is a spheroid.

>>8524678
> What makes an argument invalid in logic?
If the conclusion does not follow from the premises.

> For instance, the argument:
> (~S^L)vS, S |-- ~L
If S and L are both true, the premises (~S^L)vS and S will both be true but the conclusion ~L won't be.

> Can an argument only be valid if both S and Q are true AND the outcome is true also?
It's only valid if the outcome is always true whenever all of the premises are true. If it's possible for the outcome to be false with all of the premises true, it's invalid.

But that's not all. The outcome must follow from the premises via application of inference rules. Truth and provability aren't necessarily the same thing. They are in propositional logic, which is both sound and complete, but predicate logic is proven (by Goedel's theorem) to be incomplete (there exist true statements which cannot be proven).

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In Natural Deduction, is there a set of priority-based eliminations you should do?
For example, if you have;
E^FvE^G
Would it not be perfectly possible to eliminate the leftmost E and the rightmost G? The software I'm using doesn't allow this though, and forces me to make assumptions using the v elim.

What's the priority for elimination?

>>8525158
Im currently talking to someone who thinks its round and completely flat like a plate.... how do I prove it isn't?

>>8524454
They aren't. They're only attractive for initial values within the corresponding Julia set (which always includes zero).

>>8525197
> Im currently talking to someone who thinks its round and completely flat like a plate
No you aren't. You're talking to a troll.

> how do I prove it isn't?
"For your friends, no explanation is necessary; for your enemies, none will suffice".
Flat-earth-ism is basically just "you can't force me to accept your 'facts'; I can deny whatever I want to". It's the distilled essence of post-truth-ism.

How does
{n∈N:3n+5=5n−17}={n∈N:n=7+4}
work?
Shit is confusing.

3n+5=5n-17
so
3n-5n=-17-5
so
-2n=-22
so n=11=7+4

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Can anyone help me with this? I've been having a good time with trig and zooming through everything but I've been completely stuck on this problem.

Would really appreciate some insight on solving it.

>>8525492
I'm still not getting it.
Is 3n meant to be 3*n? Or is that not the case with set theory?

>>8525498
that should say "show", not "prove"

also, no i'm not helping, you can do this yourself, it's not hard. open a formula book (bronstein or whatever) and find the ways sines and cosines transform, you can take it from there.

>>8525517
yes 3n is 3*n

>>8525498
(sinx-cscx)/(cosx-cotx)
= (sinx-1/sinx)/(cosx-cosx/sinx)
= ((sin^2x-1)/sinx)/(cosx*(1-1/sinx))
= ((sin^2x-1)/sinx)/(cosx*(sinx-1)/sinx))
= (sin^2x-1)/(cosx*(sinx-1))
= (-cos^2x)/(cosx*(sinx-1))
= (-cosx)/(sinx-1)
= cosx/(1-sinx)

>>8525525
So if n = 2
(3*2)+5 = 11
(5*2)-17 = -3

I don't get set theory.

>>8525530
right, so n=2 isn't in the set because those numbers aren't equal...

>>8525530
you seem like a brainlet, i highly recommend actually going to class/reading your book/getting a tutor

>>8525537
Is it implying there is a number that = 3*2+5 and 5*2-17? Or that there is no number that exists so both sets must by default be equal?
>>8525539
I'm going through two textbooks as well as the lectures on this shit. Literally just started learning about this stuff last week and we're getting questions like that in our tests. I don't know if I'm just slow or it's because of a lack of teaching. Am I meant to be able to get this stuff after 2 hours + some extra hours reading?

>>8525524
The other ones I did were not hard, yes, but for some reason this one stopped me dead in my tracks.

I will look into formula books though in order to address this fault, thanks.

>>8525526
Thanks anon, I will notate this and try to digest the logic

>>8525558
you read {n∈N:3n+5=5n−17}
as "the set of natural numbers that satisfy 3n+5=5n-17"
2 doesn't satisfy this so 2 isnt in the set
11 satisfies this so its in the set

>>8523788
Danke
Btw nice hitler dubs

>>8525594
I forgot about the special sets.
Thanks, it makes a lot more sense now.

>>8525198
And how can I demonstrate that they are always attractive in zero?

Very stupid one.
Derivative of x^sinx
I know the result, I just have no clue how to get there. Must be some rule I'm missing but I can't find it.

y=x^sinx
ln(y) = (sinx)*ln(x)
(y'/y) = d/dx ((sinx)*ln(x))

How the hell do you get length contraction from the Lorentz transformation?? I keep getting x'=gamma*x.

>>8524203
bumpity

>>8522096
How would you prove this?
[math] \displaystyle \sum_{x = 1}^{\infty} x \cdot (1 - p)^{x-1} = \frac{1}{p^{2}}[/math]

>>8525619
f[c](z)=z^2+c has an attracting fixpoint if c=(μ/2)(1-μ/2) for some μ with |μ|<1, and f[c]^n(0) will converge to it.

That corresponds to the main cardioid shape on the right, while the circular lobes all correspond to cyclic orbits.

z is a fixpoint of f[c] if f[c](z)=z, i.e.
z=z^2+(μ/2)(1-μ/2)
=> z-z^2=(μ/2)(1-μ/2)
=> z(1-z)=(μ/2)(1-μ/2)
=> z=μ/2 or z=1-μ/2

Also:
f(z)-(μ/2) = z^2-(μ/2)^2 = (z-(μ/2))(z+(μ/2))
=> (f(z)-(μ/2))/(z-(μ/2)) = (z+(μ/2))
and:
f(z)-(1-μ/2) = z^2-(1-μ/2)^2 = (z-(1-μ/2))(z+(1-μ/2))
=> (f(z)-(1-μ/2))/(z-(1-μ/2)) = (z+(1-μ/2))

As |a*b|=|a|*|b|:
|z+(μ/2)|<1 => |f(z)-(μ/2)|<|z-(μ/2)|
|z+(1-μ/2)|<1 => |f(z)-(1-μ/2)|<|z-(1-μ/2)|

i.e. so long as the condition on the left holds, iteration will result in z getting closer to the corresponding fixpoint.

In the |z+(μ/2)|<1 case, clearly this holds for z=0 as |μ|<1.

The tricky part is showing that it continues to hold as z->(μ/2).

That's as much as I can be bothered working out right now.

>>8525700
>>8525711

Shit, I forgot to quote you.

>>8525814
Take the derivative of the geometric series and evaluate it at (1 - p).

