Math (2017/06/18)

アインシュタイン曰く

本やノートに書いてあることをどうして憶えておかなければならないのかね?

というわけで、よく使われるが、すぐ忘れてしまう公式などを覚えないで済むよう、メモとして残す。

随時追加する。

Early math

Arithmetic

式の計算

部分分数分解の計算例

\[ \frac {x - 25} {x^{2} + 5x - 24} \] \[ = \frac {x - 25} {(x - 3)(x + 8)} \] \[ = \frac {A} {x - 3} + \frac {B} {x + 8} \] \[ = \frac {A(x + 8)+B(x - 3)} {(x - 3)(x + 8)} \] \[ = \frac {Ax + 8A + Bx - 3B} {(x - 3)(x + 8)} \] \[ A + B = 1 \] \[ 8A - 3B = -25 \] \[ 8A - 3B + 3A + 3B = -25 + 3 \] \[ 11A = -22 \] \[ A = -2 \] \[ -2 + B = 1 \] \[ B = 1 + 2\] \[ B = 3 \] \[ \frac {x - 25} {x^{2} + 5x - 24} = \frac {-2} {x - 3} + \frac {3} {x + 8} \]

図形

正四面体の高さ

\[ h = \sqrt {\frac {2} {3}} a \]

角の二等分線定理

\[ \triangle ABC, \angle DAB = \angle DAC \Rightarrow \frac {AB} {BD} = \frac {AC} {CD} \]

球の表面積

\[ S = 4\pi r^{2} \]

放物線の方程式の計算例

\[ 焦点: (-8, -1) \] \[ 准線: y = -4 \] \[ 焦点との距離: \sqrt{(x + 8)^{2} + (y + 1)^{2}} \] \[ 准線との距離: \sqrt{(y + 4)^{2}} \] \[ \sqrt{(y + 4)^{2}} = \sqrt{(x + 8)^{2} + (y + 1)^{2}} \] \[ (y + 4)^{2} = (x + 8)^{2} + (y + 1)^{2} \] \[ y^{2} + 8y + 16 = (x + 8)^{2} + y^{2} + 2y + 1 \] \[ y^{2} -y^{2} + 8y - 2y = (x + 8)^{2} + 1 - 16 \] \[ 6y = (x + 8)^{2} - 15 \] \[ y = \frac {(x + 8)^{2}} {6} - \frac{15} {6} \] \[ y = \frac {(x + 8)^{2}} {6} - \frac{5} {2} \]

楕円の焦点

\[ \frac {x^{2}} {a^{2}} + \frac {y^{2}} {b^{2}} = 1 \ (a > b > 0) \] \[ 焦点: (\pm \sqrt{a^{2} - b^{2}}, 0) \]

\[ \frac {x^{2}} {a^{2}} + \frac {y^{2}} {b^{2}} = 1 \ (b > a > 0) \] \[ 焦点: (0, \pm \sqrt{b^{2} - a^{2}}) \]

\[ \frac {(x - p)^{2}} {a^{2}} + \frac {(y - q)^{2}} {b^{2}} = 1 \ (a > b > 0) \] \[ 焦点: (\pm \sqrt{a^{2} - b^{2}} + p, q) \]

双曲線の焦点

\[ \frac {x^{2}} {a^{2}} - \frac {y^{2}} {b^{2}} = 1 \] \[ 焦点: (\pm \sqrt{a^{2} + b^{2}}, 0) \] \[ \frac {y^{2}} {a^{2}} - \frac {x^{2}} {b^{2}} = 1 \] \[ 焦点: (0, \pm \sqrt{a^{2} + b^{2}}) \]

論理

イプシロン-デルタ論法

\[ \forall \varepsilon \gt 0, \ \exists N \in \mathbb{N}, \ \mbox{s.t.} \ n > N \Rightarrow |a_n - L| \lt \varepsilon \]

対数

底の変換

\[ \log_b(a) = \frac {\log_x(a)} {\log_x(b)} \]

三角関数

三角関数の関係

\[ \csc{\theta} = \frac {1} {\sin{\theta}} \] \[ \cot{\theta} = \frac {1} {\tan{\theta}} \]

三角関数の性質

\[ \sin{\theta} = \sin{(180^{\circ}-\theta)} \]

\[ \sin{\theta} = \sin{(-180^{\circ}-\theta)} \]

\[ \cos{\theta} = \cos{(-\theta)} \]

加法定理

\[ \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta \]

\[ \cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta \]

