Two days after this dinner, Gen and Krista meet again at Krista's home, visibly hungry for mathematical knowledge that they didn't want to touch until such time comes to do so. Complex matrices, eigenvalues, eigenvectors...
"Look at this... a matrix of eigenvectors, eigenvalues that we find by setting the determinant of A-λI equal to zero, but what we must solve for lambda..." an awed Gen then talks about linear algebra to Krista.
"Once we solve for lambda, we have, in general, after substituting in the equation, that the lines of the matrix are equal, up to a multiplicative constant, and these are our eigenvectors. But you know that, with matrices, you must pay attention to multiplication order" Krista then tells her.
"Good Lord! You know that the VMC is in two weeks and we're studying linear algebra? That won't help us! Neither for the AIME, nor for the VMC. We may calculate two or three eigenvalue and eigenvector problems for form, as well as their diagonalizations"
"Remember what you said at the dinner..."
"The VMC is a curricular contest. But because it's a curricular contest does not mean we mustn't train for it. I believe it's appropriate to learn other mathematical notions that make use of derivatives and integrals. I don't know... curvature and torsion"
Oh... the temptation is far too great to binge for the VMC, especially with Krista who, like me... However, it's there I think I lost my mind. Last summer we spent some time on differential equations, Fourier series, and now we are doing differential geometry? Already there was someone with me at YSP that found me lead-footed, and from what I see we are running into a wall! Geneviève thinks, wondering why she uttered the words curvature and torsion.
Krista's parents watch both girls perform calculations that bewilder them. Curvature calculations, torsion, Frenet frames, osculating circles and spheres... never would they have imagined people in high school calculate such mathematical objects.
"What the hell is this? What are they learning?" Krista's father asks both girls, who, for as fanatical a Venomous Agendas fan as he is, can't help himself but look at their calculations.
"All these calculations assume we can parametrize a curve as C(t) = (x(t),y(t),z(t)), with C(t) being thrice-differentiable. We thus calculate three derivatives, even though the torsion is the only calculation that uses the third derivative" Gen answers Krista's father.
"Differential geometry. We are training for the Vans Math Contest by learning other mathematical notions that are new to us but uses what the VMC uses" Krista follows up with her own answer.
"Have you lost your minds? I know you are both mathletes but are you sure that you picked the right methods to train? You're both in the AP Physics C course, you should have enough for the VMC with that" Krista's dad scolds the two girls.
"The other three will proceed like this, but sometimes audacity can pay off, and also, in the current case, in the long run" Gen responds to this charge.
Meanwhile, the other three that entered the VMC for the Venomous Agendas are studying integrals in more detail than what the AP Physics C course did. And single-variable integrals. And with Randy, who could benefit as well, even though he's not a contestant at the VMC.
"Randy, the only reason why I brought you here is because you take the calculus BC course. You can stop going to the math team practices after the AIME, if you want" Marcia explains to Randy, before turning to Cory. "I hope that Gen doesn't hold it against you that you're training in the homestretch for the VMC with me rather than with her"
"No. She masters the material of both VMC rounds better than I do, so she feels like training with Krista will be more enriching to her than training with you three" Cory answers Marcia.
"By listening to you maybe it's just a question of time before I can use my mathletic prowess to get a girlfriend" Vontae comments.
"We wasted enough time, we will distribute last year's regular final for calculus BC, that I obtained even though I didn't take it. Please ignore the differential calculus part because you took a half-year exam on it" Marcia explains, with three copies of this exam on hand.
Unsurprisingly, according to Cory and Vontae, they are at the top of the calculus BC class as of December 26, but it's a little early to talk about multi-variable content.
And Zack comes back to see Tara that same day; he doesn't even have the slightest idea of the opportunity he has to both get mathematical help and to potentially sabotage the preparation of an opposing player for the VMC.
"Hi, Tara"
"Hi, Zack. Remember the first question asked during the dinner?"
