#Single Variable Calculus

[stemacademics]

08.07.2024 | more integral calculus.

today's energy was ~45%, I'd say.

I'm almost at the end of my single variable calc revision. I would've been done by tomorrow, if not for the not-so-greatly-productive weekend :'(

On the upside, I spent more time with my family. And I've only got a little less than a month at home before I get back to uni for my monsoon semester. I think I should pace myself carefully, lest I fall into the dark, bleary burnout phase. And we definitely do not want that happening (because then my bed and I become a single entity - either physically or in a state of mind; it's bad). anyways!!

Today I worked on problem sets and methods of integration. I'm thinking to start the 7-day challenge again! it helps me stay aware of each day, rather than the days simply blending together, which I feel is often the case with me. Well now I know better.

[emxie]

Asymptote Attraction

What happens when your long-time rival snaps? His frustrations may lead to something more.

Cyno x GN! Reader

Warnings: None

Yes, I’m writing a modern au story with Cyno based on a stupid math joke. I found someone suggesting something similar, in that he would totally be the nerd who would try and flirt with you through math puns. Me, being the nerd and student I am, actually appreciate these pick up lines, and therefore, would like to write about you and Cyno.

Also, this is a gratitude piece because CYNO CAME HOME!!!! I was so nervous I was going to lose the 50/50, but I got him! And I also got his weapon!!!!! I’m so happy that I need to write him a cute short story. Also…I’m aware this is like 3 months late since Cyno’s banner was in September-October. I was very busy, so I apologize for this being so behind schedule. Happy reading!

The hallways, not usually quiet, were no less loud today. You pressed your book bag closer to your chest, passing the jocks, primarily led by Itto, who were crowded at his locker.

“What’s up?” He nodded at you, raising a hand in greeting, before resuming his conversation with Kujou Sara. Surrounding him were Kuki, Beidou, and Kaeya, to name a few.

You returned his gesture with a timid wave, scuttling past them. It was almost third period, which meant the commencement of your calculus class was fast approaching.

Walking through the doorway of the classroom, you navigated toward your seat. Plopping down in your chair, you awaited the start of the mathematics class that you both despised and enjoyed.

You loved math itself. The intricacies of finding an answer through set formulas and having to try various things to get to the solution were never enough to satiate your love for learning. Being able to input variables and get a fixed outcome was the most comforting thing to you. The constancy of the subject made it one of your favorites.

Now, the resentment of the class. It was not due to any dislike or distaste for the subject itself. Oh no, not at all. In fact, you would have thought at the beginning of the school year that such partnership would have been beneficial. How wrong you were.

The presence of your enemy, archenemy as you liked to view him, continually dampened your enthusiasm for the class.

Cyno. The smartass of Teyvat High School. His social blunders never failed to enrage you, much less how he seemed to be doing these things on purpose. Just to get a reaction out of you.

Having been paired up with him at the beginning of the year, as the teacher thought that since you two were certainly the most intelligently gifted in mathematics, you would be a perfect match, you were looking forward to a brilliant and upstanding year with a solid and competent partner.

That, of course, was not the case. On the first day of class, he beat you to every single answer on the worksheet. And, to top it all off, he always ranked above you on class tests. Always. Even if he had scored higher than you by half a percent, he still maintained a perfect reign over the advanced subject.

And he would always tease you about your inferiority. Pointing out just how much better he was, even if he tried to appear unaware of the effect of his words, began to get tiresome. But, what could you do?

And so, you put up with his insufferable personality, no matter how much it pained you. Today would be no different.

Ah, there was the bane of your existence. He walked into the classroom, dressed in a dark purple sweatshirt and black jeans. His white hair possessed the volume of a cloud, fluffy and puffing out as if he hadn't cared to brush it before coming to school. The tips of his hair curled up, and through the thick wavy bundle of hair, you could barely see his face.

Dropping his backpack alongside his seat, he smirked at you.

“How are you?”

“Spare me the pleasantries.”

Rustling through his backpack, he glanced up at you briefly.

“You couldn’t pretend to be even a little excited to see me?”

“How could I? After you snatched the top score right out from under my nose, after that stupid word problem about surface area and volume, I can’t believe you think I’d forgive you.”

He shrugged, placing his folder on the table. “That’s your loss. It’s not my fault someone doesn’t understand how to set up two separate equations. You must have really enjoyed geometry.”

“For your information, I got As both semesters in Accelerated Geometry. Meanwhile you were struggling so hard in that class, I had to literally haul your ass behind mine in order to get you a passing grade. How’s that for stupidity?”

He narrowed his eyes at you, scoffing and flipping a strand of hair over his right eye.

“What, can’t handle the heat?”

“No.” He looked away, scanning his homework.

“You’re unbelievable,” you said, before directing your attention to your own homework.

The teacher started class, and not a word was said between you two for most of the period.

You furiously scribbled down the equation, writing it out to find the derivative. Looking at his paper, you scowled a little as you realized he was ahead of you by one problem.

You messily wrote the solution before moving onto the next one. Why, oh why, did it have to be a graph?

As you worked out the equation, grumbling under your breath as you had to derive a trigonometric function multiplied by a natural logarithm, you certainly weren’t focused on what Cyno was doing in front of you.

