The Quest for High School Mathletic Glory

Chapter 20: Back to the excesses

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.