余弦定理

\[ a^2 = b^2 + c^2 - 2bc\cos A \]

ラジアン

\[ \frac {180 ^ {\circ}} {\pi} = 1rad \]

シヌソイドの中間線

\[ f(x) = a\sin(bx + c) + d \]

\[ f(x) = a\cos(bx + c) + d \]

数列

数列の和

\[ S_n = \frac {a_1(1 - r ^ n)} {1 - r} \]

数列の計算

\[ a_n = S_{n} - S_{n-1} \]

分数の数列の和

分数を以下のように変換する。

\[ b < c, \ \frac {1} {(ax+b)(ax+c)} = \frac {1} {(c-b)(ax+b)} - \frac {1} {(c-b)(ax+c)} \]

無限級数

\[ \sum_{n=0}^\infty ar^n = \frac {a} {1 - r} \]

テイラー展開

\[ f(x) = \sum_{n=0}^\infty \frac {f^{(n)}(a)} {n!}(x - a)^n \]

マクローリン展開

\[ f(x) = \sum_{n=0}^\infty \frac {f^{(n)}(0)} {n!}x^n \]

\( \sin{(x)} \)のテイラー展開

\[ \sin(x) = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \frac {x^7} {7!} + ... \]

\( \cos{(x)} \)のテイラー展開

\[ \cos(x) = \sum_{n=0}^\infty \frac {(-1)^n} {(2n)!} x^{2n} \]

\[ \cos(x) = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} + ... \]

\( \tan{(x)} \)のテイラー展開

\[ \tan(x) = x + \frac {x^3} {3} + \frac {2x^5} {15} + ... \]

\( \arctan{(x)} \)のテイラー展開

\[ \arctan(x) = x - \frac {x^3} {3} + \frac {x^5} {5} - \frac {x^7} {7}... \]

\( e^{x} \)のテイラー展開

\[ e^x = \sum_{n=0}^\infty \frac {x^n} {n!} \]

極限

様々な極限

\[ \lim_{x \to 0} \frac {\cos(x) - 1} {x} = 0 \]

ロピタルの定理

\[ \lim_{x \to a} \frac {f(x)} {g(x)} = \lim_{x \to a} \frac {f'(x)} {g'(x)} \]

\( \lim_{x \to \infty} x^{\frac {1} {x}} \)の計算例

\[ \lim_{x \to \infty} x^{\frac {1} {x}} \] \[ y = x^{\frac {1} {x}} \] \[ \ln{(y)} = \ln{(x^{\frac {1} {x}})} \] \[ \ln{(y)} = \frac {\ln{(x)}} {x} \] \[ \ln{(y)} = \frac {(\ln{(x)})'} {(x)'} \] \[ \ln{(y)} = \frac {\frac {1} {x}} {1} \] \[ \ln{(y)} = \frac {1} {x} \] \[ \lim_{x \to \infty} \frac {1} {x} = 0 \] \[ \ln{(y)} = 0 \] \[ e^{\ln{(y)}} = e^{0} \] \[ y = 1 \] \[ \lim_{x \to \infty} x^{\frac {1} {x}} = 1 \]

微分

積の微分

\[ (f(x)g(x))' = f'(x)g(x) + f(x)g'(x) \]

商の微分

\[ \left( \frac {f(x)} {g(x)} \right)' = \frac {f'(x)g(x) - f(x)g'(x)} {(g(x))^{2}} \]

合成関数の微分

\[ (f(g(x)))' = f'(g(x))g'(x) \]

\( \tan{(x)} \)の微分

\[ (\tan(x))' = \frac {1} {\cos^2(x)} \]

\( \sec{(x)} \)の微分

\[ (\sec(x))' = \sec(x)\tan(x) \]

\( \csc{(x)} \)の微分

\[ (\csc(x))' = -\cot(x)\csc(x) \]

\( \cot{(x)} \)の微分

\[ (\cot(x))' = -\csc^2(x) \]

\( \arcsin{(x)} \)の微分

\[ (\arcsin(x))' = \frac {1} {\sqrt{1 - x^2}} \]

\( \arccos{(x)} \)の微分

\[ (\arccos(x))' = -\frac {1} {\sqrt{1 - x^2}} \]

\( \arctan{(x)} \)の微分

\[ (\arctan(x))' = \frac {1} {1 + x^2} \]

\( a^{x} \)の微分

\[ (a^{x})' = a^{x} \ln(a) \]