Having been a little ahead of you, he had already finished the problem. Glancing over his work, he huffed.

Red eyes concentrated on your hunched over form, currently clenching a mechanical pencil so hard that it seemed about to break. A tanned hand reached out, tapping your own in the middle of you drawing out a graph.

“What?” you snapped, looking up at him with annoyance glimmering in your gaze.

With a stoic expression, he lifted up his paper to show you something. It was the graph that you were aiming to draw. And…he was giving you the answer?

“Is this a joke?” You exclaimed.

He met your accusatory expression with something not condescending. Instead, he delicately pointed at the line on the graph.

“You’re the asymptote, and I’m the function.”

You blinked. What was he trying to say?

“I’m attracted to you. My love goes on for infinity.”

You scoffed at this declaration. “Nice joke, Cyno.”

His brilliant red eyes darkened. Huffing, he got up from his seat and walked straight out of the classroom.

Did you say something wrong? He had never reacted this badly to something before.

Your upper teeth tugged at your lower lip as your mind darted between countless possibilities. Should you go after him, or would that make things worse? Should you stay in here, but risk making him angrier?

Steadying your resolve, you got up from your chair and went out into the hallway, following the tracks of the white haired student.

Before long, you were able to locate him. He was leaning against a locker, slumped over so that the fluffy mane of hair shielded his eyes from your view.

“Cyno?” you called out cautiously.

A flash of red locked onto your figure through the white curtain shielding his tan skin.

He scoffed. “What do you want? Isn’t it enough to see me humiliated?”

“Well, I’ll admit that it is surprising to see you so worked up over me, the person you hate the most in the whole wide world, rejecting what said person believes is a joke just from how absurd it is.”

He muttered something.

“What?”

“I said, to you it may be a joke, but to me it meant my entire world crashing down.”

“Wait, so you actually like me?” You really couldn’t believe this boy who definitely didn’t make your heart flutter when he gazed at you with that intense scarlet stare, had the capacity to reciprocate even an inkling of what you felt.

“What do you think, you idiot?” His raspy voice did nothing to drown out the crashing waves of surprise cascading through your mind.

“I’ll be honest. I thought you despised me, Cyno.”

He turned his head so all you could see was the soft voluminous white hair that you had caught yourself daydreaming into a bit too often.

“I didn’t know how to tell you otherwise, so I challenged you at every turn. And now I feel like an idiot. It’s infuriating!”

He turned towards you, eyebrows furrowed and expression somber, trembling slightly as emotion threatened to spill over.

“Well, I guess that makes you my function. After all, I have to be the asymptote here, certainly infinite and with no end to my love for you.”

He shrunk back, looking for all intents and purposes like a small dog that had just been kicked.

“You do?”

Rolling your eyes, you reached out, grabbing his hand and placing it in your own.

“If this was what you were so worried about, you should have just told me straight up.”

Looking him dead in his stunning eyes, you explained, “I reciprocate all of your feelings. Honestly they’re probably tenfold to yours if how you’re acting is any indication.”

As he stood there, looking at a loss for words, you stepped forward, cupping his face with your hand. Your eyes flickered down to his lips before meeting his eyes once more.

He huffed a little, murmuring, “To hell with this.”

Cyno leaned in, capturing your lips in his. It was an innocent enough kiss, bordering on the edge of propriety as you felt his tongue flicker at the sides of your mouth.

You pushed him against the locker, pinning his shoulders to the metal surface as you leaned further into him. His hands roamed around your back before settling in your hair, gently twirling the locks.

It was everything you had dreamed of and more. His heavenly eyes were fluttering as you pushed him further into the locker, deepening the kiss.

“Cyno! Y/N!”

You shrank back immediately, dusting off your clothes and attempting to look as nonchalant as possible. Cyno looked just as flustered, his hair a mess and lips puffy. He looked down, hiding his face in the mass of white hair you had come to love.

One of your classmates rounded the corner. “There you two are! The teacher was wondering where you went. You guys better come back in soon. We have a quiz in a couple of minutes.”

Giving your assurances to them, you waited until they had gone out of sight before turning back to Cyno.

“So, what does that make us now?”

“Friends? More than that.” His gaze sharpened. “I’m not sure if we’re even dating. How about…frenemies?”

“I have something better. Adversaries in Affection.”

“I like that.” He smiled at you. What a beautiful smile. He looked so handsome whenever he wasn’t scowling at you.

You pursed your tender lips. “Are you ready to crush that quiz?”

“Only if I beat you,” he smirked, leaning forward to flick your forehead.

“Ouch!” you exclaimed. “Wait!” He had already bolted down half of the hallway.

“You idiot!” you screeched.

He only chuckled in reply as you ran to catch up with him, interlacing your fingers with his after only a moment of hesitation.

Math really could work wonders.

[lunarsilkscreen]

√(-4)=±2 (trying to quantify i)

A typical positive square root results in an answer of (+ or -) a number. Therefore; a square root of a negative must be both (+ and -).

The issue is: "how do we quantify a number that is both positive and negative?" It's hard to conceptualize because we've never had to use something like that, and don't really have a frame of reference to proof it.

In an early post I attempted to by suggesting that i must be adding AND subtracting a number in part. And that "i" is closer to a sign than a number.

