On April 27. two weeks after the end of the church's tax clinic, not even the auditorium sufficed to hold the entire crowd that came to support the Venomous Agendas from across the town, and even from elsewhere in the parish (the county equivalent in Louisiana), the day before the national final of the Square Root of the Answer contest. All the mathletic teams of the region's schools are attending and, for the first time of her life, she sees, before her eyes, a mathletic competition that holds a community together. If, during the victory announcement, the school was cheering, but few people outside of the school system seemed interested, on the day before the big day, the town believes that Louisiana, represented by their Venomous Agendas, will be the Cinderella team of this tournament. But will people remember later?
"Three months ago, no one could have seen coming the victory of our school at the state stage of the Square Root of the Answer. This season has been exceptional for the Venomous Agendas, with historical records on the AIME which, even though our performances have not inspired this town as much as the SRA, do not detract from their exceptional character. Tomorrow, the biggest mathematics competition in parish history begins and, regardless of what happens next, let's enjoy our trip under the limelight because it may very well be the only time in our lives that we will be able to obtain public recognition for having talent in multi-variable calculus!" Éliane addresses the crowd. For me this represents the ultimate hour of glory before my departure for Duke this fall!
Underdogs at the national level? I would love to believe that Louisiana has never been a mathletic hotbed, but for the past three months, the whole town pinned its hopes and its adulation in us! Geneviève thinks, while knowing that she isn't the only one competing that isn't a senior. I may be the youngest, maybe it's for this reason I am the mathletics mascot here! But also because the people around here seemed to imagine girls as lacking in mathematical talent until now...
"Show to these Northern states that we shall no longer consider the Deep South as a weak region academically!" Cory howls while music starts to be played.
"I don't want to be a killjoy, but what makes the town so proud of its multi-variable calculus kids?" Shane asks, sticking to reviewing single-variable calculus, at least at first.
"For far too long did we place our pride on our sports teams without realizing that it numbed our minds. Credit must be given to these people who engage in activities of the mind!" Cory's father explains, while the other mathletic teams of the parish eagerly await the review of multi-variable calculus.
"Most people among you never did multi-variable calculus. For you it will be entertaining, nothing more. But today, you shall only know the truth, nothing more than the truth, the whole truth about what this contest is about! I feel that so many people will return from this with a shovelful of questions, so please, do not hesitate to ask them by email, but we will only be able to respond on Saturday" one of the male members of the team announces
"In one variable we learn variable change in a general fashion, and we never obtained vanishing Jacobians. In multiple variables, one must calculate the partial derivatives in terms of our old variables, and calculate the determinant of the partial derivatives matrix, the Jacobian. A vanishing Jacobian means that we try to reduce a n-variable problem in n-1 variables. The two most important Jacobians are cylindrical and spherical" the other male team member explains.
Yes, the mathletic teams of the other parish schools all attend, but Geneviève is interested by one of the most exotic questions from a member of one of those teams:
"Hello, my name is Krista, and I think I will transfer to your school next year if my parents allow me to. We briefly talked about differential equations in class; how does that change in several variables?"
"With multiple variables, one must first ask if we can separate the variables, so that we can decompose a partial differential equation in two ordinary differential equations with a little algebra. To do so we suppose that the solution takes the form H(x,y) = F(x)G(y), and we insert it into the partial derivatives in the equation. But, with one or multiple variables, one must pay attention to both initial conditions and boundary conditions"
"Wow! Can you solve a PDE in front of the crowd?" Marcia asks, while knowing almost no one would understand anything to this demonstration.
"The great classic, the heat equation. We thus have"
"That the equation dG/G =-m dt has for solution ln G = -mt+C I accept, but where does the other equation's solution come from?" Trevor, a student on a mathletic team, asks [https://img.wattpad.com/70f4bb988bf8552f64d6f4d926ea6e6acbbe3375/68747470733a2f2f73332e616d617a6f6e6177732e636f6d2f776174747061642d6d656469612d736572766963652f53746f7279496d6167652f507859456568614e77706a5858413d3d2d313239303036313739302e313732623536613466653537633939353437373034393531363036342e706e67?s=fit&w=1280&h=1280]
"That the equation dG/G =-m dt has for solution ln G = -mt+C I accept, but where does the other equation's solution come from?" Trevor, a student on a mathletic team, asks.