\( \log_{a}{(x)} \)の微分

\[ (\log_{a}(x))' = \frac {1} {x\ln(a)} \]

\( \ln{(x)} \)の微分

\[ (\ln(x))' = \frac {1} {x} \]

\( e^{ax} \)の微分

\[ (e^{ax})' = ae^{ax} \]

\( e^{x^{2}} \)の微分

\[ (e^{x^{2}})' = 2xe^{x^{2}} \]

逆関数の微分

\[ h(x) = f^{-1}(x) \] \[ h'(x) = \frac {1} {f'(h(x))} \]

\( x^{x} \)の微分の計算例

\[ y = x^{x} \] \[ \ln{(y)} = \ln{(x^{x})} \] \[ \ln{(y)} = x\ln{(x)} \] \[ (\ln{(y)})' = (x\ln{(x)})' \] \[ \frac {1} {y} \frac {dy} {dx} = 1 * \ln{(x)} + x * \frac {1} {x} \] \[ \frac {1} {y} \frac {dy} {dx} = \ln{(x)} + 1 \] \[ \frac {dy} {dx} = y(\ln{(x)} + 1) \] \[ \frac {dy} {dx} = x^{x}(\ln{(x)} + 1) \]

\( x^{\ln{(x)}} \)の微分の計算例

\[ y = x^{\ln{(x)}} \] \[ \ln{(y)} = \ln{(x^{\ln{(x)}})} \] \[ \ln{(y)} = \ln{(x)} * \ln{(x)} \] \[ \frac {1} {y} \frac {dy} {dx} = \frac {1} {x} * \ln{(x)} + \ln{(x)} * \frac {1} {x} \] \[ \frac {1} {y} \frac {dy} {dx} = 2 * \frac {1} {x} * \ln{(x)} \] \[ \frac {dy} {dx} = 2 * \frac {1} {x} * \ln{(x)} * y \] \[ \frac {dy} {dx} = 2 * \frac {1} {x} * \ln{(x)} * x^{\ln{(x)}} \] \[ \frac {dy} {dx} = 2 * \ln{(x)} * x^{\ln{(x)}} * \frac {1} {x} \] \[ \frac {dy} {dx} = 2 * \ln{(x)} * x^{\ln{(x)}} * x^{-1} \] \[ \frac {dy} {dx} = 2 * \ln{(x)} * x^{\ln{(x)}-1} \]

\( \frac {dy} {dx} \)の計算例

\[ 2xy + x^3 - 3y^2 = 5 \] \[ \frac {d} {dx} (2xy + x^3 - 3y^2) = \frac {d} {dx} (5) \] \[ \frac {d} {dx} (2xy) + \frac {d} {dx} (x^3) - \frac {d} {dx} (3y^2) = \frac {d} {dx} (5) \] \[ \frac {d} {dx} (2xy) + 3x^2 - \frac {d} {dx} (3y^2) = 0 \] \[ 2\frac {d} {dx} (xy) + 3x^2 - 3\frac {d} {dx} (y^2) = 0 \] \[ 2(\frac {d} {dx} (x) * \frac {d} {dx} (y)) + 3x^2 - 3\frac {d} {dx} (y^2) = 0 \] \[ 2(1 * y + x * 1 \frac {dy} {dx}) + 3x^2 - 3\frac {d} {dx} (y^2) = 0 \] \[ 2y + 2x \frac {dy} {dx} + 3x^2 - 3\frac {d} {dx} (y^2) = 0 \] \[ 2y + 2x \frac {dy} {dx} + 3x^2 - 3\frac {d} {dx} (1 * y^2) = 0 \] \[ 2y + 2x \frac {dy} {dx} + 3x^2 - 3\frac {d} {dx} (x^0 * y^2) = 0 \] \[ 2y + 2x \frac {dy} {dx} + 3x^2 - 3(\frac {d} {dx} (x^0) * \frac {d} {dx} (y^2)) = 0 \] \[ 2y + 2x \frac {dy} {dx} + 3x^2 - 3((0 * y^2) + (x^0 * 2y \frac {dy} {dx})) = 0 \] \[ 2y + 2x \frac {dy} {dx} + 3x^2 - 3((0 * y^2) + (1 * 2y \frac {dy} {dx})) = 0 \] \[ 2y + 2x \frac {dy} {dx} + 3x^2 - 3(0 + 2y \frac {dy} {dx}) = 0 \] \[ 2y + 2x \frac {dy} {dx} + 3x^2 - 6y \frac {dy} {dx} = 0 \] \[ 2y + 3x^2 + 2x \frac {dy} {dx} - 6y \frac {dy} {dx} = 0 \] \[ 2y + 3x^2 + (2x - 6y) \frac {dy} {dx} = 0 \] \[ (2x - 6y) \frac {dy} {dx} = -(2y + 3x^2) \] \[ \frac {dy} {dx} = -\frac {2y + 3x^2} {2x - 6y} \]