And giving {i^x*a=[a^(1-x)-a^x]}. If x is 1; then the total is 0. But that's just a theory, and an unsupported one at that.

If we were to look at this with negative instead of i. It would be;

([-1^1]*a)=-a

([-1^0]*a)=a

Then

([-1^0.5]*a)=±a=ia

So we can infer that an exponent (or root) of a negative symbol acts as a sort of numeric potentiometer between positive and negative.

Using Euler's formula gives this;

[e^ix]=cos x + i sin x

The inference here is that "i" doesn't get *more* complex. It just is.

<aside>I'm insinuating that the "i,j,k" used in quaternion math is a completely different thing than the complex number "i" derived from sqrt.</aside>

It is used in 2D graphing to suggest a number represented in 2D, instead of the 1D line that math is known for.

I think that a single number or math formula represents a singular "total" it is limited to 1D.

For example; if we find the area of a square a*a=a^2

It reduces both dimensions into a single lump sum.

And this is true for every geometric equation, 3D geometry, trig, calculus, and beyond.

Therefore, every variable can be assumed to be its own dimension with this thinking. And therefore; "i" isn't needed to represent multiple dimensions.

But again, what does it represent? A number that is neither positive or negative, but can be used to create positive or negative sums.

As in [e^(iπ)=-1]

Why does the exponent iPI cause a number to be negative when it's i^2 that creates a negative?

The exponent i must do what a negative exponent cannot.

Axiom; Any exponent that isn't prefaced with the symbol "i" must always be greater than 0, unless the number is zero itself.

And therefore any exponent "i" must be less than zero unless the number is Zero itself.

This isn't to distance [i] from trigonometry. This suggests that it is an essential part of trigonometry.

I feel like; if I were a better mathematician I'd be better able to explain how though.

[improper-integral]

I had my calculus 1 final last Thursday, and it was the only time I actually stayed for the full 2 hours allotted to the exam so I wasn't feeling too great about it, but the grades were released today and it went pretty well!

The final was split into 2 parts, one was multiple choice with a scantron (unlike the previous exams) where we couldn't use a calculator, and then the "open answer" part where we turned in our work like usual. I surprisingly got the highest grade in the class on the multiple choice part! And the open answer part wasn't too far behind the top grade so still pretty good!

She also posted our official grade for the entire course and I got an A!! tbh my grade in Canvas was a low A (~91%) for most of the semester so I honestly wasn't sure if I'd pull off an A or not but I did!!

This was a pretty fun experience overall, I'm glad I decided to do this! Now I get to wait until February to take Single Variable Calculus II, so expect to see these types of posts come back then!

Finally, since the final wasn't based out of 100 points it's not easy to convert the scores into grades at a glance, so I did the calculations here:

Multiple Choice - My Grade/High Grade: 112/128 = 87.5% - Upper Quartile: 88/128 = 68.75% - Mean: 74.29/128 = 58% - Median: 80/158 = 62.5% - Lower Quartile: 64/128 = 50%

Open Answer - My Grade: 104/122 = 85.25% - High Grade: 107.5/122 = 88.11% - Upper Quartile: 74.25/122 = 60.86% - Mean: 62.93/122 = 51.58% - Median: 61.25/122 = 50.2% - Lower Quartile: 53.5/122 = 43.85%

So while I did pretty well, the rest of the class mostly did not 😭 I never ended up setting up a study group. I think next semester I'll try to start one right off the bat!

[armandposting]

i hate being interested in math now because what I'm ACTUALLY interested in is vector analysis, fluid dynamics, and ultimately naval/ocean architecture, but my background in basic math is so dogshit that i keep having to zoom back. like ok I cant do vector analysis without multivariate calculus which means I need single variable calculus which means I need precalculus because let me tell you I did NOT retain anything from my six weeks of precalculus in summer 2015

[notarealwelder]

@caprice-nisei-enjoyer said:

Well, at least give us a taste of what those reduction strategies look like so we can be suitably impressed when the full thing comes out?

Ok, very sketchily. I'd like to talk about five; two classical (normal- and applicative-order), two applied (Haskell's and HVM's), and one hypothetical.

---

So there's lambda calculus and a notion of β-reduction of its terms. It's unspecific! You can't directly translate it into code; it does not tell you which redex to reduce first. Consider a term: (\n -> n * n) (3 * 5). Which do you reduce first: the top-level application or the argument? The choice matters, computationally: top-level gives you (3 * 5) * (3 * 5) ( ==> 15 * (3 * 5) ==> 15 * 15 ==> 225), while argument produces (\n -> n * n) 15 ==> 15 * 15 ==> 225.

The first strategy the literature calls normal-order; the second, applicative-order. We can formalize them approximately as follows: both reduce the first reducible node of the AST of the lambda-term they find, but their lookup is different: normal-order checks the root for reducibility first and looks in subtrees second; applicative-order tries subtrees first and root as the final resort.

(We'll assume both reductions know which subtree to consider first when there's choice; e.g. that they prefer leftmost subtrees.)