"Yes, but spare me the mathematical details; I won't understand anything"
"What was the aim to date a girl specifically because you wanted to date a girl that..." Tara loses her train of thought while her face turns red.
"It's complicated... and this question is just as complicated for me. The limit of x going to zero of a fraction. My other teammate said the solution to it came from a film after shouting The limit does not exist"
"If we calculate the limit directly, we have an indeterminate form zero over zero. There are two ways to lift an indeterminate form. Either we examine the behavior of both levels of the fraction, but since you asked me to spare the mathematical details, I won't talk about the second method. The hospital rule. If we examine the numerator's behavior in the neighborhood of zero... it will be simpler for you"
"The denominator will be even simpler: it will always be between zero and one"
"Start by taking the numerator for small positive values of x... x is in radians, by the way. Pay attention as to whether the values are smaller or greater than zero"
All that Tara knows is that one would pretty much need to be a Venomous Agenda to treat a mathlete as a romantic trophy. Yet most of the time we respect that so and so is in a couple, and we won't try to force a boyfriend or girlfriend to break up, but all three of them must pay attention to their respective relationships, he thinks, while Zack is in pre-calculus like many senior footballers. And he is thinking of what happens to the numerator on each side of zero; it seems to be a little slow to him.
"Yes. Is it me or the sign of the numerator changes at zero?" Zack asks about his observation.
"Now you know why the limit does not exist"
"I have a question for you: why did you not want to answer practice questions for the VMC during the dinner? You could have won that gift card... but without you the public was completely powerless against even Cory or Vontae over calculus"
"I didn't want it to transform into a showdown between them and me; I already had a similar experience at the Duel of the Parishes. In fact, no, the public wasn't totally powerless over these questions, but Randy, in all likelihood, could have and should have answered some of these questions. And it's not true that you don't understand anything"
"You know that, in football we work much harder in class in the second half of the year. But math is my big weakness so... I must stop envying my teammates"
"Good, very good, Zack. But I think it's not time to date; for the entire duration of the holidays, maybe after the VMC we might go on a date. I don't expect much from the AIME because I narrowly qualified so just being there is very good for me"
"Thank you"
A bit disappointing for Zack, but he realizes his girlfriend cares more about the VMC than the AIME. And we're talking about a player in the Math Madness playoffs in the top division!
On New Year's Eve, it's the turn of Geneviève's parents to ask questions about the preparation that she and Krista are making regarding the single-variable calculus phase of the VMC while differential geometry is a multi-variable calculus application; they know the town has very high expectations regarding this competition, but the level of the calculations and the mathematical objects they manipulate leave them perplexed and speechless.
You could be reading stolen content. Head to Royal Road for the genuine story.
"What?" Geneviève's mother shouts in surprise. "Christoffel... symbols? I'm lost, honey, I know you care about the VMC, and the rest of the town as well, but what's the aim of proceeding like this? You know what the competition is made of, you know most of your opponents will only do it once"
"Yet last spring you said absolutely nothing when I was solving PDEs in open air, and now that I am covering both fundamental forms, the first representing the intrinsic properties of a surface while the second depends on the surface's plunge in space..." Gen answers her mother.
"There's something wrong. What motivates you to proceed like this?" Geneviève's father asks after having interrupted her.
"We want to get ahead and learn something new for college" Krista responds.
"You said it so yourself that you solved PDEs in open air last spring. How many high schoolers would be able to do this?" Geneviève's mom asks her.
"Not a whole lot. Most of those who do so are good mathletes, and nearly all of them will be able to qualify for the AIME" Geneviève makes her estimate.
"Surely you know that, for most people, even being able to do single-variable calculus in high school represents a very high level of mathematical talent. Maybe not for us, but it's still something that I respect" Krista continues on.
In the previous days, they calculated objects about curves in space, osculating spheres, planes, evolutes and involutes (i.e., reverse evolutes) as well as pedal curves, by using problems appearing on undergraduate differential geometry exams from past years to practice.