定積分の微分の計算例

\[ \left( \int_{0}^{x ^ {2}} \sin(t) dt \right)' \] \[ = \frac {d} {dx} \int_{0} ^ {x ^ {2}} \sin(t) dt \] \[ = \sin(x ^ {2}) \cdot \frac {d} {dx} (x ^ {2}) \] \[ = \sin(x ^ {2}) \cdot 2x \] \[ = 2x \sin(x ^ {2}) \]

オイラー法

\[ y_{n+1}= y_{n} + \Delta y_{n} \]

平均値の定理

\[ \frac {f(b) -f(a)} {b - a} = f'(c) \]

二階微分

\[ x = f(t), \ y = g(t) \] \[ \frac {d^{2}y} {dx^{2}} = \frac {f'(t)g''(t) - f''(t)g'(t)} {(f'(t))^{3}} \]

積分

\( \sin{(ax)} \)の積分

\[ \int \sin(ax) dx = -\frac {1} {a} \cos(ax) + C \]

\( \sec^{2}{(ax)} \)の積分

\[ \int \sec^2(ax) dx = \frac {1} {a} \tan(ax) + C \]

\( \csc^{2}{(x)} \)の積分

\[ \int \csc^2(x) dx = -\cot(x) + C \]

\( e^{ax} \)の積分

\[ \int e^{ax} dx = \frac {e^{ax}} {a} + C \]

\( \frac {1} {1 + x ^ {2}} \)の積分

\[ \int \frac {1} {1 + x ^ {2}} dx = \tan ^ {-1}(x) + C \]

曲線の長さ

\[ L = \int_a^b \sqrt{1 + (f'(x))^2} dx \]

y軸周りの回転体の体積(バームクーヘン積分)

\[ V = 2\pi\int_a^b x f(x) dx \]

関数の平均値

\[ \frac {1} {b - a} \int_a^b f(x) dx \]

分数関数の積分

\[ \int \frac {f'(x)} {f(x)} dx = \log |f(x)| + C \]

分数関数の積分の計算例

\[ \int \frac {a} {b(x + c)} dx = \frac {a \ln |x + c|} {b} + C \]

放物線の積分

\[ A = \int_\alpha^\beta \frac {1} {2} r^2 d\theta \]

リーマン積分

\[ \int_a^b \sin(x) dx = \lim_{n \to \infty} \sum_{i = 1}^{n} \sin \left(\frac {b - a} {n} i\right)\cdot \frac {b - a} {n} \]

置換積分の計算例

\[ \int \frac {x} {\sqrt{16 - x ^ 2}} dx \] \[ u = 16 - x ^ 2 \] \[ \frac {du} {dx} = -2x \] \[ -\frac {du} {2x} = dx \] \[ \int \frac {x} {\sqrt{u}} \cdot (-\frac {du} {2x}) \] \[ = -\int \frac {1} {2\sqrt{u}} du \] \[ = -\int \frac {1} {2} u ^ {-\frac {1} {2}} du \] \[ = -u ^ {\frac {1} {2}} + C \] \[ = -\sqrt{u} + C \] \[ = -\sqrt{16 - x ^ 2} + C \]

部分積分の計算例

\[ \int x \cos(\pi x) dx \] \[ \int u(x)v'(x)dx = u(x)v(x) - \int u'(x)v(x) dx \] \[ \int u dv = uv - \int v du \] \[ u = x, dv = \cos(\pi x) dx \] \[ du = dx, v = \int \cos(\pi x) dx = \frac {\sin(\pi x)} {\pi} \] \[ \int x \cos(\pi x) dx \] \[ = \frac {x \sin(\pi x)} {\pi} - \int \frac {\sin(\pi x)} {\pi} dx \] \[ = \frac {x \sin(\pi x)} {\pi} - \frac {1} {\pi} \left( -\frac {\cos(\pi x )} {\pi} \right) + C \] \[ = \frac {x \sin(\pi x)} {\pi} + \frac {\cos(\pi x)} {\pi ^ 2} + C \]