Both of these are bad! Consider the following two expressions: ((\x -> 0) 100!) and ((\x -> x + x) 100!). From the point of view of these two reduction orders, they're identical: both are of form ((\x -> %irreducible lambda body%) %reducible arg%); so normal-order will reduce the top-level application in both, (getting 0 and 100! + 100!) and applicative-order will reduce arg in both. But we'd rather never reduce the expensively-reducible argument in the first — it's ignored later! — and only reduce it once in the second.

---

Haskell's a little cleverer, and won't do excess work in either of those cases. AFAIU it effectively does normal-order, but instead of substituting the body of the argument it will substitute a ≈reference to the body of the argument and store the body elsewhere:

((\x -> x + x) 100!) ==> let ref = 100! in (ref + ref)

Then, whenever the resulting ((+) ref ref) is reduced, reducing the first argument will force a single reduction of the stored-elsewhere body, and reduction of the second will simply look up the already-fully-reduced body:

let ref = 100! in (ref + ref) ==> let ref = 93326215.... in (ref + ref) ==> let ref = 93326215.... in (93326215... + ref) ==> let ref = 93326215.... in (93326215... + 93326215...) ==> etc.

(NB: let .. in syntax here is a metaphor to conveniently represent reduction semantics, not literal haskell expressions that occur during execution.)

---

Ok, what about lambdas and their subexpressions? When should we reduce these? Consider:

let f = \n -> 100! in f 0 + f 1

We'd rather reduce the expensive constant in the lambda's body only once, right? But normal-order doesn't care; topmost reducible subexpression is (f 0), and reduced it's gonna be:

let f = \n -> 100! in f 0 + f 1 ==> let f = \n -> 100! in 100! + f 1 ==> let f = \n -> 100! in 93326215... + f 1 ==> let f = \n -> 100! in 93326215... + 100! ==> etc.

And indeed (interpreted) Haskell does that. Observe:

import Debug.Trace thing :: Integer thing = let f n = trace "expensive" 2 in f 0 + f 1 ==> λ> thing expensive expensive 4 λ>

I think compiled haskell tries to do better, extracting subexpressions into lets and floating lets outside of their original lambdas, as far up as makes sense. Something like:

let f = \n -> 100! in f 0 + f 1 ==> let body = 100! ; f = \n -> body in f 0 + f 1 ==> let body = 100! ; f = \n -> body in body + f 1 ==> let body = 93326215... ; f = \n -> body in body + f 1 ==> etc

Haven't checked that though.

---

HVM claims to do this systematically, without any ad-hoc optimizations on top! It does several unintuitive things to get there though.

First, it disallows double use of variables, categorically; any time a programmer wants to use a variable twice, this desugars into a duplication primitive (whose reduction semantics will do the trick).

let f = \n -> n + 100! in f 0 + f 1 ~~> (desugar) let f = \n -> n + 100! ; g, h <~~ dup f in g 0 + h 1

(NB: HVM's HOW.md uses dup g h = f notation for this)

Second, it refuses to duplicate lambda's body without reducing it, and refuses to reduce it before it's obvious that reduction will be necessary. Therefore it....routes the duplication of the body through the dup primitive, again:

let f = \n -> n + 100! ; g, h <~~ dup f in g 0 + h 1 ==> (dup-lam (false)) let body = n + 100! ; g = \n -> body_g ; h = \n -> body_h ; body_g, body_h <~~ dup body in g 0 + h 1 ==> (g and h, used directly, can and should be substituted) let body = n + 100! ; body_g, body_h <~~ dup body in (\n -> body_g) 0 + (\n -> body_h) 1

Except that this is weird: n is now unbound in body, and there are two applications that intend to bind it (to 0 and 1 respectively). Is this legal?

HVM bites the bullet on the first subject, declaring that variables, being single-use, are bound as long as there's a single lambda anywhere that uses them. Two lambdas binding the same variable is illegal though, so instead introduces two new variables, and remembers that the old n is supposed to be replaced by those; by the first in the first copy of the body and by the second in the second copy:

let f = \n -> n + 100! ; g, h <~~ dup f in g 0 + h 1 ==> (dup-lam (true)) let n <- {n1 n2} ; body = n + 100! ; g <- \n1 -> body_g ; h <- \n2 -> body_h ; body_g, body_h <~~ dup body in g 0 + h 1 ==> (subst substitutables; n, g, h) let body = {n1 n2} + 100! ; body_g, body_h <~~ dup body in (\n1 -> body_g) 0 + (\n2 -> body_h) 1 ==> (force dup body to be able to reduce (\n1 body_g) 0) let body_g <- n_g + expr_g ; body_h <- n_h + expr_h ; n_g, n_h <~~ dup {n1 n2} ; expr_g, expr_h = dup 100! in (\n1 -> body_g) 0 + (\n2 -> body_h) 1 ==> (subst body_g & body_h) let n_g, n_h <~~ dup {n1 n2} ; expr_g, expr_h = dup 100! in (\n1 -> n_g + expr_g) 0 + (\n2 -> n_h + expr_h) 1 ==> (dup-{}; to duplicate a remembered-to-be-duplicate variable we just provide the two variables it's remembered as) let n_g <- n1 ; n_h <- n2 ; expr_g, expr_h = dup 100! in (\n1 -> n_g + expr_g) 0 + (\n2 -> n_h + expr_h) 1 ==> (subst) let expr_g, expr_h = dup 100! in (\n1 -> n_1 + expr_g) 0 + (\n2 -> n_2 + expr_h) 1 ==> (β-reduce, finally) let expr_g, expr_h = dup 100! in (0 + expr_g) + (1 + expr_h)

---and from there duplication of 100! reduces it only once, approximately as above.

