利用者:知識熊/sandbox/自然法則

自然科学（数学・物理学・電磁気学・力学・熱力学・熱化学・化学）

$f(x)={\tfrac {\ln {x}}{4\pi ^{3}x}}$

関数$f(x)$についての微分の定義 ${\dot {f}}(x)=f'(x)={\tfrac {d}{dx}}f(x)\equiv \lim _{\Delta {x}\to {0}}{\tfrac {f(x+\Delta {x})-f(x)}{\Delta {x}}}$

関数$f(x,y)$についての偏微分の定義。

1. ${\tfrac {\partial f}{\partial x}}=f_{x}(x,y)=\left({\tfrac {\partial f}{\partial x}}\right)_{y}\equiv \lim _{\Delta {x}\to {0}}{\tfrac {f(x+\Delta {x},y)-f(x,y)}{\Delta {x}}}$
2. ${\tfrac {\partial f}{\partial y}}=f_{y}(x,y)=\left({\tfrac {\partial f}{\partial y}}\right)_{x}\equiv \lim _{\Delta {y}\to {0}}{\tfrac {f(x,y+\Delta {y})-f(x,y)}{\Delta {y}}}$

$\left(x^{n}\right)'=nx^{n-1}$

$\left(\ln {\left[f(x)\right]}\right)'={\tfrac {f'(x)}{f(x)}}$

ニュートン法 $x_{n+1}=x_{n}-{\tfrac {f\left(x_{n}\right)}{f'\left(x_{n}\right)}}$

3次元ベクトルに関するベクトル解析の演算子及び作用素。

1. ナブラ $\nabla \equiv {\tfrac {\partial }{\partial x}}\,\mathbf {i} +{\tfrac {\partial }{\partial y}}\,\mathbf {j} +{\tfrac {\partial }{\partial z}}\,\mathbf {k} =\left({\tfrac {\partial }{\partial {x}}},{\tfrac {\partial }{\partial {y}}},{\tfrac {\partial }{\partial {z}}}\right)$
2. 勾配 ${\mathsf {grad}}\,\phi =\nabla \phi ={\tfrac {\partial \phi }{\partial x}}\,\mathbf {i} +{\tfrac {\partial \phi }{\partial y}}\,\mathbf {j} +{\tfrac {\partial \phi }{\partial z}}\,\mathbf {k} =\left({\tfrac {\partial \phi }{\partial {x}}},{\tfrac {\partial \phi }{\partial {y}}},{\tfrac {\partial \phi }{\partial {z}}}\right)$
3. 発散 ${\mathsf {div}}\,{\boldsymbol {X}}=\nabla \cdot {\boldsymbol {X}}={\tfrac {\partial {X_{x}}}{\partial {x}}}+{\tfrac {\partial {X_{y}}}{\partial {y}}}+{\tfrac {\partial {X_{z}}}{\partial {z}}}$
4. 回転 ${\begin{aligned}{\mathsf {rot}}\,{\boldsymbol {E}}&=\nabla \times {\boldsymbol {E}}={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\{\tfrac {\partial }{\partial x}}&{\tfrac {\partial }{\partial y}}&{\tfrac {\partial }{\partial z}}\\E_{x}&E_{y}&E_{z}\end{vmatrix}}\\&=\left({\tfrac {\partial E_{z}}{\partial y}}-{\tfrac {\partial E_{y}}{\partial z}}\right)\mathbf {i} +\left({\tfrac {\partial E_{x}}{\partial z}}-{\tfrac {\partial E_{z}}{\partial x}}\right)\mathbf {j} +\left({\tfrac {\partial E_{y}}{\partial x}}-{\tfrac {\partial E_{x}}{\partial y}}\right)\mathbf {k} \\&=\left({\tfrac {\partial {E_{z}}}{\partial {y}}}-{\tfrac {\partial {E_{y}}}{\partial {z}}},{\tfrac {\partial {E_{x}}}{\partial {z}}}-{\tfrac {\partial {E_{z}}}{\partial {x}}},{\tfrac {\partial {E_{y}}}{\partial {x}}}-{\tfrac {\partial {E_{x}}}{\partial {y}}}\right)\end{aligned}}$
5. ラプラシアン ${\begin{aligned}\Delta \,\phi &=\nabla ^{2}\,\phi =\nabla \cdot \nabla \,\phi ={\mathsf {div}}\,{\mathsf {grad}}\,\phi \\&={\tfrac {\partial }{\partial {x}}}\left({\tfrac {\partial \phi }{\partial {x}}}\right)+{\tfrac {\partial }{\partial {y}}}\left({\tfrac {\partial \phi }{\partial {y}}}\right)+{\tfrac {\partial }{\partial {z}}}\left({\tfrac {\partial \phi }{\partial {z}}}\right)={\tfrac {\partial ^{2}\phi }{\partial {x^{2}}}}+{\tfrac {\partial ^{2}\phi }{\partial {y^{2}}}}+{\tfrac {\partial ^{2}\phi }{\partial {z^{2}}}}\end{aligned}}$
6. ダランベルシアン $\Box \equiv \Delta -{\tfrac {1}{c^{2}}}{\tfrac {\partial ^{2}}{\partial t^{2}}}$
7. ${\begin{aligned}\nabla \times \nabla \times {\boldsymbol {A}}&={\mathsf {rot}}({\mathsf {rot}}\,{\boldsymbol {A}})=\nabla (\nabla \cdot {\boldsymbol {A}})-\nabla ^{2}\,{\boldsymbol {A}}\\&=\nabla \,{\mathsf {div}}\,{\boldsymbol {A}}-\nabla \cdot \nabla \,{\boldsymbol {A}}={\mathsf {grad}}\,{\mathsf {div}}\,{\boldsymbol {A}}-\Delta \,{\boldsymbol {A}}\end{aligned}}$