>>8525867
why does that work?

>>8525871
did you try it?

>>8525875
no, i wanted to know what that would work before i dealt with that mess

>>8525886
its a simple derivative just using the chain rule/power rule, no mess...

Is there anyway to do physics experiments at home or does it all require a lab of some sort?

Also where can I get physics equipment?

>>8525913
yeah mang i heard u can buy a particle accelerator for about 100 bucks at walmart

>>8525890
[math] \ln\left(1-p\right)\left(1-p\right)^{p+1}+\left(1-p\right)^{p} [/math]

So where do I go from here?

So I vaguely understand there are the four fundamental forces, being gravity, electromagnetism, and strong and weak nuclear interaction. How are these forces rationalized with stuff like Newtonian physics? i.e. how is motion, changes in kinetic energy and impact forces described using the fundamental forces?

>>8525925
do you know what the geometric series is? and what it equals?

>>8525933
yea, its sum is in the form of [math] \tfrac{a}{1-r}[/math]
how does that derivative relate to that sum?

>>8525933
the series starts like [math] = 1 + 2 \cdot (1 - p) + 3 \cdot (1 - p)^{2} + \cdots [/math]

>>8525983
>>8525964
do you need ur hand held brainlet?

sum_{n=0 to infinity} x^n = 1/(1-x)

so derivative gives

sum_{n=0 to infinity} nx^(n-1) = 1/(1-x)^2

so

sum_{n=0 to infinity} n(1-p)^(n-1) = 1/(p^2)

>>8525930
It's almost all electromagnetic. EM waves are obvious; anything involving commutation of energy via a normal force (i.e. objects bumping into each other) is, at the atomic scale, basically effected by their electron shells electrostatically repelling each other. Strong and weak nuclear are both magic and irrelevant until you get to quantum mechanics.

Or I'm stupid, macroscopic objects excluding each other in space somehow has to do with quantum principles based on the nuclear forces, and I'm about to get my ass handed to me and look like an idiot. But that's why I'm anonymous?

>>8525964
1 + (1-p) + (1-p)^2 + (1-p)^3 + ...
. + (1-p) + (1-p)^2 + (1-p)^3 + ...
......... + (1-p)^2 + (1-p)^3 + ...
................... + (1-p)^3 + ...
...

>>8525987
>calls someone a brainlet
>he can't into latex

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>>8526019
>implying i care enough to add latex tags when the math is this simple

>>8525987
i did. ty

Physically, why doesn't it matter whether the resistor is put on the anode or cathode. I know KCL and that the current is the same in series but don't know physically why it doesn't matter.

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>>8526129

>>8526129
A better question is: why /would/ it matter?

It doesn't matter ... because it doesn't. There really isn't any deeper explanation.

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PLEASE HELP!!!
I'm in Circuits 1 class right now and need an idea of a "neat" circuit to make that has any sort of use please ANY ideas are helpful. I do not have to make it if I don't want to, but I have to model it in multisim or pspice

also we've only covered circuits that include resistors, capacitors, inductors, and RLC circuits with one of each. op amps and ac circuits have been covered very basically

Test

What's better, Tableau or R?

Could you make a sound that's quiet only at close range with the following method?
>ring of speakers that project sound out away from the ring, creating beams of sound
>ring rotates at a frequency that is the frequency of the speakers divided by the number of speakers, and as this means beams of sound sweep over a nearby listener, this rotation creates a sound frequency too, that matches that of the speakers
>two frequencies cancel out for a nearby listener
>due to diffraction, as you move away the sound becomes no longer in distinct beams, and thus the secondary frequency is lost, so the sound does not cancel out

>>8526305
Something that sometimes has to be done to boats is to completely drain their batteries, then completely recharge them.
This is to extend battery life.

What would be useful is a circuit which could make this process more efficient if the boat has multiple batteries.

Instead of draining all batteries equally, it would be better to drain only one battery at a time.
Not only does this mean that regular use of the boat would completely drain batteries passively, one at a time; rather than only partially draining all the batteries at once; this also means that if you weren't using the boat you could upkeep the batteries fairly efficiently by recharging the batteries with themselves.

It could start with battery 1 being drained with 2 3 and 4 being full, then battery 2 is completely drained to recharge 1 (losses compensated for of course), then 3 recharges 2, then 4 recharges 3.

>>8524203
here's a sufficient condition:
https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem

i don't know if there's a necessary condition or why this would be useful. what's the context of this question?

>>8522096
What does the "C" in "C(n,r)" stand for when you're designating a particular element in a row of Pascal's Triangle? I know (n) is the row, (r) is the element in that row, but what is the "C" for?

>>8526477
I think it stands for "choose" as in "n choose r". there are other notations without the C, like [math] {n\choose k}[/math]

How do I prove that a set of operations is a boolean algebra?

>>8526522
>>8526477
when you read it out loud you would say n choose r but i think the C stands for combinations,because when you similarly use P(n,r) meaning permutations when the order of the elements you choose matter

In natural deduction, when introducing or eliminating a negation, can you literally use anything prior to the assumption to get a contradiction?
So if you assume A, you can use B,C,D,E... To get a contradiction and therefore prove ~A? Or does it have to be related to A somehow?

Any professors or teaching assistants in here? Why do some classes make attendance part of the grade? I would have an A in my computer programming class but attendance is worth 10% so that alone lowers me to a B. I got a perfect score on the final exam btw too.

>>8526749
do you go to a uni or a daycare?

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>>8526749
>10 free % of your grade
>doesn't want them

I'm having trouble with surface integrals, /sci/. Help an autist out. I get that to project onto a plane I can derive a "dA" factor from a vector "dS" because vector dS = normal vector * dA. But how do I derive the normal? I know I can cross two partial derivatives, but I have trouble parametrizing surfaces. Why can't I just take the gradient in example 2 like they do in example 1?

http://tutorial.math.lamar.edu/Classes/CalcIII/SurfIntVectorField.aspx

>>8526730
You can use anything indeed.

In boolean algebra, how would you simplify this? I just want to double check my work since I haven't done this for a while in the semester, and after doing my kmaps, I have to design a circuit based on it.

~(AB) + (B~C~D) + (AB) + (B~C~D) + (AC)

Do you cancel out AB with ~A~B? Also, what do you do when the same expression comes twice? Ignore one of them?

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What would happen if a tentacle actually, y'know, penetrated your brain? Would it be possible without doing any fatal damage, and would things like memory, bodily control, and rational thought be instantly wrecked or what?