Yet to calculate a Christoffel symbol it requires the first fundamental form, and it appears in the calculation of geodesics, or the shortest paths between two points on a surface (the word manifold appears too often on their worksheets to the parents' tastes)
"You're fantastic, Gen, you see realistically who exactly would be able to do what we are doing. We see all the time examples of people who, in high school, take night classes in college, especially those who dream of Ivies, and the most extreme mathematics cases are close to us. What courses they take exactly depend on the profile"
"You too, Krista. But some mathletes talk about the calculus trap"
"You fell headlong in it" Geneviève's father comments, before turning towards his wife. "Look at the speed the two seem to fly through a differential geometry textbook! Last week they barely learned curvature, torsion and the Frenet frame; now they're talking about Christoffel symbols? And they claim to be doing this to train for the VMC? I wouldn't be able to calculate anything they are talking about, and you either"
"You go way too fast in your training regimen for the VMC. I know you want to make the town proud of its mathematical prowess, but I think you had a point during the dinner" Geneviève's mother points out to both girls while they keep calculating Christoffel symbols, Gauss-Codazzi equations...
Even though it's not a common debate to have, both mathletes are in a funny situation where it might be relevant for them to have. The infamous acceleration vs enrichment debate. The calculus trap Gen alluded to, and that some mathletes could condemn in them.
"I know that, in all cases, regardless of the proportion of acceleration vs enrichment we have, we must pay attention to the people with whom we're studying. That's why I'm very close to Krista, and she's very close to me. That's also why, if we add Marcia, all three of us banged our heads with college admissions the way other people didn't"
"No need to remind us, all three of you are very good. But what makes the calculus trap a trap exactly?" Gen's mon asks her daughter.
"It's not as relevant for us, but for someone younger, and I will be honest, it can be a source of false challenge to some, in the sense that the problems will still remain rather computational, just with more advanced notions. Also, even though differential geometry might seem new to us, what actually constitutes a novelty for us is the topology behind it" Gen explains.
"From what I've seen you can make these wads of calculations without complication" Gen's father then comments.
"I come back to what I said at the dinner. Exploiting the calculus trap, as Gen alluded to, might be a good idea if the ultimate objective is more applied. But for someone dreaming of pure math, going in greater depth in more elementary topics can be a better idea, and for them thinking and solving complex problems is important. The debate of acceleration vs enrichment does not comprise the same dimensions for us than for people in elementary or middle school. In elementary and middle school, one needs to consider the psychosocial dimension on top of that" Krista comments on the calculus trap.
But what happened? Would things have happened differently for the three of us if there wasn't that lockdown that straddled our eighth and ninth grades? This geometry summer course was equivalent to doing a whole year of mathematics in a month, already that I was made to skip a year of math and a year of science in middle school so that I am taking Algebra I in eighth grade? Up to a few details all three of us followed that trajectory. The parish, in general, accepts to accelerate students on a per-subject basis without skipping an entire year across the board, but it takes a student like me for it to work, she thinks, while what Krista had to say about the calculus trap was actually more general than the calculus trap alone. It was also a gifted children's education issue.
Meanwhile, Marcia continues to teach single-variable integral calculus intensively while abstaining to touch multi-variable integral notions. By reviewing key concepts the dinner questions covered. Like integration by parts.
"If we solve this, what do we get?" Marcia asks [https://img.wattpad.com/a5541ddaca46acaac37e466503acb0fa6be76fbb/68747470733a2f2f73332e616d617a6f6e6177732e636f6d2f776174747061642d6d656469612d736572766963652f53746f7279496d6167652f71567247485055366337796569413d3d2d313239323131323335302e313732643764393836396239303461653335303539303632393430352e706e67?s=fit&w=1280&h=1280]
"If we solve this, what do we get?" Marcia asks.
"Regardless of if we take sin x as u or as dv it will repeat itself after doing the process twice" Vontae points out.