関数の定積分$\int _{a}^{b}(x^{n})\,dx=\left[{\tfrac {x^{n+1}}{n+1}}\right]_{a}^{b}$

閉曲線Cの線積分$\oint _{C}{f(x)}\,dx$

重積分。

1. 平面Dの二重積分$\iint _{D}{f(x,y)}\,dx\,dy$
2. 立体の三重積分$\iiint \limits _{\text{cuboid}}{f(x,y,z)}\,dx\,dy\,dz$

数列の総和。

1. $\sum _{k=0}^{n}{k}={\tfrac {n(n+1)}{2}}$
2. $\sum _{k=0}^{n}{k^{2}}={\tfrac {n(n+1)(2n+1)}{6}}$
3. $\sum _{k=0}^{n}{k^{3}}={\tfrac {n^{2}(n+1)^{2}}{4}}$

$6{\tfrac {\partial ^{2}f}{\partial x^{2}}}+5{\tfrac {\partial f}{\partial x}}-2f(x,y)=0$

$6{\tfrac {d^{2}y}{dx^{2}}}+5{\tfrac {dy}{dx}}-2y=4e^{3x}$

$6y''+5y'-2y=4e^{3x}$

フックの法則。

1. $F=-kx$
2. $U={\tfrac {1}{2}}kx^{2}$

ニュートンの提唱した古典力学。

1. 速度 ${\vec {\boldsymbol {v}}}\equiv \lim _{\Delta {t}\to {0}}{\tfrac {{\vec {\boldsymbol {r}}}(t+\Delta {t})-{\vec {\boldsymbol {r}}}(t)}{\Delta {t}}}={\tfrac {d{\vec {\boldsymbol {r}}}}{dt}}$
2. 加速度 ${\vec {\boldsymbol {a}}}\equiv \lim _{\Delta {t}\to {0}}{\tfrac {{\vec {\boldsymbol {v}}}(t+\Delta {t})-{\vec {\boldsymbol {v}}}(t)}{\Delta {t}}}={\tfrac {d{\vec {\boldsymbol {v}}}}{dt}}={\tfrac {d^{2}{\vec {\boldsymbol {r}}}}{dt^{2}}}$
3. 運動量 ${\vec {\boldsymbol {p}}}=m{\vec {\boldsymbol {v}}}$
4. 運動方程式 ${\vec {\boldsymbol {F}}}={\tfrac {d{\vec {\boldsymbol {p}}}}{dt}}=m{\vec {\boldsymbol {a}}}=m{\tfrac {d{\vec {\boldsymbol {v}}}}{dt}}=m{\tfrac {d^{2}{\vec {\boldsymbol {r}}}}{dt^{2}}}$
5. エネルギー $E=m{\boldsymbol {g}}h+{\tfrac {1}{2}}m{\boldsymbol {v}}^{2}$