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What is this symbol called and what does it represent? I teach myself pretty much everything and have solved a few of them. I have never came across a written theorem or name for it but I guess it has something to do with distances and restrictions? I think if I knew what the help I was solving for, I would "get" it more? Air brushed in paint with bad aesthetics for extra stupidity.

>>8526997
Of course it would be possible for the person to survive and yeah, there is a more likely than not they will lose important function(s). We have already had cases of steel bars and other objects fully penetrating the brain and people surviving. The brain is a wonderful thing and many of these people are able to live "normal" lives overtime thanks to the wonders of neuroplasticity. I'm far from a neuroscientist but I was interested in the subject for years reading tons of books and other shit on it in my free time. Surprisingly, my interest in neuroscience was brought on by psychedelics and my curiosity of how and why they work the way they do.

>>8527090
I never exactly solved that last part of my sentiment but my "hallucinations" were wonderful at times. I know many will not believe me, especially those who have not taken psychedelics or had a full blown experience but I experienced perfect geometric patterns flash before my eyes that I had never seen before. It actually got my ass back in shape to continue to pursue academics seriously, get inshape, and live "life to the fullest".

>>8527076
the E? it means sum. in this case sum of the series. sum of (equation) in the range bottom number through top number.

if the bottom number was 2 and the top number was 10 and it was (2k+5)

you'd get 9 + 11 + 13 + 15.. to the 10th interation. then add them all up.

it's called arithmetic series. or infinite arithmetic series.

>>8527103
it's called sigma by the way.

Question about space/time and relativity

Time is the measurement of movement through space relative to other things and the speed of light is the maximum speed something travels at. But because things are relative, wouldn't that mean if something were traveling faster than light it wouldn't just mean that object is moving faster, but it'd also mean everything in the entire universe has slowed down. If ultra light speed objects have an effect on all matter in the universe, then shouldn't this be an observable event, IE able to test for it somehow?

If we tested for this and it proved true, it might prove there are ayyy lmaos out there some where going faster than light.

>>8527103
>>8527116
Thank you. I understand I'll be seeing sigma quite a bit in the future, just used for different purposes in most cases. What you said, definitely is right as I've been with working with sequences quite a bit.

>>8523060
Did, they didn't understand it and I don't blame them.

>>8526886
>only 4 variables
>already making a Karnaugh map
Honestly I would forget the equation and just do a simplified sum-of-products directly from the map. Are you expected to minimize it beyond even sum-of-products?

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I have a question about soft drink.
Once I open the bottle, the carbonation will slowly go away. I want to extend the carbonation shelf life.
Will it go away faster in the fridge or kept at room temperature?
Will altitude also effect it?

>>8526997
>Wanting to mix real science with /d/ sex
Anon no, that way lies misery
It's like having more than one waifu except instead of waifus it's models of physics

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If I cook rice for half the time, will it have half the calories? My theory is that the material would be less digestible and just get shit out.

>>8527315
>Will it go away faster in the fridge or kept at room temperature?
Faster at room temperature (Gases solubility in water usually decrease with temperature).

>Will altitude also effect it?
More pressure, more solubility (you can easily see that using Le Chatelier's principle).

>>8527162
Wouldn't FTL imply time moves backwards or something? I don't see how you could possibly test for that because your sensor would be an instrument that exists in time, so time would move backwards for it too.
>>8527315
Carbonation is just the releasing of dissolved gas, so more total gas would be released at altitude or room temperature than at sea level or fridge temps because there'd be a greater change in pressure from the sealed container. Similarly, keeping the liquid cold lets it release more fizz. I don't know how much longer those make it last, though, because it might also affect the rate of release.
You should be able to "pause" and restart the reaction by resealing the bottle, except it still releases until the air bubble on top of the bottle is saturated. That means you'd want a container you can change the volume of, but perturbing the soda by pouring it between containers or having it in a malleable container (such as a plastic bag) would probably cause the release of more gas than you'd save.
I'd suggest an experiment: buy two identical 2-liters (same bottling date, if it's printed on them) and pour every glass of soda half from each so that they drain at the same rate. Keep them both in the fridge right next to each other too. Crush the top of one as you drain it, taking care not to affect the seal of the cap or shake the soda too much, and leave the other unmessedwith. In theory, the crushed bottle should stay fizzier because you lose less gas to the bubble in the top of the bottle.
TL;DR keep your soda in the fridge anon
Relevant wiki:
https://en.wikipedia.org/wiki/Carbonation
https://en.wikipedia.org/wiki/Effervescence
https://en.wikipedia.org/wiki/Henry%27s_law
https://en.wikipedia.org/wiki/Le_Chatelier's_principle
>>8527333
>implying uncooked rice is acaloric

Any good introductory linear algebra texts? My textbook by Leon isn't cutting it for preparing for the final.

I just failed an exam in a basic chemistry class. It's the only chemistry I need to take and I need to pass it this next exam or else I will need to re-do this whole year.
How do I best teach myself chemistry? I was pretty far away from the passing mark(I got 22 points, needed 40 for a passing grade).
Do I just keep on studying through the book or are there any simple tips for learning basic chemistry? This is some basic shit I just failed it, no excuses honestly.
The exam is in the end of Jan.

Would really appreciate any help, I'll be watching the thread

>>8527356
Do you need more conceptual help or a better explanation of formulas and such? This should help with the former:
https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

>>8527350
>your sensor would be an instrument that exists in time, so time would move backwards for it too

I guess the only way to test for FTL anomalies is to use a device while traveling FTL then. Maybe this is how FTL capable alien life forms detect and meet each other. Kinda like star trek first contact.

https://www.wolframalpha.com/input/?i=(7%2B50%5E(1%2F2))%5E(1%2F3)+%2B+(7-5*2%5E(1%2F2))%5E(1%2F3)&rawformassumption=%22%5E%22+-%3E+%22Real%22

No matter what I do I keep getting 8 for an answer. How do I do this? I can't see step by step solutions.

>>8527584
Are you getting a complex answer for [math]\sqrt[3]{7 - 5\sqrt{2}}[/math]?

