Reiner vaguely remembered his high school mathematics teacher saying that when learning mathematics, the stupid bird flies first. People with inflexible thinking need to do a lot of training to develop their calculation and problem-solving abilities. If you are not good at mathematics, you will do less questions. Now Thinking about it, this principle is quite correct.
Of course, the math teacher later added that smart birds can fly higher and faster. This is another story.
One of the main reasons why Dana was unable to successfully construct the spell model was that she could not correctly calculate the coordinates of the spell nodes and the functional equation of the magic channel, resulting in deviations, which led to the failure.
It’s not easy for mages in this world either.
Reiner thought to himself that after he tried casting spells himself, he found that just calculating the node position and magic channel trajectory of the zero-ring spell was a headache. This was equivalent to mentally calculating the quadratic curve equation, but under the influence of magic, this process was very difficult. Miraculously, Reiner succeeded in constructing it with almost no effort. This calculation process seemed to be instinctive. If he was proficient, he wouldn't even need to put too much consciousness into it.
Having not yet experienced the spellcasting process of more powerful mages, Reiner speculated that maybe those mages could mentally calculate high-order equations and differential equations in a short time, and could be regarded as humanoid computers.
Putting these aside, facing the problem at hand, Reiner believed that the only way to improve Dana's mathematics level on the one hand, and to give her better mathematical tools on the other hand.
Picking up the test paper, Reiner compared it with Claire. It was easy to see that Dana's math differences were reflected in many aspects.
First of all, the way of thinking is inflexible, which is reflected in the inability to draw auxiliary lines in geometry questions and the inability to change conditions in curve questions.
The second is calculation ability, which is relatively basic but requires complex calculations. Although Dana was able to find a solution to the problem, she made mistakes in her calculations.
Finally, Reiner noticed that Dana still seemed to be hiding a hint of unconfidence.
Since there were also draft notes left on the test paper, it was clear that on some questions, Dana's original ideas were correct, but because the calculated results were very cumbersome, she thought she had made a mistake. Missed the answer.
There are many reasons for this mentality. It may be due to low self-esteem due to mistakes in the past, or it may be due to personality. More background information is needed.
But what makes Reiner feel strange is that since Dana was born in a magic family, she has not been exposed to it and is very unfamiliar with related magic. This is not normal.
Reiner was thinking about these things while explaining the correct way to solve problems to Dana. He was a teacher, and he couldn't help but want to teach this "poor student" in front of him well.
"You need a lot of training. If your foundation is not as good as others, you will have to work twice as hard. From today on, I will assign a similar test paper to you every day. You come to my office after dinner, and I will give you You answer."
Reiner said, making Dana shudder.
This test paper has made her feel the horror of being dominated by mathematics, and now Reiner actually wants her to write one every day. Is this person a devil
But this is not Reiner's evil deed. In fact, writing test papers is much more difficult than simply answering them. Reiner is also doing this to exercise his mathematical skills and prepare for passing the advanced examination.
At the same time, he can also test whether this educational method is effective on Dana. If the effect is good, he may extend it to the entire Crescent College.
After all, the proportion of successful mage advancements is also part of the annual assessment.
Fortunately, the mathematics skills required for low-level mages are not very deep, and they don't even need calculus. Reiner's current knowledge is more than enough.
"Can I have a few less questions..."
Dana asked timidly, but Reiner flatly refused the request, making the girl sigh.
"In addition, apart from the training of basic skills, the method of constructing a spell model is also very important."
Reiner returned to the podium, causing Dana and Claire to focus their eyes on the blackboard again, the spell model of the illumination spell.
What Reiner said at the beginning about improving the spell model came to their minds again. The two ladies looked at Reiner with curiosity, not knowing where he would start improving it.
Unexpectedly, Reiner did not continue to write on the spell model. Instead, he marked a dot with white chalk on the side.
"We create a new coordinate system."
Reiner drew a straight horizontal line, setting the origin as O and the horizontal axis as r. Of course, these are not English characters, but two letters of the common language.
But then, the vertical axis that Claire expected did not appear, as if Reiner's coordinate axis ended here.
"Huh?"
Just when the two were confused, Reiner extended a line segment from the origin, and then marked the angle between the line and the horizontal axis, which was designated as θ, and the point at the other end of the line segment was designated as A.
"In the past, the Cartesian coordinate system could use two values to determine a point on the plane. For example, if this point was on the Cartesian coordinate system, it should be A (x, y). Assuming that x and y are both 1, then A should It’s (1, 1).”
Reiner said, then changed the subject.
"But if I don't use x and y, but instead use the angle θ between the line connecting point A and the origin and the abscissa axis and the unit length r to represent this point, what will be the result?"
After giving the two of them some time to think, Reiner continued writing on the blackboard.
A(r*cosθ, r*sinθ).
This somewhat special way of expressing it made Dana a little confused, but trigonometric functions are the basis of magic. In magic, the calculation of angles is also more convenient, so she quickly understood it.
"This is a new coordinate expression method I introduced, which can be called polar coordinates."
After speaking, Reiner established a normal rectangular coordinate system next to it and drew a parabola that passed through the origin and opened upward.
"If we wanted to describe the functional equation of this curve, what would it be, Dana?"
he asked, catching Dana off guard.
But fortunately, it was relatively simple, and Dana quickly gave the answer.
"Uh, y=x^2?"
"To be precise, it should be y=2p*x^2. In this function equation, because it involves square operations, it is more complicated than the general straight line equation. If the position of the curve changes, for example, it is not at the origin If so, it will be even more troublesome.”
Reiner said and continued writing on the blackboard.
"Next we can establish two equations: y=r*sinθ, x=r*cosθ, substitute them into the original equation, and after elimination and simplification, we can get an equation, r=tanθ/cosθ."
Claire nodded, but the function equation seemed more complicated. She didn't understand why Reiner used such a troublesome way to record the trajectory of the curve.
"Of course, this is a very complicated way, but if we change the definition slightly, r is the distance between the point on the parabola and the focus, and θ is determined as the angle between the point on the parabola and the line connecting the focus with the positive direction of the vertical axis? "
Reiner's question made Claire and Dana stunned.