"If that's how it will be, my lover, then the answer will be"
"If that's how it will be, my lover, then the answer will be" [https://img.wattpad.com/fd38fecee992eb825c8c44d848fa96276bc3ecdf/68747470733a2f2f73332e616d617a6f6e6177732e636f6d2f776174747061642d6d656469612d736572766963652f53746f7279496d6167652f493856476769454237475a4641773d3d2d313239323131323335302e313732643764396265643763376539393834353135383837323239382e706e67?s=fit&w=1280&h=1280]
"Very well, Randy, we'll get to the next step... Taylor series" Marcia then continues.
And it's not just because I date Marcia that people on the football team consider me as the team's math geek; it's also because I am the player that has the best grade in calculus BC. Or rather I am the only player in calculus BC, Randy thinks, while they are getting explanations on Taylor's theorem, as well as other useful results to work on power series, such as the convergence criteria and radius.
"You're telling me that we need to calculate several derivatives of the function we want the series of about a point to obtain one?" Randy asks, a little lost.
"And if possible, evaluate them at a point" Cory adds. "Like the series of e to the x about zero"
By far the most important convergence criterion is the quotient criterion, in other words the limit of n goes to + infinity of the quotient between the n+1-th term and the nth term of the coefficient sequence. Not that the root or comparison criteria aren't important.
"You go way too fast, Marcia... With your mathlete teammates it's ok, they are used to double down to understand that, but I am not as fast as your teammates. And certainly not Ted's girlfriend"
"Krista, Vontae reminds Randy.
"Randy, I can't believe you could forget the name of Ted's girlfriend! You already met her multiple times. What's wrong?" Marcia asks, starting to see signs of dizziness, worse than in the training sessions of the math team he attended.
"Krista, what a maniac, that one..." Randy rambles while he's mentally exhausted by six days of intensive learning of single-variable integral calculus. "I do this now so I can take it easy for the rest of the year in single-variable calculus"
"We must stop here for today and tomorrow; we'll miss New Year's countdown. But you can do integration by parts, it's already progress for you" Marcia announces to the three guys before kissing the cornerback.
Speaking of the New Year's countdown, Geneviève and Krista are both around the dining table, while they attempt to remember in their heads how to calculate curls, divergences, as well as the calculation of multiple integrals, while they know they have no desire to touch that final chapter that links what they spent the last six days to study in an intensive fashion to topology, that is, the Gauss-Bonnet theorem.
"There's something I didn't understand of what you talked about earlier, Krista: the psychosocial dimension?" Gen's mom asks Krista.
"It was certainly my case, I was suggested to make me skip fifth grade, but my parents refused. It's not that I couldn't double down academically back then, but so many people in a similar situation believe the students would be too immature to function with different age groups" Gen talks about her academic past.
"For me it was seventh grade, not fifth" Krista adds another detail.
"Now it comes back. It was painful, we fought for weeks. But it's only now that this ghost from our past resurfaces, while we are dealing with two super-brilliant girls that burn through a differential geometry textbook in a week, and that are in the same academic boat?" Geneviève's mother is wondering while reminiscing these painful scenes.
"Mom, on the one hand you're saying that we're going way too fast in our VMC training, on the other hand you say that we can perform these wads of differential geometry calculations without complication. There's something wrong. If we really went way too fast in our training for the VMC, there would have been complications in our learning of differential geometry... I acknowledge, however, that it's very fast"
Logically I could and should have done more on a mathletic level in the first half of high school, but why it was only last year that I started to perform as I currently do? I am not worried about the VMC, but was it just about inadequate support during the first half of high school? If I had adequate support during the first half of high school, would I have been in position to compete in the EGMO, or the IMO even? Geneviève thinks, while also remembering the situation from which Krista sought to get out by transferring schools. I already accomplished a lot mathletically, and this season looks even better than the last; in our corner of the state there are few people that dream even of the AIME, let alone the USAMO?