>>8527584
consider writing 7+5\sqrt(2) as (a+sqrt(2)b)^3

>>8527604
>>8527619
[math]x = \sqrt[3]{7+5\sqrt{2}} + \sqrt[3]{7-5\sqrt{2}}[/math]
[math]x^3 = (\sqrt[3]{7+5\sqrt{2}} + \sqrt[3]{7-5\sqrt{2}})^3[/math]
[math]x^3 = (\sqrt[3]{7+5\sqrt{2}})^3+3\times(\sqrt[3]{7+5\sqrt{2}})^2\times(\sqrt[3]{7-5\sqrt{2}})+3\times(\sqrt[3]{7+5\sqrt{2}})\times(\sqrt[3]{7-5\sqrt{2}})^2+(\sqrt[3]{7-5\sqrt{2}})^3[/math]
[math]x^3 = 7+5\sqrt{2} +3\times(\sqrt[3]{7+5\sqrt{2}})\times(\sqrt[3]{7-5\sqrt{2}})(\sqrt[3]{7+5\sqrt{2}} + \sqrt[3]{7-5\sqrt{2}}) + 7 - 5\sqrt{2}[/math]
[math]x^3 = 14 +3\times(\sqrt[3]{49-50})(\sqrt[3]{7+5\sqrt{2}} + \sqrt[3]{7-5\sqrt{2}})[/math]
[math]x^3 = 14 +(-3)\times(\sqrt[3]{7+5\sqrt{2}} + \sqrt[3]{7-5\sqrt{2}})[/math]
Here's what I have, I don't know how to simplify this further, but when I put this in the calculator I get 8.

>>8527642
Shit I forgot to take cube root of 8. Then the answer is 2. So I'm doing this right I guess.
But I still don't know how to simplify the final roots.

>>8527642
>>8527647
I just realized I did all that for no reason, since I still have to simplify the same expression as in the first line. I just made it more complicated...

>>8526886
~(AB) + (AB) = 1
So the expression is true regardless of the variables.

> Do you cancel out AB with ~A~B?
No, with ~(AB).

> Also, what do you do when the same expression comes twice? Ignore one of them?
A+A=1
A&A=1

The mechanical approach is to convert to minterms (sum-of-products). Any conjunction containing both a variable and its negation is always false, so can be removed from the expression.

>>8527076
The symbol is a capital Greek letter sigma (S). In that context (a large sigma with expressions above, below and to the right), it indicates a summation:
https://en.wikipedia.org/wiki/Summation

https://en.wikipedia.org/wiki/Greek_letters_used_in_mathematics,_science,_and_engineering

has the Greek letters as well as common mathematical usage.

Another useful list is

https://en.wikipedia.org/wiki/List_of_mathematical_symbols

>>8527584
Post the actual expression you're trying to simplify. Copy-pasting the Wolfram Alpha link says it's 1084.5, but it seems to be converting %2B to + then converting it a second time to a space.

>>8527749
I have no idea why it's like that for you, but I just solved it so nevermind.

>>8525815
Thank you

>>8527715
A+A=A&A=A, anon. You're thinking of A+~A=1 and A&~A=0.

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Quickest way to solve this manually? (fewest steps as possible)

>>8527786
Manually enter the augmented matrix into your calculator
Then press RREF

>>8527798
We're unfortunately not allowed scientific calculators in the exam. I can do it manually in like 5 minutes but I really need to reduce the time.

What the fuck is a derivative?

I know the formulas, I know how to use them, I know how to integrate them, but I don't know what, exactly, they do.
And I mean in a practical way.
I keep seeing shit about cars and graphs and somehow, you can derive shit like velocity with it.
How do I visualize this bullshit? Please help.

>>8527786
>>8527798
So far I've got it down to

0 = 2*d2x - d3x
-5e3 = 84e6*(d3x - d2x)

d2x = -5e3/84e6 = 5.95e-7

chemical biology or biochemistry?

I love both biology and chemistry

>>8527805
A derivative is just the slope of a function. More precisely, if you have a function f(x), the derivative f'(x) is the function such that the value of f'(a) is the slope at f(a) for all a. Is the problem with one of the concepts "function", "slope", or how it applies irl?

The car thing: if you go faster, you get there sooner. Derivatives are literally just a way of quantifying that. If you go sixty miles in one hour, that means your (average) speed was sixty miles an hour, obviously. Speed is the derivative of position; your speed at any instant tells you how long it'd take to get to your destination if you keep going at that speed. So if you know the car's position at, say, 45 minutes into the drive, then at 46 minutes, you can tell its average speed. That's how GPS can tell how fast you're going. That makes sense, right? Differentiation is just telling you what that becomes as your sample rate gets infinitely fast; all the differentiation rules you learn are just shortcuts.

>>8527405
That's good, but it's missing orthogonality.

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>>8527834
It's the slope concept that throws me off. I don't understand its usefulness in real situations.
There's also this .gif. It almost made me understand something but I don't get what the derivative values are relative to. How fast it's going down/up? Relative to what?

I just feel so fucking stupid. Everyone seems to "get" it while I can only memorize the goddamn formulas. Ironically it's one of the few mathematical concepts I enjoy and I just want to understand it.

>>8522102
Practice and I.L.A.T.E.
Inverse trigs
Logarithmic
Algebraic
Trigs
Exponentials

For the int(udv) = uv - int(vdu) substitution select u as the "least easy" function to immediately integrate, that is, the first letter that appears on the ILATE prioritization.

>>8525161
ok but is it flat?

>>8527857
Ok, remember back when you started algebra, and you had to draw a bunch of lines with the formula [math]y = mx + b[/math]? M is the slope of the line you're drawing, right?

So in that animation you posted, it's got those lines sliding along the function's curve. Each of those lines also has a slope. The derivative at a given point, say x = 1, is what the slope of the tangent line at that point is.

The way this is defined is by drawing a line between f(1) and, say, f(1.5), then getting a better approximation by drawing it between f(1) and f(1.1), f(1) and f(1.01), etc., and seeing what slope it approaches. Taking the derivative by the limit definition is literally just doing this by equations instead of on a graph; the derivative is a function that tells you that at all points. So the derivative's *value* is the original function's *slope* (I think sine and cosine illustrate this well).

As for real situations -- that thing I said about the GPS, for example. Power is the time-derivative of energy, current is the time-derivative of charge, lots of other physical quantities work like that. If you have an estimate of an aquifer's total size, you can tell how much you have left by measuring the rate you pump water out of it (that's integration, but same concept). Pretty much everything you can represent as a continuous function, a derivative will be helpful sooner or later.