相対性理論に於けるエネルギーと運動量。

1. $E={\tfrac {mc^{2}}{\sqrt {1-\left({\tfrac {v}{c}}\right)^{2}}}}$
2. ${\boldsymbol {p}}={\tfrac {m{\boldsymbol {v}}}{\sqrt {1-\left({\tfrac {v}{c}}\right)^{2}}}}$
3. $E={\sqrt {m^{2}c^{4}+c^{2}{\boldsymbol {p}}^{2}}}$

• $v\ll c$の時

1. $E=mc^{2}$
2. ${\boldsymbol {p}}=m{\boldsymbol {v}}$

アインシュタイン=ド・ブロイの関係式。

1. $E=h\nu ={\tfrac {hc}{\lambda }}\equiv \hbar \omega$
2. $\omega \equiv 2\pi \nu$
3. $p={\tfrac {h\nu }{c}}={\tfrac {h}{\lambda }}\equiv \hbar k$
4. $k\equiv {\tfrac {2\pi }{\lambda }}$

アインシュタイン方程式。

1. $R_{\mu \nu }-{\tfrac {1}{2}}R\,g_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }$
2. $\kappa ={\tfrac {8\pi G}{c^{4}}}$
3. シュヴァルツシルト半径 $r_{S}={\tfrac {2GM}{c^{2}}}$

• フリードマン方程式

1. $\left({\tfrac {\dot {a}}{a}}\right)^{2}+{\tfrac {Kc^{2}}{a^{2}}}={\tfrac {8\pi G}{3c^{2}}}\rho +{\tfrac {\Lambda c^{2}}{3}}={\tfrac {8\pi G}{3c^{2}}}{\tilde {\rho }}$
2. $2{\tfrac {\ddot {a}}{a}}+\left({\tfrac {\dot {a}}{a}}\right)^{2}+{\tfrac {Kc^{2}}{a^{2}}}=\Lambda c^{2}-{\tfrac {8\pi G}{c^{4}}}P$
3. ${\tilde {\rho }}\equiv \rho +\rho _{\Lambda }\equiv \rho +{\tfrac {\Lambda c^{4}}{8\pi G}}$

宇宙の温度 $T={\sqrt[{4}]{\tfrac {3c^{3}}{128\pi G\sigma }}}{\tfrac {1}{\sqrt {t}}}$

ハッブルの法則。

1. ${\boldsymbol {v}}=H_{0}{\boldsymbol {r}}$
2. $H_{0}\equiv {\tfrac {\dot {a}}{a}}$

等加速度直線運動の公式。

1. ${\boldsymbol {v}}={\boldsymbol {v_{0}}}+{\boldsymbol {a}}t$
2. $x={\boldsymbol {v_{0}}}t+{\tfrac {1}{2}}{\boldsymbol {a}}t^{2}$
3. $2{\boldsymbol {a}}x={\boldsymbol {v}}^{2}-{\boldsymbol {v_{0}}}^{2}$

自由落下運動の公式。

1. ${\boldsymbol {v}}={\boldsymbol {g}}t$
2. $y={\tfrac {1}{2}}{\boldsymbol {g}}t^{2}$
3. $2{\boldsymbol {g}}y={\boldsymbol {v}}^{2}$

鉛直投げ上げの公式。

1. ${\boldsymbol {v}}={\boldsymbol {v_{0}}}-{\boldsymbol {g}}t$
2. $y={\boldsymbol {v_{0}}}t-{\tfrac {1}{2}}{\boldsymbol {g}}t^{2}$
3. $2{\boldsymbol {g}}y={\boldsymbol {v}}^{2}-{\boldsymbol {v_{0}}}^{2}$