"If we derive twice sin or cos of kx, we obtain negative k squared times the initial function. After all, the contest is primarily about multi-variable calculus and not on partial differential equations!"
"That's our champion!" bewildered fans exclaim, such as Dylan and Randy, without realizing that they must review problems on partial derivatives and multiple integrals, such as those on Stokes' theorem.
"Jesus! How do you study so many mathematical notions that fast?" the booster club administrator whose tax return was filed by Geneviève earlier in the month asks.
If we were in a big city, there wouldn't be a pep rally of this size for the same competition! Solving some problems on each major part of the material in front of the crowd should help me, but inevitably the public seems more impressed by the speed of the calculation than the mathematics behind them! she thinks while answering the math questions of the crowd on the board and, in addition, those of the other mathletic teams of the parish forces her to be quick with everyone, and the same goes with the other team members in front of their respective boards, as if they were exhibits.
"Oh, I am not worried about your success. Unfortunately, you cannot answer everyone, in fact, even with the entire team combined, the team will be unable to answer everyone. Today you shall see that I can do advanced math, too!" Marcia exclaims.
"Marcia, I invite you to try out for the math team and thus take part in other meets next year, especially if people are satisfied by the answers given during this pep rally" Gen issues an invitation to Marcia.
During the past three months, I lived in Geneviève's shadow; three months ago, she was nothing more than a self-assuming nerd. Now she's the one that leads the charge, and she does not make any more attempts to hide her brilliance than she used to, unlike other popular students in the other schools of the region... Marcia thinks, while she must answer questions about single-variable calculus that the audience must have. Especially people like Cory or his father, or other parent-children pairs of the same kind, Like Dylan and his father.
"Marcia, despite the impression I might have given you to always be late to do homework, to always ask others for help when there's a hint it can go wrong, and, if possible, make everyone benefit from it... you have always been here for me, even though I will take only one AP Exam this year. You and your friends are very busy so we settle for answers if we need help, if possible"
"My son briefly talked about you. Of how a project went poorly at the beginning of winter, and how she claimed my son waited weeks to do his part" Dylan's father then tells Marcia.
"Let's put that behind us and try to enjoy this peculiar occasion" Marcia sermons the tight end and his father.
"You certainly remember that a couple years ago, the football team was much better than today. That it was worthwhile to attend pre-game pep rallies back then, but this time it's a little different. We have the opportunity to see what it's actually made of; we have a chance to see the various types of problems in multi-variable calculus as we never saw them!" Dylan's father harangues Marcia.
"What the hell is an improper integral? What determines the hygiene of an integral?" Dylan asks, while he notices an allusion to an improper integral on one of the team's boards.
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"The hygiene of a definite integral, as you call it, is determined by the nature of the bounds : at least one of the bounds must either be infinite or lead to an infinite discontinuity of the function to integrate for an integral to be called improper" Marcia answers him to the best of her abilities.
Trent, however, seems to fade in the background, realizing that, even though he is part of the team's successes as much as the students themselves, the public seems to only care about the students, and people asking him questions didn't get the opportunity to ask the students first. The four team members juggle notions with such an ease that most people in the crowd finds troubling, to say the least. He still makes a reminder of an announcement for the calculus BC course.
"For those among you that are taking calculus BC, I invite you to participate, unofficially, to the Square Root of the Answer final tomorrow, as well as for those who want to be part of the mathletics team next year. You will be entitled to extra credit, if you are taking the calculus BC course, and get the opportunity to see what math tests could look like in college"
"This year is a golden opportunity for the parish, or barring that, the town, to give back education the place it deserves! What you are seeing here is the result of a handful of students that work hard in class and that make continuous efforts to learn the material! From now on, we have a clear position to fight with the Department in Bâton-Rouge for funding!" the superintendent of the parish school district harangues the crowd.