Literally draw a bunch of curves, draw tangents on them at random points, and ask yourself what "m" would be if you wanted to graph that tangent line algebra 1 style. (You don't care about b.) Then graph a bunch of functions and graph their derivatives below, with the x axes lined up so you can easily see what both f(x) and f'(x) at every x-value. Now draw some tangent lines on the top row and guess _their_ m-values. Look at the derivative's value at that point. It'll be m -- not m the slope, but m the number that'll give you that slope if you graph it. (cont'd)

>>8522576
Physics all the way if you want to get there. Even of it wasn't for that, then, physics is best scientia.

Jk, I'm also majoring EE. I'd recommend finding materials engineering options and shit for that purpose.

I need help with this IVP using Laplace transforms:

y'' + y = sin(t), where y(0) = 1, y'(0) = -1

Here's what I have:

s^2Y(s) - s + 1 + Y(s) = 1 / (s^2 + 1)

Y(s) = 1 / ((s^2 + 1)^2) + s/(s^2 + 1) - 1/(s^2 + 1)

Now when I try to use partial fraction decomposition on the first term it never works out. Do I use decomposition on this? If so, how. If not, where do I go from here?

>>8527928
Again, do that with sine and cosine. Draw little graphs so you can go sin, cos, -sin, -cos, sin all the down. I think this is the easiest way to see how the slope of a function becomes the value of its derivative, which sounds to me like your problem.

I don't know if I'm being too elementary or too complicated (or both at once because I'm unclear), but even if nothing I've said is the trigger, you WILL have that "a-ha!" moment. I can tell by the why you write about it that all the pieces are there and you just have to see how it fits together. Don't blame yourself because you don't magically already know, because that's how I've fucked up a million times; draw the pictures, think about the slopes, relax and worry about memorizing formulas instead when you feel blocked. It'll come together.

And keep asking about what doesn't make sense, as specific as you can.

>>8527755
Turned out it was NoScript causing the URL to be decoded twice.

FWIW:
(7+50^(1/2))^(1/3) + (7-5*2^(1/2))^(1/3)
= (7+5*2^(1/2))^(1/3) + (7-5*2^(1/2))^(1/3)
The radicals de-nest to 1±sqrt(2):
(1+sqrt(2))^3 = 1^3 + 3.1^2.sqrt(2) + 3.1.sqrt(2)^2 + sqrt(2)^3
= 1 + 3.sqrt(2) + 3.2 + 2.sqrt(2)
= 7 + 5.sqrt(2)
(1-sqrt(2))^3 = 1^3 - 3.1^2.sqrt(2) + 3.1.sqrt(2)^2 - sqrt(2)^3
= 1 - 3.sqrt(2) + 3.2 - 2.sqrt(2)
= 7 - 5.sqrt(2)

So their sum is (1+sqrt(2))+(1-sqrt(2))=2

General hint for de-nesting (a+b*c^(1/m))^(1/n) is to set
(a+b*c^(1/m))^(1/n) = (A+B*c^(1/m))
=> (a+b*c^(1/m)) = (A+B*c^(1/m))^n
Expand out the RHS by the binomial theorem, expand occurrences of c^((pm+q)/m) = c^(p+q/m) = (c^p).(c^(q/m), collect like terms (rational, c^(1/m), c^(2/m), etc), equate coefficients, solve the system of equations.

So e.g.
(7+5*sqrt(2))^(1/3) = (p+q*sqrt(2))
=> 7+5*sqrt(2) = (p+q*sqrt(2))^3
= p^3 + 3*p^2*q*sqrt(2) + 3*p*(q*sqrt(2))^2 + (q*sqrt(2))^3
= p^3 + 3*p^2*q*sqrt(2) + 6*p*q^2 + 2*sqrt(2)*q^3
=> p^3 + 6*p*q^2 = 7, 3*p^2*q + 2*q^3 = 5
=> p=1, q=1

>>8522576
>>8527931
Sorry, i thought i'd read majoring. My best guess would be ME but it depends specifically on what kind of job you're aiming for.

How do I solve for x in terms of y for something like this: [eqn] e^x+e^{-x} =e^y [/eqn]
Is it possible without using hyperbolic functions?

>>8527963
>How do I solve for x in terms of y for something like this:
>[math]e^{x}+e^{−x}=e^{y}[/math]
If I define z=e^{x} then first we have to solve [math]z+\frac{1}{z}=e^{y}[/math], which gives us
[math]z=\frac{e^{y} \pm \sqrt{e^{2y}-4}}{2}[/math]
Now the only thing left to do is to apply logs
[math]x=\log \frac{e^{y} \pm \sqrt{e^{2y}-4}}{2}[/math]

On the other hand you can use that [math]e^{x}+e^{−x}=2\cosh (x)[/math] therefore
[math]x=\cosh^{-1} \frac{e^{y}}{2}[/math]

I have this question:

Let f: [0,1] -> [0,1] be continuous. Prove there exists x such that f(x) = x.

Is there a way to do this without invoking the IVT? I can only do it with it.

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>>8527316

>>8527937
> Now when I try to use partial fraction decomposition on the first term it never works out. Do I use decomposition on this?
Yeah, but it's a bit tricky. You need to factor the denominator to
(s+j)^2*(s-j)^2
then decompose it to
a/(s+j)+b/((s+j)^2)+c/(s-j)+d/((s-j)^2)

Equating powers of s in the numerator gives you
a=j/4, b=-1/4, c=-j/4, d=-1/4
i.e.
(1/4) * (j/(s+j) - 1/(s+j)^2 - j/(s-j) - 1/(s-j)^2)

Then the inverse transforms are 1/(s+k) => e^-kt and 1/(s+k)^2 => t*e^-kt.

The imaginary parts cancel, leaving you with sin(t)/2 - t*cos(t)/2

>>8527963

Take the natural log of both sides:

x-x=y

y=0

ur welcome

>>8528143
This is how you wind up with genetic abominations anon
(honestly I'm just scared I'd act on my vore, slavery, or feminization fetishes if I could)

>>8528156
>implying [math]e^{x} + e^{-x} = e^{x - x}[/math]
C'mon, anon, troll harder.

>>8527816
>>8527803
Honestly I was just fucking with you about the calculator, but I would consider leaving the 84e6 in both equations and seeing if it cancels out to make that easier.

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>>8528166
>vore
I'm saving up whatever money I can in anticipation of Matrix-level VR simulations, because you're not going to be eating people IRL any time soon.

>>8528177
anon
I'm prey
that's why I don't want it to become real
>but also kinda do

Ive never taken a programming class before. Idk how computers work.

What are some good resources to use to prepare for my computer science class?