複素指数関数を使った三角関数の定義。

1. $\sin {x}\equiv {\tfrac {e^{ix}-e^{-ix}}{2i}}$
2. $\cos {x}\equiv {\tfrac {e^{ix}+e^{-ix}}{2}}$
3. $\tan {x}\equiv {\tfrac {\sin {x}}{\cos {x}}}=\sin {x}\sec {x}={\tfrac {e^{ix}-e^{-ix}}{i(e^{ix}+e^{-ix})}}$
4. $\cot {x}\equiv {\tfrac {1}{\tan {x}}}={\cos {x} \over \sin {x}}=\cos {x}\csc {x}={\tfrac {e^{ix}+e^{-ix}}{e^{ix}-e^{-ix}}}i$
5. $\sec {x}\equiv {\tfrac {1}{\cos {x}}}={\tfrac {2}{e^{ix}+e^{-ix}}}$
6. $\csc {x}\equiv {\tfrac {1}{\sin {x}}}={\tfrac {2i}{e^{ix}-e^{-ix}}}$

三角関数の関係式。

1. $\sin ^{2}x+\cos ^{2}x=1$
2. $\sin {(x\pm y)}=\sin x\cos y\pm \cos x\sin y$
3. $\cos {(x\pm y)}=\cos x\cos y\mp \sin x\sin y$
4. $\tan {(x\pm y)}={\tfrac {\tan x\pm \tan y}{1\mp \tan x\tan y}}$
5. $\sin {2x}=2\sin x\cos x$
6. $\cos {2x}=1-2\sin ^{2}x=2\cos ^{2}x-1$
7. $\tan {2x}={\tfrac {2\tan x}{1-\tan ^{2}x}}$
8. ${\begin{aligned}\sin {3x}&=2\sin x\cos ^{2}x+\sin x(2\cos ^{2}x-1)=4\sin x\cos ^{2}x-\sin x\\&=\sin x(4\cos ^{2}x-1)=\sin x(2\cos x+1)(2\cos x-1)\end{aligned}}$
9. ${\begin{aligned}\cos {3x}&=\cos x(1-2\sin ^{2}x)-2\sin ^{2}x\cos x=\cos x-4\sin ^{2}x\cos x\\&=\cos x(1-4\sin ^{2}x)=\cos x(1+2\sin x)(1-2\sin x)\end{aligned}}$
10. $(\sin x\pm \cos x)^{2}=1\pm \sin {2x}$

三角関数の微分公式。

1. $(\sin {x})'=\cos {x}$
2. $(\cos {x})'=-\sin {x}$
3. $(\tan {x})'={\tfrac {1}{\cos ^{2}x}}=\sec ^{2}x=1+\tan ^{2}x=1+\left({\tfrac {\sin {x}}{\cos {x}}}\right)^{2}$
4. $(\cot {x})'=-{\tfrac {1}{\sin ^{2}x}}=-\csc ^{2}x=-(1+\cot ^{2}x)=-\left[1+\left({\tfrac {\cos {x}}{\sin {x}}}\right)^{2}\right]$
5. $(\sec {x})'=\sec {x}\tan {x}={\tfrac {\tan x}{\cos x}}={\tfrac {\sin x}{\cos ^{2}x}}$
6. $(\csc {x})'=-\csc {x}\cot {x}=-{\tfrac {1}{\sin {x}\tan {x}}}=-{\tfrac {\cos x}{\sin ^{2}x}}$

三角関数の逆関数。即ち逆三角関数。

1. $\sin ^{-1}x=\arcsin {x}=\int {\tfrac {1}{\sqrt {1-x^{2}}}}\,dx$
2. $\cos ^{-1}x=\arccos {x}=\int {\tfrac {-1}{\sqrt {1-x^{2}}}}\,dx$
3. $\tan ^{-1}x=\arctan {x}=\int {\tfrac {1}{1+x^{2}}}\,dx$
4. $\cot ^{-1}x=\operatorname {arccot} {x}=\int {\tfrac {-1}{1+x^{2}}}\,dx$
5. $\sec ^{-1}x=\operatorname {arcsec} {x}=\int {\tfrac {1}{x^{2}\,{\sqrt {1-x^{-2}}}}}\,dx$
6. $\csc ^{-1}x=\operatorname {arccsc} {x}=\int {\tfrac {-1}{x^{2}\,{\sqrt {1-x^{-2}}}}}\,dx$

