"Fuck you!" hisses the student who was talking.  "That's averting drama, you bitch!"

"It's fine so long as I'm doing it because I hate you."

"Thanks!" Opalyn whispers to the student who helped her out. She gives a tiny wave and moves across the room away from those two, in case it's about to escalate.

Well, not getting to take the paper with her is unfortunate, but if it turns out class is boring maybe she can take notes about all the other things she wants to think about while the teacher drones on? Wouldn't be the first time!

Speaking of which... could the teacher please show up now?

That happens just about as soon as the previous class has finished clearing out.

The man who walks into the classroom, alongside a mug that hovers obediently along through the air near his mouth, seems to have turned himself into a catboy at some point.  Or maybe just put on some very realistic cat ears.  He is wearing black armored robes and bears a smile that some might consider... unnatural.

He is followed by a long-suffering-looking lady dressed in similar black armored robes, who has either also been turned into a cat-girl, or is also wearing a pretty decent set of cat-ears.

"Greetings, my vulnerable apprentices--oh hey new face.  What's your deal?"

Opalyn hesitates just a moment in case there is some other new pupil this -- catwizard -- might be addressing, perhaps some other student directly behind her?

When no one else speaks up, rather than let the silence get awkwardly long, she offers, "Hello, I'm here to learn some wizardry!"

"You're in the wrong place then, this is my class on cupcake-baking."

Oh, fantastic. Banter. Cross-cultural banter when Opalyn is at the bottom of the power differential. She'll opt to play along.

"Well, looks like we'll have to bake the cupcakes with wizardry, since I don't see any ovens or muffin tins here, so I'll stick around and see if I can learn something!"

Opalyn hopes they use ovens and muffin tins to make cupcakes in this universe, or else she just outed herself as very foreign.

Well, at least this one can respond to banter literally at all, and has any life at her literally at all.

"As you're new to my teachings, you can start off by asking me any one question about wizardry.  It doesn't need to be restricted to what you think is the class topic; neither are we, really.  I won't pretend that I won't judge you on your choice of question and it is definitely possible for you to end up spanked here."

Opalyn has had it with playing it safe; she'll just ask what she really wants to know.

"So, I'll need to explain a few things first."

"I've never been trained in magic at all. I've only gotten to use magic... " Opalyn starts to count and realizes she can't really tell the truth. "... a few times."

"And the only time anyone provided me with any real instruction was also the one time I charged the lightlines, the day of my lightline test."

"When I charged the lightlines, the monitor kept yelling at me, for lots of things. I didn't understand half of it. There was a lot of confusion about wizardry vs. sorcery because I didn't know the difference. He explained that when I'm doing sorcery I'm just supposed to will the magic to be a certain way, and after a lot of misery I eventually managed to do that."

"He also said that some of what I was doing was... I don't know. It had too much wizardry in it, somehow?"

"I gather that wizardry involves thinking math at the magic to get the magic to change what it's doing. With this collar on, I never get a chance to try it, though, and none of the books in the library give simple examples."

"So instead of a question, what I really want is some very basic instruction in the most basic possible unit of wizardry, before we do anything fancy. Like, what is the sort of thing you have a five year old do when they are first learning wizardry. I can probably extrapolate from there once I've seen even one basic thing."

"You don't think math at -- important question, do you know math beyond arithmetic, at all?  This class presumes calculus and the ability to differentiate a formula from scratch.  This is a quality that should have been screened; and if it wasn't, my secretaries get spanked about it more than you, and I forward you to a different teacher and you come back later."

Opalyn is not sure what "differentiate from scratch" means and if she's being honest it has been quite a while since she's differentiated anything at all, but she is not going to leave and go to some other teacher, she's going to get what she came for. She has a lot of differentiations memorized even after all these years, and she can probably estimate it with some quick and dirty graphing and spitballing if she has to.

"Yes, I know loads of math beyond arithmetic, I'll be fine with the math."

Besides, why should she have to share the spankings with the secretaries?

"Derivative of a sine function?"

"Cosine."

Jiminy Cricket, these people have a weird idea of how to test people about math.

"I have no idea where in a thousand worlds they manage to fuck up education on that level, of successfully managing to teach you the derivative of a sine function, but leaving you with the impression that you think math at magic to enact wizardry.  But wow if I was the Dread Emperor I would tell some places to improve their educational systems or die."

Gaaaah.

On Earth, math is used for many applications. If someone on Earth came up to Opalyn and said they knew math but did not know any economics, or they knew math but didn’t know any engineering, or they knew math but didn’t know any physics, that would be utterly unsurprising.

Surely Eldrida has at least something that passes for economics and engineering and (mundane) physics.

So why is magic so special? Why is everyone constantly sputtering that Opalyn knows math without magic?

WHY DOES THIS KEEP HAPPENING?

"It is not that strange to know math and yet know nothing of magic! Why do people keep saying this to me?"

And... wait. What else did he say?

"Also... what do you mean you don't think math at magic? What do you actually do with the math?"

"I shall hold off on the first question until I have answered the second one, your original question, for it is prerequisite to the first."

"A wizard knows math.  Math describes magic.  Sorcerers tell magic what to do after it leaves their body, and it does that.  Wizards have to figure out what different possible magical actions will do, and do the right one.  They figure it out using math."

What.

Well that sure doesn't match any of what Opalyn had been thinking since the first time Fdera yelled at her about it.

This class is already incredibly educational.