(Q: why are now writing substitutables with <-, and why isn't body substitutable? A: I said this is gonna be sketchy, ok.)

---

Before the final reduction order, let's talk about strictness.

In a lazy language, bottoms (also called ⊥) can occur — terms that can't be reduced to an irreducible form. (let bool = not bool in bool, e.g.) It might be nice, then, if a function could finish computing even if some of its arguments is ⊥: who cares if trying to compute b will hang; True || b is True regardless!

We say that a function is strict in its argument if passing a bottom in that argument will make the function hang.

In Haskell, the default boolean or (||) is nonstrict in the second argument but strict in first one: True || ⊥ = True, but ⊥ || True = ⊥. Obviously we can define an or that's strict in the second but not the first (flipping (||) will suffice). However:

A symmetric nonstrict ||, with True || ⊥ = ⊥ || True = True, is impossible. (to define in Haskell. I think.)

The problem is that a single reduction step in haskell's execution model isn't small enough: any attempt to discern the form of a value (whether it's True, or False, or what) forces a reduction of that value to WNHF. Bottoms have no WNHF — or rather, trying to compute it hangs.

To define a symmetric nonstrict || we need a language whose smallest reduction steps are finer-grained: not "reduce to wnhf, no matter what", but "please make some progress towards wnhf; here's what to do if you don't get there". Then we'd be able to write:

or x y = case (reduce_slightly x) of True -> True False -> y x_thunk -> case (reduce_slightly y) of True -> True False -> x_thunk y_thunk -> or x_thunk y_thunk

And that, at last, is enough reduction.

[chronomally]

I remember a while ago someone saying they'd finished The Silmarillion and were wondering if they should pick up LOTR or The Hobbit next, and someone jokingly responding with "Hey guys, I just finished teaching myself multi-variable calculus, should I learn colors or animal sounds" but like as dense as The Silmarillion is, it's not that hard to follow. I think the confusion comes from its somewhat archaic language and the fact that it's a mythology and multi-generational saga, compared to a single story like in a traditional novel

[sevicia]

HIIII today's exercise haul is the first 47 questions of the 1.1 section of Single Variable Calculus (7th edition) by Stewart, except I skipped 16 thru 22 cause they were repetitive & did examples 1 thru 10 of the chapter in addition to the questions themselves. So 50 total if u count the examples (which are solved right in the book but I still did them cause u know.) It took me ummm 5 hrs with looots of breaks maybe too many breaks TBH. Also they were all about representing functions so I did a lot of graphs which is mostly easy when it comes to describing situations that don't really involve actual numbers. I was getting to the ones that do involve them but my brain is kinda fried because of how repetitive it was @_@ still I'm happy with today :thumbsup:

[topoillogical]

From Real's math asks: 9, 47, 52 :)

Thanks for the ask!

9. Do you have any favorite theorems?

The hairy ball theorem is pretty good. I also like the fundamental theorem of calculus, and nonexistence of a quintic formula (although that one is more like a love-hate relationship)

47. Just how big is a big number?

They start around 70 factorial or so

52. Do you have favorite math textbooks? If so, what are they?

I don't have any huge favorites, but there are ones I like more than others. Spivak's "CALCULUS" is a good introduction to single-variable real analysis, and was one of my first higher math textbooks, so I have a soft spot for it. Fun fact, I did every single integration question in that book over winter break in my first year

[stemacademics]

day 2/7 | 14.07.2024

some trees. not nearly as lush green as I'd seen.

a bit lengthier update, I suppose.

well, hello again! today was only marginally better than yesterday (i am still not greatly proud of myself, not at all) but. we did some work. might not be reassuringly solid, but maybe in the non-newtonian fluid typa tricky solid feeling, i guess?

well. anyhow. more calculus today (of course). but am near the end of single-variable. will start multivariable tomorrow. also a friend texted me saying that they'd finally started their summer internship report that they'd been sitting on for too long, which frankly, kinda called me out in a not-so-great way, but hey, at least there was some extra motivation to drag myself up today. yeah. we haven't been having the best of days these days.

my prof is yet to reply me back for updates about our meeting (my anxiety is through the roof, with a velocity>>>earth's escape velocity; one might assume that that is the point of Numbness but no. No. sometimes i forget to breathe and i panic-press my Violin for the Desolate playlist in sorry attempts to fend off my panic attacks). i also am planning to maybe talk to my old therapist. but i am slightly (Very) nervous about that too. because it is highly likely that he has forgotten me (i saw him last year, around summer. and i'd already done some tough sessions with him. redoing them would be... yeah, even more tough. also. i have observed a pattern; my summers have been Difficult for different reasons each year since middle school, but that's a story for another day. i should come up with a name for that. and no, Summertime Sadness just sounds too good.)

anyhoo. today's focus time was 3hrs and a bit more. yesterday's was 38 minutes. hmm. there's still a whole mighty mountain, sending me into panic. wish me luck guys, i think i might need it.