逆三角関数の微分公式。

1. $(\arcsin {x})'={\tfrac {1}{\sqrt {1-x^{2}}}}$
2. $(\arccos {x})'={\tfrac {-1}{\sqrt {1-x^{2}}}}$
3. $(\arctan {x})'={\tfrac {1}{1+x^{2}}}$
4. $(\operatorname {arccot} {x})'={\tfrac {-1}{1+x^{2}}}$
5. $(\operatorname {arcsec} {x})'={\tfrac {1}{x^{2}\,{\sqrt {1-x^{-2}}}}}$
6. $(\operatorname {arccsc} {x})'={\tfrac {-1}{x^{2}\,{\sqrt {1-x^{-2}}}}}$

双曲線関数の定義。

1. $\cosh {x}\equiv {\tfrac {e^{x}+e^{-x}}{2}}$
2. $\sinh {x}\equiv {\tfrac {e^{x}-e^{-x}}{2}}$
3. $\tanh {x}\equiv {\tfrac {\sinh {x}}{\cosh {x}}}=\sinh {x}\operatorname {sech} {x}={\tfrac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}$
4. $\coth {x}\equiv {\tfrac {1}{\tanh {x}}}={\tfrac {\cosh {x}}{\sinh {x}}}=\cosh {x}\operatorname {csch} {x}={\tfrac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}$
5. $\operatorname {sech} {x}\equiv {\tfrac {1}{\cosh {x}}}={\tfrac {2}{e^{x}+e^{-x}}}$
6. $\operatorname {csch} {x}\equiv {\tfrac {1}{\sinh {x}}}={\tfrac {2}{e^{x}-e^{-x}}}$

双曲線関数の関係式。

1. $\cosh ^{2}x-\sinh ^{2}x=1$
2. $\sinh {(x\pm y)}=\sinh x\cosh y\pm \cosh x\sinh y$
3. $\cosh {(x\pm y)}=\cosh x\cosh y\pm \sinh x\sinh y$
4. $\sinh {2x}=2\sinh x\cosh x$
5. $\cosh {2x}=1+2\sinh ^{2}x=2\cosh ^{2}x-1$
6. ${\begin{aligned}\sinh {3x}&=2\sinh x\cosh ^{2}x+\sinh x(2\cosh ^{2}x-1)=4\sinh x\cosh ^{2}x-\sinh x\\&=\sinh x(4\cosh ^{2}x-1)=\sinh x(2\cosh x+1)(2\cosh x-1)\end{aligned}}$
7. ${\begin{aligned}\cosh {3x}&=\cosh x(1+2\sinh ^{2}x)+2\sinh ^{2}x\cosh x=4\sinh ^{2}x\cosh x+\cosh x\\&=\cosh x(4\sinh ^{2}x+1)\end{aligned}}$
8. $(\cosh x\pm i\sinh x)^{2}=1\pm i\sinh {2x}$

双曲線関数の微分公式。

1. $(\cosh {x})'=\sinh {x}$
2. $(\sinh {x})'=\cosh {x}$
3. $(\tanh {x})'={\tfrac {1}{\cosh ^{2}x}}=\operatorname {sech} ^{2}x=1-\tanh ^{2}x=1-\left({\tfrac {\sinh {x}}{\cosh {x}}}\right)^{2}$
4. $(\coth {x})'=-{\tfrac {1}{\sinh ^{2}x}}=-\operatorname {csch} ^{2}x=1-\coth ^{2}x=1-\left({\tfrac {\cosh {x}}{\sinh {x}}}\right)^{2}$
5. $(\operatorname {sech} {x})'=-\coth {x}\operatorname {csch} {x}=-{\tfrac {1}{\tanh {x}\sinh {x}}}=-{\tfrac {\cosh {x}}{\sinh ^{2}x}}$
6. $(\operatorname {csch} {x})'=-\tanh {x}\operatorname {sech} {x}=-{\tfrac {\tanh {x}}{\cosh {x}}}=-{\tfrac {\sinh {x}}{\cosh ^{2}x}}$