"Even though your highlights from high school will be more emotional in nature, such as a first kiss, a first date, an extracurricular success, or even an extremal grade, or a course we liked or hated, you must not see the material strictly as a body of knowledge. What will stay with you once this is over is not so much the material itself, but the by-products of their learning such as logic and critical thinking. But tomorrow here's our chance to show not only to these Northern states, but also to the people in the big cities that it's not because we live in a rural parish in the Deep South that we are condemned to academic mediocrity! That excellence does not know regional borders nor environments! Nevertheless, what matters to us is participating and doing our best. No matter what happens, I wish simply that you do not hold it against us: you must not see a bad rank as an indictment of any lack of skill on our part! After all, we compete against the best teams, the best schools in the nation! Long live the Venomous Agendas!" Geneviève exclaims while the town's euphoria is peaking.
"Well said!" Cory's father exclaims.
But once she returns home, she really wonders if she is up to the town's expectations. I really hope that the people in the area will really not hold it against us should we fail to perform! I must not waste these three months of fame; I must succeed! I may say one thing in front of the public, I cannot assume that they will really listen! And what would the University of Chicago think of this? Or of any other university of this kind; they will definitely want to see me perform for me to even put that on an application! Up to this point, it was easier for me to do mathletic competitions without having to worry about what's happening. One bad performance and I am done! she thinks while she tries to sleep, even though none of her worries are about any gaps in her knowledge of multi-variable calculus. She feels that it's not what's going to stop her.
The following day, she can't seem to focus in class, with the memories of this pep rally of the previous day and her memories of the material on one hand, on the other hand, the implications of her performance on the grand final tonight. Especially when she meets with her friends after the first period:
"For my teammates, this tournament represents their swan song in mathletics; they can hold their heads high regardless of what happens. But me? This contest carries its load of implications in college. Who will forgive me should I fail tonight? Certainly not everyone in this town!" Gen asks, while trembling like a leaf.
"Me. Bad performances happen to everyone. If it can cheer you up, I will take part in it, too, unofficially, however. You must have accumulated enough extra credit to give you a safety cushion for the final exam!" Marcia explains herself.
"Me too. My dad and I support the team until the very end, for better or for worse. He organized the pep rally, he has complete confidence in the team, you included" Cory encourages her.
"When I saw you last night solving PDEs, partial derivatives and multiple integrals... everything seemed so natural to you. I will forgive you and I will judge neither Marcia nor you based on their performances in this event" Dylan adds.
"I can't believe we're talking about the same girl, a girl that can perform advanced mathematics in front of an audience, and suddenly you think that you will fail? I might be a cornerback myself, and play in front of the same crowd size as you last night, the difference is simply one of activities" Randy continues.
"I feel like I need to go to the bathroom..." Gen warns her friends.
Poor lass! Her nose starts bleeding while she attempts to contain her stress towards the sheer number of mathematical notions that she must ensure she is capable of remembering for the grand final of the Square Root of the Answer and especially the implications of the result. More than simply nosebleeds, she starts crying. What did I do to get there? I should be happy to compete at this level, and I would love to think that we started from almost nothing, but never did I find myself with my nose bleeding! she thinks while her teammates are at the bathroom's exit.
"You must stop thinking about this competition and the material before it's time!" Éliane comments.
"All right..."
At night, rather than to simply have four students come take it, i.e., the team members, Trent is surprised to see there are other students, and not just Marcia and Shane, who took the extra credit bait, and even some that came from elsewhere in the parish, such as Krista, without the extra credit promise applying to them.
Three hours, twelve questions. Nothing out of the ordinary when one compares the content of these exam questions to typical questions of multi-variable calculus final exams of target universities of the participants, such as the University of Chicago, Duke, and so on. In fact, the contest's organizers simply took an amalgam of these multi-variable calculus final exams while ensuring that no university is represented more than once.