[all-her-stars]

Spring Semester Schedule

Successful Starts in Science // W // 5:30-6:30pm // 1 credit

Intro to Programming in Python // MWF // 12:20-1:10pm // 3 credits

Calculus of a Single Variable // MWF // 11:15am-12:05pm // R // 12:30-1:20pm // 4 credits

Introduction to Astronomy // MWF // 1:25-2:15pm // 3 credits

Astronomy Lab // R // 8:00-9:50pm // 1 credit

Introduction to Science, Technology, and Society // TR // 2:00-3:15pm // 3 credits

[links-studies]

Hello! I’ve been wanting to get into astrophysics when I start uni. What’s it like studying astrophysics ? I’m curious,,

Hi! I'm so sorry I'm so late to answering this: I only just finished my semester and I was trying to lock in to make sure my grades stayed fine through my senioritis ^^;

To answer your question: Hmmmm I suppose the exact experience depends on whether or not you have a specific specialty of astrophysics you want to go into, and what school you'd like to study at! If you have something specific you're looking for in your studies/you'd like advice on, I'm happy to answer further questions （*＾-＾*） But I could also speak on some general things I've noticed and experienced in my studies that a lot of people don't seem to know before going into the field!

Generally speaking, astrophysics (and physics in general, really!) is a very math-heavy field! I find a lot of people go into physics not expecting the amount of mathematics knowledge that's required. Do not be fooled, you will be doing lots of math! Not just punching numbers- physics gets into some of the grittier parts of higher level mathematics applications. There is a lot of calculus (single variable and multivariable) and linear algebra in particular. Personally, most people I know who are graduating now are either math double majors or were not terribly far from having a math degree with the number of math courses they took. If you enjoy applied math, you'll probably have a great time! If you're not a fan of math, then there could still be other parts of studying astrophysics you enjoy. But astronomy is another great option that might be better, depending on what you're looking for! (❁´◡`❁)

Another thing I'd like to note is that it is unlikely you will get to the astrophysics specific classes until your later years in uni (unless you decide to take some astronomy classes first). There's a lot of physics you need to learn before you get to all of the other physics, and learning it all takes a lot of time! Many of the specific concentrations and specializations stuff doesn't come up until much later on, so you gotta keep in mind what your goals are and try not to lose your motivation for studying all the other stuff first. Especially because when get around to taking your astrophysics classes, it quickly becomes apparent why you needed to learn it all to begin with! It requires understanding of most of the concepts you learn prior to it (classical mechanics, electromagnetism, special relativity, thermodynamics, etc). But believe me, it's fun when you finally get to it! It's like a puzzle piece coming together, and I can say I certainly gained a much deeper appreciation for how the universe works afterwards.

Additionally- and I cannot say this enough because some people don't seem to realize it until they're in pretty deep- if you're not personally a fan of data analysis and having to work with at least some coding, you might want to consider going into a teaching application of astrophysics or contemplating a different specialization! Most astrophysics research involves working with a ton of data, always, all the time. Even in fields which could be considered to be "untraditional" fields of astrophysics still require working with data. If you find data analysis fun, you'll probably really enjoy doing astrophysics research! If it's not up your alley, it may not be a great time. You could probably still study astrophysics without having to do data analysis, but personally I don't see that being worth it unless you plan to stick to teaching physics/astrophysics outside higher academia. I'd suggest taking some data analysis courses if you have the opportunity and seeing how you feel about them.

Those are kind of the key points I can think of to mention when it comes to what's important to know about the ✨astrophysics experience✨. From a more personal standpoint: I am obviously biased, but I've had a great time so far. Part of me just kind of thought space was cool going into it and I still totally think space is cool, but from a much more appreciative and knowledgeable standpoint now. Not to mention, I would say the astrophysics community is very open and welcoming. There's lot of wonderful organizations you can join, and I've had a wonderful time at every conference I've been to. Not only that, but I feel like the community really puts effort in to welcome and accommodate people of underrepresented backgrounds- there's lots of events geared towards people from all kinds of walks of life. Such things are important as well, because the astrophysics community is incredibly collaborative! Most research is done at least in a team, if not part of a much bigger organization. This might depend on what specifically you want to research, obviously, but for the most part it holds true. And that experience is a lot more fun than you'd expect it to be! I've had many wonderful collaborators and team members in my time doing research, and I expect I'll still continue to meet much more. That sense of community that I've gained in my studies is personally invaluable to me. Again- biased!- but I'd say on some level astrophysics is worth studying just for that feeling alone.

Aaah sorry I realize I'm rambling a little bit. As aforementioned, I'm more than happy to answer any more questions you might have (´▽`ʃ♡ƪ) My background is personally from a smaller school going into graduate studies (I'm going off to get my PhD focused on gravitational physics research, specifically!) so I might not be able to speak much on studying physics at most bigger schools or on going into industry after your uni studies, but I'm happy to answer questions about generally what it's like studying/doing research in physics, as well as questions about preparing for grad school 💚

[zooterchet]

Quantum Calculus (Charlebois Proof)

Imaginary Number: Three degrees of dimension, to remove one, to calculate timed interval.

Sigma Proof: Contradiction of unfound variable, x, y, z, is solution to variable.

Triple Formula Enactor: Three formulas are taken, on x, y, and z, to solve.