双曲線関数の逆関数。即ち逆双曲線関数。

1. ${\begin{aligned}\cosh ^{-1}x=\operatorname {arcosh} x&=\ln {\left|x\pm {\sqrt {x^{2}-1}}\right|}\\&=\ln {\left|x\pm {\sqrt {x+1}}{\sqrt {x-1}}\right|}=\int {\tfrac {1}{\sqrt {x^{2}-1}}}\,dx\end{aligned}}$
2. $\sinh ^{-1}x=\operatorname {arsinh} x=\ln {\left|x+{\sqrt {x^{2}+1}}\right|}=\int {\tfrac {1}{\sqrt {1+x^{2}}}}\,dx$
3. $\tanh ^{-1}x=\operatorname {artanh} x={\tfrac {1}{2}}\ln {\left({\tfrac {1+x}{1-x}}\right)}=\int {\tfrac {1}{1-x^{2}}}\,dx$
4. $\coth ^{-1}x=\operatorname {arcoth} x={\tfrac {1}{2}}\ln {\left({\tfrac {x+1}{x-1}}\right)}=\int {\tfrac {1}{1-x^{2}}}\,dx$
5. ${\begin{aligned}\operatorname {sech} ^{-1}x=\operatorname {arsech} x&=\ln {\left|{\tfrac {1}{x}}\pm {\sqrt {{\tfrac {1}{x^{2}}}-1}}\right|}\\&=\ln {\left|{\tfrac {1}{x}}\pm {\sqrt {{\tfrac {1}{x}}+1}}{\sqrt {{\tfrac {1}{x}}-1}}\right|}=\int {\tfrac {\mp 1}{x\,{\sqrt {1-x^{2}}}}}\,dx\end{aligned}}$
6. $\operatorname {csch} ^{-1}x=\operatorname {arcsch} x=\ln {\left|{\tfrac {1}{x}}+{\sqrt {{\tfrac {1}{x^{2}}}+1}}\right|}=\int {\tfrac {\mp 1}{x\,{\sqrt {1+x^{2}}}}}\,dx$

逆双曲線関数の微分公式。

1. $(\operatorname {arcosh} x)'={\tfrac {1}{\sqrt {x^{2}-1}}}$
2. $(\operatorname {arsinh} x)'={\tfrac {1}{\sqrt {1+x^{2}}}}$
3. $(\operatorname {artanh} x)'={\tfrac {1}{1-x^{2}}}$
4. $(\operatorname {arcoth} x)'={\tfrac {1}{1-x^{2}}}$
5. $(\operatorname {arsech} x)'={\tfrac {\mp 1}{x\,{\sqrt {1-x^{2}}}}}$
6. $(\operatorname {arcsch} x)'={\tfrac {\mp 1}{x\,{\sqrt {1+x^{2}}}}}$

オイラーの公式 $e^{\pm ix}=\cos x\pm i\sin x$ より導出される世界で最も美しい数式の一つとされるオイラーの等式 $e^{\pi i}+1=0$

虚数 $i={\sqrt {-1}}$、即ち $i^{2}=-1$。

x に関する二次方程式 $ax^{2}+bx+c=0$ について

1. 判別式 $D=b^{2}-4ac$
2. $x={\tfrac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$

• b が偶数の場合 $b=2b'$ として

1. 判別式 ${\tfrac {D}{4}}=b'^{2}-ac$
2. $x={\tfrac {-b'\pm {\sqrt {b'^{2}-ac}}}{a}}$

万有引力の法則。

1. $\mathbf {F} =G\,{\tfrac {m_{1}m_{2}}{r^{2}}}\mathbf {\hat {r}}$
2. $U=G\,{\tfrac {m_{1}m_{2}}{r}}$
3. $\mathbf {\hat {r}} ={\tfrac {\mathbf {r} }{|\mathbf {r} |}}={\tfrac {\mathbf {r} }{r}}$

熱力学第一法則 $dU=\delta {Q}-\delta {W}=d'Q-pdV=TdS-pdV$

熱力学第二法則及び第三法則に関わるエントロピー。

1. $S=k_{\mathrm {B} }\ln {W}$
2. $\Delta S=nR\ln \left({\tfrac {V_{2}}{V_{1}}}\right)$
3. $dS={\tfrac {dQ}{T}}$

エンタルピー。

1. $H=U+pV$
2. $dH=dU+pdV+Vdp=TdS+Vdp$

• 等圧過程の場合

1. $dH=TdS=dQ$

自由エネルギー。

• ヘルムホルツエネルギー

1. $F=U-TS$
2. $dF=dU-TdS-SdT=-SdT-pdV$

• ギブズエネルギー

1. $G=U-TS+pV=H-TS=F+pV$
2. ${\begin{aligned}dG&=dU-TdS-SdT+pdV+Vdp=dH-TdS-SdT\\&=dF+pdV+Vdp=-SdT+Vdp\end{aligned}}$

