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【数学】劇場版まどマギに登場するホワイトボードに書かれていた数式について

本日、劇場版 魔法少女まどかマギカ[新編]叛逆の物語のBlu-ray Discが届いたので、映画を見ていた時にとてもとても気になっていた数学の問題について紹介したいと思います。再生時間で言うと29分頃だったと思います。再現ホワイトボードもつくろうかと思いましたが、今日作ることは諦めました。文字及び数字は全て私の推測ですのでご了承下さい。結構長いので先に感想言います。

感想:積分の問題は大学レベルで最初は難しいと思ったが、高校までの知識でなんとかなるものもあり良かった。また、文字式、二次関数から三角関数、積分と幅広い範囲をカバーしていることが分かった。(数学ⅠからⅢまで??)なお、鹿目まどかの持っていた数学の教科書は、東京書籍"風"の数学Ⅱであることが分かった。(絵柄は一致していないがレイアウトはよく似ている)

では、問題一覧です。

謎の式の羅列X1(ホワイトボード左上)

\[\frac{****}{*}=\frac{****}{*}=\frac{****}{*^*}\\
**\cdot\frac{****}{*}=**\cdot\frac{****}{*}=**\cdot\frac{****}{*}\\

*\cdot(****)=*\cdot(****)=****
\]

二次方程式の解の公式(ホワイトボード左下)
\[ax^2+bx+c=0\\
a(x^2+\frac{b}{a}x+\frac{c}{a})=0\\
x^2+2\cdot\frac{b}{2a}x+(\frac{b}{2a})^2-(\frac{b}{2a})^2+\frac{c}{a}=0\\
(x+\frac{b}{2a})^2-\frac{b^2-4ac}{4a^2}=0\\
(x-\frac{b}{2a}+\frac{\sqrt{b^2-4ac}}{2a})(x+\frac{b}{2a}+\frac{\sqrt{b^2-4ac}}{2a})=0\\
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

\]

謎の式の羅列X2(ホワイトボードやや左上)
\[\left\{
\begin{array}{l}
x^2+y^2=\alpha\beta\frac{*}{**}\\
y=\frac{*}{*}**^*
\end{array} \right.\]


ひし形の面積と辺の関係など(ホワイトボードやや左真ん中)
7

\[S=ah=\frac{d_{1}{\cdot}d_{2}}{2}=a^2\cdot\sin\alpha\\
d_{1}^2+d_{2}^2=4a^2
\]実際はsinのsが大文字に見える。

3つのグラフ(ホワイトボードやや左下)


謎の積分の式の羅列X3(ホワイトボードやや右上)
\[\int^*_* *^* dx=(*-*)*^*\\
\int *^**^* dx=*^**^*-*\int *^{*-1}*^*dx \\
\int * * dx=*^** * *+C\\
\int *^***dx =\frac{*^{***}}{***}***-\frac{*^***}{(***)}*+C
\]

積分の問題(ホワイトボードやや右下)
\[a)\int\frac{\arcsin^3 x}{\sqrt{1-x^2}} dx\]
\[b)\int x\ln(x^2+y) dx\]
\[c)\int\frac{x^3+2x^2+10x}{x^2-x+1} dx\]


解答はこちらです。(Cは積分定数とする。)解説については、こちらから。
まず、\[a)\int\frac{\arcsin^3 x}{\sqrt{1-x^2}} dx=\frac{1}{4}\arcsin^4 x+C\]
\[b)\int x\ln(x^2+y) dx =\frac{1}{2}((x^2+y)\ln(x^2+y)-x^2)+C\]
\[c)\int\frac{x^3+2x^2+10x}{x^2-x+1} dx =\frac{1}{2}x^2+3x+6 \ln(x^2-x+1)+2\sqrt{3} \arctan{\frac{2x-1}{\sqrt{3}}}+C\]


直方体の表面積と体積の公式(ホワイトボード右上)
3

\[S=2(ab+ac+bc) もしくは S=2(ab+ca+bc)\]
\[V=abc\]