Singular Benefactor: One formula, is the equation to all three formulas, with a single, x, y, or z, unsolved.

Matrix Proof: Matrix sheet, on row, is gathered research, and on column, is Sigma Proof operations. Single column across, is unsolved variable.

Operations Register: Each type of operation on Matrix Proof, is given function of abstract, the word problem.

Constant Sheet: Each constant has to be assigned, through Matrix Proof, to solve, x, y, or z, until Sigma, can be applied through Singular Benefactor, to leave one, x, y, or z, unsolved.

[dingdangit]

taking single-variable calculus through MITs open courseware and let me tell you. the quality of their chalkboards is borderline sexual to me

[myrahasan]

Computer Science - Dickinson College

Coming into Dickinson and not knowing which direction to take in terms of a major, I wanted to take classes in my first semester that served as potential routes I wanted to explore further. And so, I took a class in the Political Science department, a class in the Computer Science department and a class in the Philosophy department. Although each of these classes expanded my knowledge in different ways, the one class that I enjoyed the most was my Computer Science class, Introduction to Computing. This class environment was not only the most welcoming but the most rewarding too in terms of being able to write lines of code and see that it did computations, created graphics and had an immediate effect and product that came from it. Being an international student unfamiliar with the American culture and on top of that, college culture, being able to work on labs and projects with other people in the class helped me ease into this change comfortably, whereas the other classes primarily emphasized individual learning. Following that first Computer Science class, I was sure that I wanted to major in Computer Science. Hence, the following semester, I was able to take Calculus I, which I needed as a prerequisite to the math that I needed for the major. I was also able to take Principles of Object-Oriented Design. Calculus I was beneficial in laying down and reiterating the mathematical concepts that would be necessary for Single Variable Calculus. Principles of Object-Oriented Design was interesting because while Intro to Computing was primarily focused on Python, this class was focused on Java. It was interesting in this way to notice the similarities and differences between the two languages and how they serve different and similar purposes. It also delved into some more detailed projects and coding methods while still allowing collaboration with others in the major, which was a rewarding and beneficial aspect to the class to be able to build relationships that have lasted throughout my college career. In the subsequent semester, I was able to take up Single Variable Calculus, Foundations of Higher Mathematics and Computing Abstractions in the major. All three of these classes were vividly different from one another but have all connected in the classes in the major. Single Variable Calculus helped solidify some of the mathematical concepts and computations that come up in coding and the principles of coding. Foundations of Higher Mathematics helped detailed ways to write mathematical proofs to be able to prove and disprove elements within Computer Science, namely the computability and complexity of programs. Computing Abstractions laid down the foundation of coding and computing and the most basic elements of what make up a computer, which explained how programs are run and how their efficiencies can be maximized. 

For the following two semesters, I had the privilege to study abroad in Norwich, England at the University of East Anglia where I was able to continue to take classes in my major that translated to Dickinson’s requirements for courses. This transition was stimulating as well as constructive as it offered me an experience extremely diverse from Dickinson, with Dickinson having small class sizes and encouraging group learning, whereas UEA had much larger class sizes and had more of an emphasis on individual learning in ways that I had never experienced before. In England in the Fall, I took Systems Analysis class which stood for the special topics 200-level elective where we focused on the development of computer-based systems. In the Spring, I was able to take Software Engineering, which stood for the half credit courses taken at Dickinson, Comp 190 and Comp 290, which segue into Senior Seminar. In this class, we worked on creating a POS system for a grocery store, with first implementing a class model design and then transitioning that into a working software system, focusing on both, the frontend and backend aspects. I also took Networks, which counted for the Abstraction Implementation requirement for the major. In this class, we focused on encrypting and decrypting a Voice Over IP system as well as making sure the transmission of packets was seamless. We also focused on Network Monitoring, including tracking latency and other components and seeing what aspects affected these elements. Finally, I also took Data Structures which translated to the Data Structures class at Dickinson and helped to familiarize us with the different structures that exist to model computer programs as well as implement them, which was crucial to understand for other classes in the major, such as Complexity and Computability and Analysis of Algorithms. 

Coming back from being abroad, I was able to take Analysis of Algorithms and Senior Seminar, which takes course over a year. Analysis of Algorithms helped build our knowledge on how to analyze how efficient programs are in terms of time and memory and how to improve efficiency. In Senior Seminar, we delved into the principles of Software Engineering, including looking at some practical efficiency techniques used within companies such as Scrum and Kaizen. We also analyzed Open Source projects and software in order to gauge our interest in them as well as our ability to contribute to them and began to work on these in closing issues and adding to them in meaningful and useful ways. Finally, this semester, I am taking Complexity and Computability, which looks into how certain programs are uncomputable and proving this as well as the complexity of programs and modelling programs through Turing machines. I am also taking Senior Seminar, where we are continuing to work on and contribute to our open-source projects as well as looking into the ethics and morals of Computer Science related topics. Lastly, I am also taking Database Systems where we are working with databases and learning to how construct and retrieve information from databases using SQL. 