比熱。

1. 定圧比熱 ${\bar {C_{P}}}=\left({\tfrac {\partial H}{\partial T}}\right)_{P}$
2. 定積比熱 ${\bar {C_{V}}}=\left({\tfrac {\partial U}{\partial T}}\right)_{V}$

• 比熱比

1. $\gamma ={\tfrac {\bar {C_{P}}}{\bar {C_{V}}}}$
2. 液体や個体の場合 $\gamma \approx 1$
3. 単原子気体の場合 $\gamma \approx {\tfrac {5}{3}}$
4. 二原子気体の場合 $\gamma \approx {\tfrac {7}{5}}$
5. 多原子気体の場合 $\gamma \approx {\tfrac {4}{3}}$

ポアソンの関係式。

1. $PV^{\gamma }=k$
2. $TV^{\gamma -1}=k$

マイヤーの関係式 ${\bar {C_{P}}}-{\bar {C_{V}}}=R$

理想気体に関する法則。

1. ボイルの法則 $PV=k$
2. シャルルの法則 ${\tfrac {V}{T}}=k$
3. ボイル＝シャルルの法則 ${\tfrac {PV}{T}}=k$
4. 状態方程式 $PV=nRT$

ファン・デル・ワールスの状態方程式 $\left(p+{\tfrac {an^{2}}{V^{2}}}\right)(V-nb)=nRT$

アレニウスの式。

1. $\ln {C}=-kt+\ln {C_{0}}$
2. 速度定数 $k=A\cdot e^{-{\tfrac {E_{a}}{RT}}}$
3. $k=A\cdot e^{-{\tfrac {E_{a}}{k_{\mathrm {B} }T}}}$

• アレニウスプロット

1. $\ln {k}=-{\tfrac {E_{a}}{RT}}+\ln {A}$
2. $\ln {k}=-{\tfrac {E_{a}}{k_{\mathrm {B} }T}}+\ln {A}$

ミカエリス・メンテン式 $V={\tfrac {V_{max}C}{K_{m}+C}}$

ラインウィーバー＝バークプロット ${\tfrac {1}{V}}={\tfrac {K_{m}}{V_{max}}}{\tfrac {1}{C}}+{\tfrac {1}{V_{max}}}$

ランベルト・ベールの法則 $I=I_{0}\,e^{-\varepsilon cl}\approx I_{0}\,10^{-\varepsilon cl}$

吸光度 $A_{\lambda }=-\log {\left({\tfrac {I}{I_{0}}}\right)}=-{\tfrac {\ln {\left({\tfrac {I}{I_{0}}}\right)}}{\ln {10}}}=\varepsilon cl$

散乱に関する公式。

1. レイリー散乱の散乱強度 $I=I_{0}{\tfrac {1+\cos ^{2}\theta }{2R^{2}}}\left({\tfrac {2\pi }{\lambda }}\right)^{4}\left({\tfrac {n^{2}-1}{n^{2}+2}}\right)^{2}\left({\tfrac {d}{2}}\right)^{6}$
2. トムソン散乱の公式 ${\begin{aligned}{\tfrac {d\sigma _{\text{T}}}{d\Omega }}&=\left({\tfrac {q^{2}}{4\pi \varepsilon _{0}mc^{2}}}\right)^{2}\cdot {\tfrac {1}{2}}(1+\cos ^{2}\theta )\\&=\left({\tfrac {q^{2}}{4\pi \varepsilon _{0}mc^{2}}}\right)^{2}\left(1-{\tfrac {\sin ^{2}\theta }{2}}\right)\end{aligned}}$
3. 電子によるX線のコンプトン効果の公式 ${\begin{aligned}\Delta \lambda =\lambda '-\lambda &={\tfrac {h}{m_{\mathrm {e} }c}}(1-\cos {\theta })=\lambda _{\mathrm {e} }(1-\cos {\theta })\\&={\tfrac {2h}{m_{\mathrm {e} }c}}\sin ^{2}\left({\tfrac {\theta }{2}}\right)=2\lambda _{\mathrm {e} }\sin ^{2}\left({\tfrac {\theta }{2}}\right)\end{aligned}}$
4. クライン＝仁科の公式 ${\begin{aligned}{\tfrac {d\sigma }{d\Omega }}&=\left({\tfrac {\alpha }{2\pi }}\right)^{2}{\tfrac {{\lambda _{\text{e}}}^{2}}{2}}\left({\tfrac {\lambda }{\lambda '}}\right)^{2}\left[{\tfrac {\lambda }{\lambda '}}+{\tfrac {\lambda '}{\lambda }}-\sin ^{2}\theta \right]\\&=\left({\tfrac {\alpha }{2\pi }}\right)^{2}{\tfrac {{\lambda _{\text{e}}}^{2}}{2}}(2-\sin ^{2}\theta )\\&=\left({\tfrac {\alpha }{2\pi }}\right)^{2}{\tfrac {{\lambda _{\text{e}}}^{2}}{2}}(1+\cos ^{2}\theta )\\&={\tfrac {{r_{\text{e}}}^{2}}{2}}(1+\cos ^{2}\theta )\\&={r_{\text{e}}}^{2}\left(1-{\tfrac {\sin ^{2}\theta }{2}}\right)\end{aligned}}$