直角三角形に垂線を下ろした時の辺の関係と三角関数(ホワイトボード右下)
4

\[a^2+b^2=c^2  h^2=c\alpha\cdot c\beta\\
a^2=c\alpha(※)   b^2=c\beta\cdot c\\
\sin\alpha=\frac{a}{c} \cos\alpha=\frac{b}{c}\\
\tan\alpha=\frac{a}{b}
\]ここの式に出てくる全ての$\alpha,\beta$が、$a,b$に見える。
※おそらく、$\times c$が抜けている。さらに、$\alpha$が、角度と辺の違う意味で2回使われているので本当は正しくない。


積分の問題の解説
a)置換積分を使って求めることにする。 \[\begin{eqnarray}
u&=&\arcsin xとおくと,\\
\frac{du}{dx}&=&\frac{1}{\sqrt{1-x^2}}より,\frac{1}{\sqrt{1-x^2}}dx=du\\
よって,\int\frac{\arcsin^3 x}{\sqrt{1-x^2}} dx&=&\int\arcsin^3 x\cdot \frac{1}{\sqrt{1-x^2}} dx\\
&=&\int u^3du=\frac{1}{4}u4+C\\
&=&\frac{1}{4}\arcsin^4 x+C
\end{eqnarray}\]



b)部分積分を使って求めることにする。
 
\[\begin{eqnarray}
\int x\ln(x^2+y) dx &=&\int \ln(x^2+y)\cdot (\frac{1}{2}x^2)' dx \\
&=&\ln(x^2+y)\cdot \frac{1}{2}x^2-\int (\ln(x^2+y))'\cdot \frac{1}{2}x^2 dx ←部分積分partial integration
\\
&=&\frac{1}{2}x^2\ln(x^2+y) -\int \frac{2x}{x^2+y}\cdot \frac{1}{2}x^2 dx \\
&=&\frac{1}{2}x^2\ln(x^2+y) -\int \frac{x^3}{x^2+y} dx \\
&=&\frac{1}{2}x^2\ln(x^2+y) -\int x-\frac{yx}{x^2+y} dx \\
&=&\frac{1}{2}x^2\ln(x^2+y) -\int x-\frac{\frac{1}{2}y(x^2+y)\prime}{x^2+y} dx \\
&=&\frac{1}{2}x^2\ln(x^2+y) -(\frac{1}{2}x^2-\frac{1}{2}y\ln(x^2+y))+C \\
&=&\frac{1}{2}x^2\ln(x^2+y)+\frac{1}{2}y\ln(x^2+y) -\frac{1}{2}x^2+C \\
&=&\frac{1}{2}((x^2+y)\ln(x^2+y)-x^2)+C
\end{eqnarray}\]

c)いろいろな方法を駆使する。
\[\begin{eqnarray}
\int\frac{x^3+2x^2+10x}{x^2-x+1} dx &=&\int(x+3+\frac{12x-3}{x^2-x+1}) dx \\
&=&\int(x+3+\frac{6(2x-1)}{x^2-x+1}+\frac{3}{x^2-x+1}) dx \\
&=&\int(x+3+\frac{6(2x-1)}{x^2-x+1}+\frac{3}{x^2-x+1}) dx \\
&=&\frac{1}{2}x^2+3x+\int\frac{6(x^2-x+1)\prime}{x^2-x+1}+\frac{3}{x^2-x+1}) dx\\
&=&\frac{1}{2}x^2+3x+6 \ln(x^2-x+1)+\int\frac{3}{x^2-x+1}) dx\\

&=&\frac{1}{2}x^2+3x+6 \ln(x^2-x+1)+2\sqrt{3} \arctan{\frac{2x-1}{\sqrt{3}}}+C
\end{eqnarray}\]
cの一番最後の式は、微分すればそうなるとわかるが、逆がちょっと…。
なお、英語版まどかWikiと啓林館『Focus Gold数学Ⅲ+C  解答編 新課程用』を参考に解説を作成しました。