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>>8528025
>draw picture
>pick f(0)>0 (otherwise we're done)
>try to draw line to f(1) without crossing the red line
>realize you can't

>>8528284
How competitive is your school?

>>8528284
I wouldn't worry too much, to be honest. Intro programming courses tend to be pretty elementary (and I'm not just saying that because of muh "computer science is for brainlets" meme). You should be able to follow along. You aren't expected to know how computers work; they'll teach you the basics.

As for preparation, I'd suggest starting to fiddle around with whatever programming language they're going to teach you. Pick up a beginner's book and start working through it. I'd also figure out how to set up the development environment before you start the course (i.e. configuring an IDE, using the compiler, using the command line, etc.), because they probably won't go into too much detail on that. In the course I took, the focus was on learning how to program, but they left ourselves on our own when it came to setting up the environment in which we program.

>>8522096
Tectonic Plate Theory is just that!
A Theory!

The plates arent moving!

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how does he go to the line in the end of the picture? z(x-r(1))=...? how?

>>8522096
Why does the difference of cubes: (A-B) (A^2+AB+B^2) work whenever applied to certain polynomials?

My textbook shows me the steps in which to take to factor these types of polynomials, but if were to just be told to factor out a cubed polynomial, then I'd be left confused.

I tried to do it myself without thinking about the steps, and I failed to factor it out correctly.

I am just attempting to wrap my head around the concept better.

>>8528293
>realize the only guarantee you have that the line and diagonal intersect is the intermediate value theorem

>>8528330
https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem

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>>8522096
Does anyone give a fuck about this?

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>>8528335
no because its "fake news"

>>8528334
Why even draw it up as intersections of lines if you're just going to nuke a babby calculus question with a fixed-point theorem

don't be a faggot anon

>>8528302
Thanks, i think its JAVA and they make us use python in a group project (our group has to figure it out on our own).

Thanks for the advice.

>>8528298
Not too bad, its a growing state college

>>8528384
Then you'll be able to keep up, don't worry about it.

>>8528328
Not really sure what you're asking, but a few facts about a^n-b^n:

1. If a^n-b^n=0 then clearly a=b is always a solution, so a-b is a factor.

2. For n=p*q, a^pq-b^pq = (a^p)^q-(b^p)^q = (a^q)^p-(b^q)^p, meaning that it can be factored as A^p-B^p with A=a^q,B=b^q or as A^q-B^q with A=a^p,B=b^p. So it will have (a^q-b^q) and (a^p-b^p) as factors, which can in turn be factored.

3. All of the coefficients in the resulting factorisation will be +/-1. More precisely, they'll either be +1, or the factors will occur in conjugate pairs where one has coefficients of +1 and the other alternates +/-1. The latter arises from factoring (a^2n-b^2n) into (a^2-b^2) and a polynomial involving only even terms, which can then be factored into conjugate terms.

4. Multiplying sum[i=0:n](a^i*b^(n-i)) by (a-b) gives the difference of two polynomials, one with all of the a exponents incremented by one, one with all of the b exponents incremented by one. All of the terms except the first and last cancel, leaving a^(n+1)-b^(n+1)

>>8522557
My guess is no. If you were in water, you would still be slammed against the wall of the tank. And a thicker fluid would be like a chair basically, so the same effects would be present.

>>8527899
It's L.I.A.T.E.
not I.L.A.T.E

>>8523582
This is pretty brutal I think. Also avoid red and green as it is the most common type of colour blindness

"The subsets of {{R}} are: {} and {{R}}."

Why aren't the subsets {} and {R}?

>>8528406
{{R}} is a set with one element, namely the set {R}

a subset of a one element set either has 0 elements (empty set) or 1 element, of which you can only choose {R}, which placed in a set looks like {{R}}

>>8528406
Because {R} is an *element* of {{R}}, not a subset. The set containing {R}, which is denoted {{R}}, is a subset of {{R}}.

Can someone explain this? Find "P(P({o}))", where
>P = power set
>o = empty set

The answer is this: P(P({o})) = {o, {o}, {{o}}, {o, {o}}}

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Ive been stuck on this and I'm getting nowhere. Why the hell do you do whats in the red circle I really dont understand what the heck is going on there. I'm assuming they multiplied k by 2 before because of 2^k was multiplied by two.

Can you guys tell me if how I'm working it instead is valid?

heres what I got:

Prove P(5)
>2^5 > 5^2
36 > 25 which is true. Easy enough. Next

If P(k) is true then that implies P(k+1) is also true
P(k):
>2^k > k^2

Lets assume P(k) is true then following must be true
P(k+1):
>2^(k+1) > (k+1)^2
2*2^k > k^2 +2k +1
This must be true for all k>4 as well
2*2^5 > 5^2 +10 +1
64 > 36 which is true

Would this be ok?

>>8528471
They factored it.

You might want to brush up on your algebra.

>>8528461
do you know how to find P({o})?

>>8528480
yes

it would be {o, {o}}

How specialised is the study of non-Euclidean geometries? How purely mathematical is it?

Does anyone know of any scientists/mathematicians who put extremely high stock in the philosophical/epistemological dimensions of projective geometry?

>>8528471
>2*2^k > k^2 +2k +1
>This must be true for all k>4 as well
This is very false. The inequality you're assuming is true is the exact inequality you're supposed to be proving.

The reason they did the thing in the red circle is because they need to find something in terms of (k+1)^2. In order to get their k^2 to turn into a (k+1)^2 they had to add those factors using the inequality shown.

There's no deep reason to that other than fiddling with the algebra looks how you want it to.

>>8528471
It's just a clever trick. The goal is to conjure up the inequality [math]2^{k + 1} > (k + 1)^2[/math], so when they get to [math]k^2 + k^2[/math], they, after a bit of thinking, come up with an expression that's greater than [math]k^2 + k^2[/math]. In this case [math]k^2 + k^2 > k^2 + (2k + 1)[/math] because [math]k^2 > 2k + 1[/math] for k > 5. And this is convenient because [math]k^2 + (2k + 1) = (k + 1)^2[/math] like we need.

>>8528461
If a set contains N elements, its powerset contains 2^N elements (i.e. all possible subsets).

The empty set, written O or {}, contains no elements. Its powerset contains one element (the empty set).