微分方程式

微分方程式の計算例

\[ \frac {dy} {dt} = 3y \] \[ \frac {dy} {y} = 3 dt \] \[ \int \frac {dy} {y} = \int 3 dt \] \[ \int \frac {1} {y} dy = \int 3 dt \] \[ \ln |y| = 3t + C \]

分離可能な微分方程式の形

\[ \frac {dy} {dx} = g(x) \cdot h(y) \] \[ \frac {1} {h(y)} dy = g(x) dx \]

複素数

長方形の計算例

\[ |z| = \sqrt{a^{2} + b^{2}} \]

\[ \theta = \arctan \left ( \frac {b} {a} \right )\ \]

\[ a = |z|\cos{\theta} \]

\[ b = |z|\sin{\theta} \]

偏角

\[ \tan{\theta} = \frac {Im(z)} {Re(z)} \]

複素数の積

\[ z_{1} = r_{1}[\cos{(\theta_{1})} + i \sin{(\theta_{1})}], \ z_{2} = r_{2}[\cos{(\theta_{2})} + i \sin{(\theta_{2})}] \] \[ z_{1} \cdot z_{2} = r_{1}r_{2}[\cos{(\theta_{1} + \theta_{2})} + i \sin{(\theta_{1} + \theta_{2})}] \]

複素数の商

\[ z_{1} = r_{1}[\cos{(\theta_{1})} + i \sin{(\theta_{1})}], \ z_{2} = r_{2}[\cos{(\theta_{2})} + i \sin{(\theta_{2})}] \] \[ \frac {z_{1}} {z_{2}} = \frac {r_{1}} {r_{2}}[\cos{(\theta_{1} - \theta_{2})} + i \sin{(\theta_{1} - \theta_{2})}] \]

複素数のn乗根の計算例①

\[ z^{3} = -512, \ 270^{\circ} \le \theta \le 360^{\circ} \] \[ z^{3} = r^{3}[\cos{(3 \cdot \theta)} + i \sin{(3 \cdot \theta)}] \] \[ r^{3}[\cos{(3 \cdot \theta)} + i \sin{(3 \cdot \theta)}] = 512[\cos{(180^{\circ} + k \cdot 360^{\circ})} + i \sin{(180^{\circ} + k \cdot 360^{\circ})}]\] \[ r^{3} = 512 \] \[ r = 8 \] \[ 3 \cdot \theta = 180^{\circ} + k \cdot 360^{\circ} \] \[ \theta = 60^{\circ} + k \cdot 120^{\circ} \] \[ \ 270^{\circ} \le \theta \le 360^{\circ} \] \[ \theta = 300^{\circ} \] \[ z = r[\cos{(\theta)} + i \cdot \sin{(\theta)}] \] \[ z = 8[\cos{(300^{\circ})} + i \cdot \sin{(300^{\circ})}] \] \[ z = 8\cos{(300^{\circ})} + 8\sin{(300^{\circ})} \cdot i \] \[ z = 4 - 6.928i \]

複素数のn乗根の計算例②

問題が間違っているかもしれない。「270度より大きい」だったかも。

\[ z^{5} = -7776i, \ 270^{\circ} \le \theta \le 360^{\circ} \] \[ z^{5} = r^{5}[\cos{(5 \cdot \theta)} + i \sin{(5 \cdot \theta)}] \] \[ r^{5}[\cos{(5 \cdot \theta)} + i \sin{(5 \cdot \theta)}] = 7776[\cos{(270^{\circ} + k \cdot 360^{\circ})} + i \sin{(270^{\circ} + k \cdot 360^{\circ})}]\] \[ r^{5} = 7776 \] \[ r = 6 \] \[ 5 \cdot \theta = 270^{\circ} + k \cdot 360^{\circ} \] \[ \theta = 54^{\circ} + k \cdot 72^{\circ} \] \[ \ 270^{\circ} \le \theta \le 360^{\circ} \] \[ \theta = 342^{\circ} \] \[ z = r[\cos{(\theta)} + i \cdot \sin{(\theta)}] \] \[ z = 6[\cos{(342^{\circ})} + i \cdot \sin{(342^{\circ})}] \] \[ z = 6\cos{(342^{\circ})} + 6\sin{(342^{\circ})} \cdot i \] \[ z = 5.706 - 1.854i \]

行列

2×2行列の逆行列

\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \] \[ A^{-1} = \frac {1} {ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]