Morbledainen raises one finger in the air, and a light is born upon its tip, accompanied by magic.  A flick of his other hand's finger sets a second orb of light to orbit the first.

"You asked for a children's exercise.  This moving mana-bundle, circling the mana-source on my finger, is attracted to that mana-source with a force that varies as the inverse square of its distance from that mana-source."

"I don't make that be true by thinking math at it.  It's just always been the case that the moving type of magic is attracted in that exact way, well-described by math, to the magic I'm radiating from my finger."

"Since I know this, I can figure out how fast I need to flick the magic, at what distance, to make it orbit in a rough circle.  I also know that since I didn't flick it perfectly to make a perfect circle, the actual orbit will be an ellipse."

Though Morbledainen is trying to use simple words, Opalyn's senses will provide her with additional and more complicated information.  She can see that there's a Type A of mana flowing out of his finger, constantly radiating and losing its energy, and a type B of mana he's also emitting from his finger that just immediately turns into visible light.  The orb is made up of a different type C of mana that's orbiting the A-attractor and not losing energy; along with a bundle of type-B energy that's closely bound to the C-type mana and gradually losing energy as it turns into light.

(In other words:  The two simple kinds of mana here, type A at the center and type C orbiting it, might by default be invisible to ordinary eyes, if mere Baron-level sorcerers had weaker senses especially when collared.  But Morbledainen is helpfully tacking on B-type mana to both the central A-mana and the orbiting C-mana, so that they'll be visible to the naked eye.)

"Okay. Looks like gravity to me, and the equations seem to be the same shape as gravity. Is there a well-known constant associated with those types of mana that's analogous to the gravitational constant? And does that constant change if you change the types of mana involved?"

... Is there a gravitational constant here or is she speaking alien right now?

"Some kinds of mana pass right through each other.  Some are attracted inverse-linear, or inverse-cube.  Some repel each other instead of attracting."

He flicks his finger again, and another orb of light starts orbiting the star on his finger, at a wider distance, but circling faster.

"But in terms of finding another kind of mana attracted to my finger's mana inverse-square, but with a different constant -- well, there you go.  Can you say to me whether the constant of attraction, analogous to gravitation, must be higher for the new mana-bundle, or lower?"

There's now a new D-type of stable mana that's orbiting the A-type mana radiating from his finger, along with its own B-type rider to make it be visible to the naked eye.  The way the D-type mana holds its bound B-type mana is different from how the C-type mana keeps its own B-type mana along for the ride; but possibly that is not the most important thing for Opalyn to be thinking about right now.

The earlier orb (C+B) is circling at a distance of a few inches, the width of a human hand or so, and circling once per four seconds.  The new orb (D+B) orbits two hand-widths away from his finger, but completes an orbit every three seconds, outpacing the first despite the greater distance.

She told him the math wouldn't be a problem, time to cough up some ancient, dusty recollections of physics!

Let's see. Step one is to remember how orbital mechanics works at all, without even varying the gravitational constant or bringing magic into it. Just regular old mundane satellites orbiting a planet will do just fine.

This is about when Opalyn would ordinarily just... look it up. Alas, she is utterly without the internet and must mentally muscle through this on her own, whether by recall or by calculation.

She can remember that comets typically zoom very fast when they do their fly-by of the sun, and then amble lazily when they're at aphelion. Do the same speed calculations just work for circular orbits, or does anything break when you take away the slingshot effect?

Somehow the shape of the circle needs to be maintained. A small orbit has four times the gravitational pull of one double its radius, so the satellite has to go twice as fast to avoid falling. Yep. Checks out.

Yet when she looks at Morbledainen's dancing lights, the one with the larger orbit is actually going faster. Much faster. Therefore, it must be counteracting a stronger pull toward his finger.

"The new constant-of-attraction must be higher, and the new mana-bundle must go correspondingly fast to avoid collapsing into your finger."

"Yes.  And so that is an example of the relationship between math and magic.  Knowledge of math grants wizards power, because through that knowledge, wizards can predict the results of their own magical actions, and calculate which magical actions they ought to take.  Not because math grants them power by mere fiat of life, like some mere sorcerer who gains more power over magic when their emotions rise.  I suppose it's no one's fault to be born a sorcerer, but still, learn the damn math."

"As you apparently did, without knowing anything at all of magic."

"Now, it's not an incoherent state of knowledge for someone to know advanced math -- and have never once heard anything about the single most prestigious and powerful application of math, that is responsible for virtually all children first getting interested in math, that would appear in a supermajority of all children's books about math, that would be used in at least some illustrations and examples in all textbooks covering fields of math that can be used in wizardry at all -- the other fields of math being far less studied -- and that you'd expect a teacher to show off by magical demonstration when they taught some kid about inverse-square attractions in the first place.  Yours used gravity and that, you know, is kind of hard to demonstrate in a classroom if you want actual orbits."

"None of this matters at all to this classroom, but yes, strange.  You come from some very strange background, young lady, and stranger yet is that you do not know how strange it was.  I am genuinely unsure whether or not I ever want to hear about this again."

"If you'll permit me one more question... do you distinguish between gravity -- which, to be clear, I think of as the attraction of mass upon other mass -- and other kinds of attraction, as in your examples with mana? Or are these all just shades of the same thing to you?"

Perhaps Opalyn should stop thinking of magic as being magical, and start thinking of it as an extension to physics. It does get much more weird to say you know calculus without knowing, say, Newtonian physics, and maybe that's why everyone keeps giving her funny looks.