{O} contains 1 element, O (the empty set).
P({O}) contains 2^1 = 2 elements, {O} and O.
P(P({O})) contains 2^2 = 4 elements, O, {O}, {{O}} and {O,{O}}.
I.e.
P(P({O})) = { O, {O}, {{O}}, {O,{O}} }

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What's the difference between the power set of R, and the power set of R^2? Where R = all real numbers.

If they were both looked at as coordinates graphed on a plane don't both P(R) and P(R^2) contain the same exact points? Currently reading through this, and I'm not really understanding it. They're saying the power set of R^2 can contain every possible point on a plane, but couldn't the power set of just R contain every possible point on a plan as well? I just don't really get the significance of it.

Also, same with P(P(R^2)). I don't see how the set of all real numbers is any different from the set of all real numbers squared, since they both contain all real numbers.

>>8528570
>I don't see how the set of all real numbers is any different from the set of all real numbers squared, since they both contain all real numbers.
No, that's not the meaning of R^2. Go back and read the explanation of the cartesian product of sets. The set R^2 is the set of all ordered pairs of real numbers, not the set of all real numbers squared.

So, graphically, the set R of all real numbers would just be one axis, while R^2 would be a plane, R^3 a 3-dimensional space, and R^n an n-dimensional space.

>>8528588
So the set R^4 graphed would be (w,x,y,z) and would be 4-dimensional? This is the same for all natural numbers, complex numbers, etc.? Like N^3 = {(x,y,z) : x,y,z -> N} would be a 3 dimensional space containing coordinates of all the natural numbers?

What would be P(P(R^2)) graphically then?

>>8528475
I know that but I was wondering where it came from

k^2 + k^2 comes from 2(k)^2
>>8528503
so it just comes from making something thats less than k^2 + k^2

why not k^2 + (k+1)? I mean it wont work out because it doesn't factor to (k+1)^2

is it just because P(k+1) makes k^2 =(k+1)^2 which is k^2 +2k+1

Ended up with a B in Calc I this semester. I was doing well until we got to Related Rates, Implicit Differentiation, and applying integration to physics problems (like deriving the position function from velocity). Basically fucked up the whole second half of the course and made it out cause of the curve.

Should I retake it before going to Calc II?

>>8528620
>why not k^2 + (k+1)?
They could... but that wouldn't get them any closer to their goal, like you said. They wanted to get (k + 1)^2, so they used a specific algebraic manipulation to get to that point. Now, how they thought of that idea is a different question. I think the best way to learn how to come up with ideas like this is to try doing proofs yourself - do the exercises. Working backwards is often a good strategy, but you have to be careful that the implication works both ways, etc.

>>8528612
I dont actually know what P(R^2) nor what P means because its not standard notation and its not mentioned anywhere in your picture formally. I can only refer P(X) means the set of all subsets of X. In this case P(R^2) is the collection of all R^2 subsets. If you graphed all subsets of R^2, you would just get R^2. But this isnt really a proper way to think about it because the elements are themselves sets, whereas thinking of the sets representation on a graph is really considering the points inside that set. In other words you consider the graph of P(R^2) and R^2 to be the same thing despite them being entirely different sets. For the same reason P(P(R^2)) has no graphical analog. Many sets have no graphical analog. For example consider the set of the binary operations of the reals, the operations addition and multiplication. You cant graph this in any useful way.

>>8528612
Yes, this is all correct.

So you should be able to figure out now that P(R^2) is the set of all subsets of R^2... which graphically means it's the set of all possible shapes, curves, etc. in the plane. P(P(R^2)) is just a little "blow your mind" exercise. It's all the subsets of P(R^2), or all the possible sets containing those shapes, curves, etc. in the plane.

>>8528646
Thanks a lot! That cleared up everything

>>8528637
Eh, I don't think it's necessary. Just spam a bunch of exercises on Khan Academy or whatever over the break and make sure you really understand how to do all that stuff you fucked up on. Especially implicit differentiation. Some of the "applications" problems don't show up lot in later calc courses, so I wouldn't worry too much about them.

Honestly just prepare your ass for calc 2. It's not that tough but it's a lot of shit thrown at you very fast. Think about what you did wrong this semester in terms of managing your time and studying properly and fix that.

Okay so I have a 3x3 Matrix in matlab,

How do I equate rows 1,3 and columns 1,3 to 0?

I mean this generally.

All I know from a worked example to make row and column 1 =0 is using this;

K_bc( [1], : )=[];
K_bc( :, [1] )=[];

how, for example, would I make this applicable to row 3 and column 3 as well? ( I mean this generally)

>>8528825
lol, okay yeah it's just

K_bc( [1,3], : )=[];
K_bc( :, [1,3] )=[];

Well I guess we are in stupid questions thread...

I'll be taking my first quantum mechanics course next semester and I'd like brush up my math background on this holidays. Which book(s) should I work through to be comfortable with the math we'll be using?

Main textbooks are Saxon and Zettili.

Stats question:

I did an eeg experiment. People were shown word pairs and had to press a button if they were related or unrelated to each other (~150 trials). Electrodes were on C3 and C4 and we measured the average amplitude for related word pairs and unrelated word pairs for each subject. Would I use a paired samples t-test to test the difference between related word pairs and unrelated word pairs or an independent samples t-test (assuming parametric)

>>8528903
At the risk of sounding obvious, use a pair t-tests if you have clear pairs of observations (e.g., amplitude if paired + amplitude if unpaired, FOR THE SAME PERSON) and an independent t-test otherwise.

Of course the best option would depend on how you set up your experiment (e.g., did you ensure that every person was shown related and unrelated word pairs, were there word pairs that were not shown to everybody, etc.)

>>8528908
That's what I tended to think, I've just always seen people use paired for temporal differences with the same subject, not comparing two sets of scores taken at the same time.

Also my undergrad professor has told us in a similar experiment to use an independent, but they've also said "lol just lie and say your data was normal and had homogeneity" before so...

how do I become smart

>>8528918
dewd do acid it opens up the connections and makes u rlly see the world and how we're like all one being connected and shit maaan.

Read. Study. Create. Repeat

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>>8522096
How do I test to see what specific biological gender I have? Aren't there actually 8 biological genders? Not that lgtbq bullshit, but my chromosome type gender.

>>8528926
You would almost definitely know if you were not a stereotypical male or female.

Kleinfelters is not a gender, if that's what you're worried about.

>>8528946
What is kleinfelters

>>8528950
XXY

Bitchtits, fat, low t, tall, infertile.

Not always extreme though and often it only sucks during puberty and slightly after.

>>8528918
Read.