Phew! No differential equations or PDEs on the menu! she thinks, sighing, while the students that unofficially take the test are not all on her level. Those that can follow are not numerous outside of the team, but there are two that seem to be able to amount to something among the "unofficials": Krista and Marcia. The others, depending on whether they normally attend the school or not struggle a bit more, but unevenly. One struggles more on Lagrange multipliers, another one on Stokes' theorem, but Geneviève responds to a curvilinear integral question.
If a vector field can be expressed as
Ifa vector field can be expressed as [https://img.wattpad.com/200a712257a05085923e553c8511a863873d1b82/68747470733a2f2f73332e616d617a6f6e6177732e636f6d2f776174747061642d6d656469612d736572766963652f53746f7279496d6167652f462d7130465352624264654148773d3d2d313239303036313739302e313732623536636338396230303430393336343736343635333339342e706e67?s=fit&w=1280&h=1280]
then we have
We can either try several parametrizations y=f(x) and insert these in the curvilinear integral, which reduces a double integral problem into a single integral one or reverse a gradient but it requires as many integrals as there are gradient compon... [https://img.wattpad.com/900d3118241b50dd0b62a933f2f08b1cff13ceef/68747470733a2f2f73332e616d617a6f6e6177732e636f6d2f776174747061642d6d656469612d736572766963652f53746f7279496d6167652f3573462d32536147714d526f6e773d3d2d313239303036313739302e313732623536656565353634386138663735353632333432383238352e706e67?s=fit&w=1280&h=1280]
We can either try several parametrizations y=f(x) and insert these in the curvilinear integral, which reduces a double integral problem into a single integral one or reverse a gradient but it requires as many integrals as there are gradient components. If the solution exists, then there won't be any complication. But what am I doing? This is not a question of determining whether the initial function is a gradient field or not, but to calculate a curvilinear integral while we are given a parametrization y=f(x)! she thinks while everything clicks, and she seems to be back to her previous day's self. Less frightening than I thought; I may as well be asking why I had so many panic attacks earlier today!
Moral of the story: multi-variable calculus reuses a lot of single-variable concepts. But still: each assignment contained a few multi-variable problems clearly labeled that gives extra credit.
Even though I hate Geneviève since the past three months because the town seems to devote an excess of attention to her, I couldn't resist the temptation to obtain what dangerously looks like an exemption from the calculus final. Given the amount of extra credit that I could obtain on harder content than the course normally have, I religiously did all the problems given during the semester to leave nothing to chance, Shane thinks, while attempting to solve the Lagrange multiplier problem. Unfortunately, I could see it in everyone's eyes that it was the reason most came here tonight: extra credit. Knowing the 6 best students of the class in this course, which are all here tonight, and the best 4 are all on the team, I know at least 5 of them can realistically be assured of an A in the course without having to do anything on the final exam, while the next ten to fifteen will definitely get an A on it so long as they pass both the AP Exam and this test.
Speaking of AP Exams, the season begins the following Monday, and, with the AP Exam schedule, and the finals season after that, it's almost a month of tests. For both Marcia and Geneviève this means 4 AP Exams, because the parish allowed them to take a maximum of 4 advanced placement courses per year.
But at the end of the three hours, everyone sighs. Nevertheless, Trent must separate the copies belonging to the officially registered team and those belonging to the others, knowing that he will only grade the copies belonging to those taking it unofficially. After tasting the intensity of studying simultaneously single and multi-variable calculus, I no longer need to fear anything from the AP Calculus BC Exam, which is single-variable only, Geneviève thinks, while the accumulated fatigue from several months catches up and weighs down against her. The parish will reimburse 50% of the AP Test fees if we obtain a 3, 75% for a 4 and 100% for a 5, she continues in her train of thought, while she doesn't have a whole lot of juice left.