Reflecting back at all of the Computer Science classes now, it is evident how they all connect and build on each other and it is rewarding think about all the coding languages that we have covered, all the analysis skills we have acquired and can now think critically about matters related to Computer Science, such as computability and complexity but also ethics and morals related to the field. Looking beyond the major, it was also been rewarding to not only be able to gain knowledge related to my major but take classes in all the fields that interested me, including Psychology and Anthropology and understand these and be able to relate these back to Computer Science as well. The combination of the disciplines has helped me to think more critically about the world around me and approach it through different methodologies and as a consequence, act in ways that are influential and make a difference. The study abroad opportunity played an immense role in taking risks and seeing things from different perspectives and how they apply in different contexts and to be comfortable in unfamiliar settings and use this to enhance my skills and knowledge. Within Computer Science, I have learned to adapt and solve problems and transform them to be more efficient, with one specific example being to reduce problems to other uncomputable problems to show an initial problem is uncomputable or to try different approaches to solving one particular issue. The structure of Dickinson’s classrooms also helped me to become comfortable in working with different people and aligning our skillsets to solving issues and making progress. The contrast between Dickinson’s classroom environment and UEA’s classroom environment also gave me the flexibility to adapt in learning environments. I consider it an immense privilege to have had the ability to learn all that I have learned in a short four years, within the classroom as well as beyond it and with skills that can be applied to my major as well as beyond it. It is extremely rewarding to look back and reflect on my progress and how Dickinson has shaped me. 

[mreiyouscience]

　学ぶということは過去の偉人が築いた軌跡を辿ることを意味します。このことは数学の一分野である微積分学にも当てはまります。ニュートン及びライプニッソによって誕生した微積分学は当初数学的厳密性に欠けるとの批判を受けましたが、条件として誤差がイプシロン以内であるならば極限による近似を正しいと定めることで厳密に理論を構築することが可能となりました。これがいわゆるε-δ論法です。このε-δ論法によって局所的な微分及び大域的な積分を数学的にかつ統一的な視野から記述可能となりました。本書ではこの統一された視野から俯瞰して微積分学を学ぶことができると思います。

次に学習のロードマップについてお話ししたいと思います。まずは、YouTubeの微分積分の動画を観ながら、マセマシリーズを”ボロボロ”になるまで読みましょう。

微積分入門

実はマセマシリーズを完璧にマスターすればA以上を目指せるだけでなく、高嶺の花だとされている斎藤正彦先生のテキスト「微分積分学」を楽々と読みこなすことも夢ではないのです。だからこそ、微分積分学の基礎となる土台の足固めをマセマシリーズで学ぶことが大切なのです。（逆に言えば、斎藤正彦先生のテキストが難しく感じるのはマセマシリーズを完璧にマスターしていない何よりの証なのです。）

なので、どんなに時間がかかってもいいのでマセマシリーズを完璧にしましょう。その後、演習書としてチャート式を解きこなし、体で覚えましょう。

また、英語が得意で意欲的な人はYouTubeで「MIT 18.01 Single Variable Calculus, Fall 2007」、「MIT 18.02 Multivariable Calculus, Fall 2007」と検索して世界トップクラスの大学の授業を体感していただけたらと思います。（英語字幕あり、実はそれほど難しくありません。）

追記

夢の実現方法―PDCAサイクルを実践するー

１,Plan ；具体的に目標をイメージしてそれをノートに書きとめる。

↓

2,Do ；目標を達成するために努力する

↓

3,Check；現実と目標のギャップを受け入れて何を改善すればいいか考えてノートに書きとめる。

↓

4,Action；Checkでの反省を活かして努力する。

↓

1,Planに戻る。

～追記～

研究者もしくはエンジニアを目指す方はぜひ、こちらの書籍と動画を参考にしてこれからのキャリアを築いて下さい。

～エンジニアになるためのコツ～

「YouTube」で「デビッド・ケリー 「自分のクリエイティビティに自信を持つ方法」」と検索する。

↓

「YouTube」で「The surprising habits of original thinkers | Adam Grant」と検索する。

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「NHK」の「デデデデザインて何？！」を一年間見る。（YouTubeにも動画あり。）

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雑誌「日経デザイン」をたまに読む。

↓

有名デザイナー（「ジャスパー・モリソン」や「深沢直人」、「佐藤オオキ」などのホームページを調べる。またはグーグルの画像検索で検索してみる。）

↓

「太刀川英輔」の「進化思考――生き残るコンセプトをつくる「変異と適応」」を読む。

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「Jeff Patton」の「ユーザーストーリーマッピング」を読む。

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「James Kalbach」の「マッピングエクスペリエンス ―カスタマージャーニー、サービスブループリント、その他ダイアグラムから価値を創る」を読む。

～研究者になるための書籍～

「近藤 克則」の研究の育て方: ゴールとプロセスの「見える化」

　～研究開発組織をイノベーティブにするための書籍～

「佐宗 邦威」の「直感と論理をつなぐ思考法 VISION DRIVEN」

「佐宗 邦威」の「ひとりの妄想で未来は変わる　VISION DRIVEN INNOVATION」

「野中 郁次郎」の「直観の経営 「共感の哲学」で読み解く動態経営論」

「エイミー・C・エドモンドソン」の「恐れのない組織――「心理的安全性」が学習・イノベーション・成長をもたらす」

「ピーター M センゲ」の「学習する組織――システム思考で未来を創造する」