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I'm doing some WebGL programming, and I suck at linear algebra.

Pic related is this set of rotation matrices (and a translation matrix) that I just found.

When trying to determine where an object and its vertices should be in 3D space, I do:
>start with translation matrix, with the variables filled in appropriately
>multiply by xRotation matrix, angle being the xRotation of the object
>then multiply by yRotation
>then zRotation
But that doesn't work properly, everything moves strangely.

Which confuses me, because in 2D it worked perfectly fine when I just did:
>start with translation matrix
>multiply by rotation matrix

>>8528955
Thank you. I should come to this board more often so the world has one less idiot.

>>8528972
>But that doesn't work properly, everything moves strangely.
>Which confuses me, because in 2D it worked perfectly fine when I just did:
Yeah it is a little strange.
Your problem is a common one.
A rotation about 3 axes depends on the order that you do it. And the axis you rotate around is different after you apply a rotation (if that makes any sense)
I wish I had good advice for you, but I figured this stuff out, piece by piece, after reading like 10 papers on people doing similar stuff.

Wildberger has a series of 4 (ha because quaternions!) lectures, for a total of 4 hours, that gave me the greatest amount of geometrical intuition.
first one is: https://www.youtube.com/watch?v=uRKZnFAR7yw

>>8528926
Sex would be a more biological term than gender. Gender originates from linguistics (literally "he" vs. "she", "actor" vs. "actress", etc., which can obviously be way more complex in other languages) and is associated with societal roles, presentation, etc.; the two genders are "man" and "woman" (plus whatever else you wanna recognize, I'm sure there's plenty). Sex is older and is more associated with biology, chromosomes, how babies are made, etc.; the two sexes are "male" and "female". Obviously these are related but you can be XXY (or XY, if you're trans) and still just be male-gendered.

Your sex chromosomes are probably exactly what you think they are unless you have some really confusing anatomy. What's making you ask this? If you're here I'm assuming you've at least started puberty, so you'd know if you were intersex or guevodoces or whatever. Anyways, if you're anatomically and psychologically one gender, the only relevance I could see for knowing your chromosomes is fertility, which you would get checked before any chromosome testing if that was your concern.

Unless you're just curious, in which case... I guess a general DNA test? It's basically just another part of your genome. Chromosomes (probably, usually) don't have anything (obvious) to do with being trans, and hormones and I think even anatomy can get whacked up without abnormal chromosomes (and, vice-versa, a lot of chromosomal abnormalities can be completely asymptomatic).

TL;DR your chromosomes do real but if you don't have an anatomical reason for suspecting them this is like someone born without limbs wondering if they're left- or right-handed.

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Please for the love of god, somebody help me with this torque/equilibrium problem.

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>>8529007
The problem I seem to be having is that as the camera moves, all the triangles translate based on the direction they're facing rather than in global space.
Perhaps some of the inclusions of object position and rotation isn't supposed to be here, even though this is analogous to the 2D version.

>>8529035
Don't try to construct relative transformations like that.

The simplest way to do it is:
1. Construct the camera transformation.
2. Construct the object transformation.
3. Multiply the inverse of the camera transformation by the object transformation.

In "legacy" OpenGL code, you didn't have any way to invert a transformation, or create several matrices then multiply them later. So you'd start by creating the inverse camera transformation using the identity (A.B)^-1 = (B^-1).(A^-1), i.e. performing inverse transformations in the reverse order (constructing an inverse translation, rotation or scale is trivial; just use the negation or reciprocal of the relevant parameters).

>>8529200
One advantage of constructing the camera transformation separately is that it makes it much easier to accumulate motion (walking, flying).

Constructing the entire transformation chain from scratch each frame is fine for an "orbital" camera, but for walking or flying you want to concatenate an incremental transformation onto the previous transformation. So you can e.g. move in the direction you're facing by concatenating a fixed Y-axis (or Z-axis, whichever is "forward") translation onto the current transformation.

Remember: current=current*increment if you want the increment to be interpreted in object/camera space, current=increment*current if you want it interpreted in world space.

>>8529200
That doesn't solve my issue though, because whatever mistake I'm making won't disappear as I separate matrices into camera and object transformations.

>>8529204
You aren't going to be able to get that approach working for anything but the most trivial case.

Which way do those functions multiply? If they post-multiply (with the current transformation on the left and the increment on the right, like OpenGL), then the order would typically be:
projection, inverse camera rotation, inverse camera position, object position, object rotation, object scaling
IOW:
projection * inverse-camera * object
with translation before rotation (reversed for inverse-camera).

Why do temporary dipole forces attract instead of repel?

>>8527928
>>8527939
So wait.
With a derivative, then, I actually find the slope of a line tangent to a point. And I can use it to find that for any point on a given graph?
I feel like I'm getting much closer to understand now.

However, I don't understand how it relates to other things, like power and current.
What do you mean by time-derivative?
Do you mean that with a derivative, I actually find the *rate* some function decreases/increases?

>>8529301
> With a derivative, then, I actually find the slope of a line tangent to a point.
Yes.
> And I can use it to find that for any point on a given graph?
Yes.

The derivative is defined as
lim[δx->0]((f(x+δx)-f(x))/δx)
The expression for which the limit is being taken is the slope of the line connecting <x,f(x)> and <x+δx,f(x+δx)>. As δx->0, the line approaches the tangent, and the limit is the slope of the tangent.

> However, I don't understand how it relates to other things, like power and current.
The current flowing into or out of a capacitor is proportional to the derivative (rate of change) of the voltage with respect to time, i.e. the slope of the voltage-time graph. And the voltage across an inductor is proportional to the derivative of the current flowing through it.

> What do you mean by time-derivative?
Derivative with respect to time.
> Do you mean that with a derivative, I actually find the *rate* some function decreases/increases?
Yes. The slope of a graph is the rate at which y changes with respect to x, i.e. how many units y changes for each unit that x changes.

A derivative doesn't have to be with respect to time, but rate of change over time is very common in physical models (e.g. velocity is the derivative of position with respect to time, acceleration is the derivative of velocity with respect to time, power is the derivative of energy with respect to .time, etc).

>>8529319
Thank you very much. I understand this much better now.

Is life really worth living?

>>8529446
No

>>8527906
yes

>>8528397
You're blowing my mind right now. How in the flip do I go about learning these types of connections in mathematics? Does it just come naturally, or does it have to do something with knowing more about the logic that makes up maths?

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>>8529479
>>8529479
>>8529479