魔法少女まどか☆マギカで覚える漢文の句法のイメージ


キャラクター名 句形 イメージとして 色
鹿目まどか 反語形 どうして~なの 違う ピンク
暁美ほむら 否定形 できない 紫
美樹さやか 仮定形 もし奇跡があるなら 水色
巴マミ 使役形 彼女らに勧めて魔法少女にさせる 黄色
佐倉杏子 疑問形 どうしてそんな態度 深い赤色
きゅうべぇ 限定形 ただ~なのさ 白色
志筑仁美 詠嘆形 なんて~ことよ 黄緑色
上条恭介 願望形 腕を動かしたい 灰色
早乙女和子 比較形 なんてぬかす男よりも 茶色
鹿目知久 比況形 深い青色
鹿目詢子 塁加形 ただの人ではない 赤紫色
鹿目タツヤ 受身形 まだ子供なので世話を受ける 
ショウさん 抑揚形 しょ 黒色
中沢 選択形 どっちでもいいんじゃないかと ベージュ

鹿目まどかのグラフを作ってみた

madoka_graph

グラフの式

先日、鹿目まどかのグラフを見ました。探してみたら、他のキャラクターのグラフもあるようです。初音ミクのグラフ

これらのグラフに共通して言えるのは、グラフが一筆書きということです。リサージュ図形でフーリエ変換??残念ながら私は理解することが出来ませんでした。

そこで、私はこれとは別のアプローチで鹿目まどかのグラフを表そうと考えました。以前から構想はあったので、どの姿をグラフにしようかは考えていました。こちらのページの右上の鹿目まどかです。グラフをつくるにあたって以下の3つの方法が思い浮かんだので紹介します。


①ブロックごとで考えてグラフにする

$a≦x≦b,c≦y≦d$を塗りつぶすようなグラフは、
\[(((x-a)(x-b)+|(x-a)(x-b)|)^2+((y-c)(y-d)+|(y-c)(y-d)|)^2)=0\]で表すことが出来ます。
この式を利用して最初は鹿目まどかのグラフを描こうとしました。しかしながら、必要な式の数が圧倒的に大きく、また心が折れてしまったので断念しました。以下にあるのはこの方式で作った参考画像です。

\ナッモリーン/まどか途中まで



左の画像の詳細についてはなもり先生のグラフを書いてみたに載っています。

②3次関数を組み合わせることでグラフにする

これは、座標とその点においてのベクトルを決めるだけで決まります。しかしながら、点Aと点Bの間でグラフの傾きが0になることがあるので意図していた形と違うものが出力されるのでこの案は却下しました。

具体的な方法:点A$(x_{1},y_{1})$,傾きが$z_{1}$,点B$(x_{2},y_{2})$,傾きが$z_{2}$,これを満たす3次関数$y=a_{1}x^3+a_{2}x^2+a_{3}x+a_{4}$について、$a_{1},a_{2},a_{3},a_{4}$を、$x_{1},x_{2},y_{1},y_{2},z_{1},z_{2}$を用いて表す事を考える。ただし、$x_{1}{\neq}x_{2}$一般化には成功したものの前述の理由で却下。

③2次関数にパラメーターを加えることでグラフにする

2次関数の場合、2点の傾きの合計が一定になるので、一方の点の傾きを定めれば、ある程度意図した形のグラフをつくることができます。

具体的な方法:
点A$(x_{1},y_{1})$,点B$(x_{2},y_{2})$これを満たす2次関数$y=a_{1}x^2+a_{2}x+a_{3}$について考える。この時、2点の傾きの合計$z_{1}+z_{2}$は、$2\times\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$に等しくなる。$a_{1},a_{2},a_{3}$を定めるため、点Aの傾き$z_{1}$を、$z_{1}=(1-k)\times\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$とおく。こうすると、点Bの傾き$z_{2}$は、$z_{2}=(1+k)\times\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$となり、都合が良い。ちなみに、$k=0$とおけば、直線になる。後は、$x_{1},x_{2},y_{1},y_{2},k$を用いて$a_{1},a_{2},a_{3}$を表す。ただし、$x_{1}{\neq}x_{2}$。これも一般化に成功した。

手作業でグラフの式を出すことも可能ですが、時間がかかってしまうのでプログラムをつくって、座標と$k$を入力するだけで、グラフの式を出力するようにしました。

以下が鹿目まどかの式です。コピーアンドペーストすることでグラスソフトで表示を行うことも出来ますが式の読み込みを行うと5分以上グラフソフトがフリーズすると考えてください。取り扱いには十分ご注意ください。全部で64119文字です。テキストで表示する

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