光電効果の公式 $\Delta {E}=h\nu -W$

マクスウェル＝ボルツマン分布 $f(v)=4\pi v^{2}\left({\tfrac {m}{2\pi k_{\mathrm {B} }T}}\right)^{\tfrac {3}{2}}e^{-{\tfrac {mv^{2}}{2k_{\mathrm {B} }T}}}$

ボース＝アインシュタイン分布 $f(\varepsilon )={\tfrac {1}{e^{(\varepsilon -\mu )/k_{\mathrm {B} }T}-1}}$

フェルミ＝ディラック分布 $f(\varepsilon )={\tfrac {1}{e^{(\varepsilon -\zeta )/k_{\mathrm {B} }T}+1}}$

黒体及び熱放射に関する法則。

1. ウィーンの変位則 $\lambda _{\text{max}}={\tfrac {b}{T}}$

• ヴィーンの放射法則

1. $R(\lambda ,T)={\tfrac {c_{1}}{\lambda ^{5}}}e^{-{\tfrac {c_{2}}{\lambda T}}}$
2. $R(\nu ,T)={\tfrac {c_{1}\nu ^{3}}{c^{4}}}e^{-{\tfrac {c_{2}\nu }{cT}}}$
3. $I(\lambda ,T)={\tfrac {1}{\pi }}R(\lambda ,T)={\tfrac {c_{1L}}{\lambda ^{5}}}e^{-{\tfrac {c_{2}}{\lambda T}}}$
4. $I(\nu ,T)={\tfrac {c_{1L}\nu ^{3}}{c^{4}}}e^{-{\tfrac {c_{2}\nu }{cT}}}$
5. $c_{1}=2\pi hc^{2},~c_{1L}=2hc^{2},~c_{2}={\tfrac {hc}{k_{\text{B}}}}$
6. $u(\nu ,T)={\tfrac {8\pi h\nu ^{3}}{c^{3}}}e^{-{\tfrac {h\nu }{k_{\text{B}}T}}}$

• レイリー・ジーンズの法則

1. $u(\lambda ,T)={\tfrac {8\pi k_{\text{B}}T}{\lambda ^{4}}}$
2. $u(\nu ,T)={\tfrac {8\pi \nu ^{2}k_{\text{B}}T}{c^{3}}}$

• シュテファン＝ボルツマンの法則

1. $I=\sigma T^{4}$
2. $\sigma ={\tfrac {2\pi ^{5}k_{\mathrm {B} }^{4}}{15c^{2}h^{3}}}={\tfrac {\pi ^{2}k_{\mathrm {B} }^{4}}{60c^{2}\hbar ^{3}}}$

• プランクの法則

1. $I(\nu ,T)={\tfrac {2h\nu ^{3}}{c^{2}}}{\tfrac {1}{e^{h\nu /k_{\mathrm {B} }T}-1}}$
2. $I'(\lambda ,T)={\tfrac {2hc^{2}}{\lambda ^{5}}}{\tfrac {1}{e^{hc/\lambda k_{\mathrm {B} }T}-1}}$
3. $u(\nu ,T)={\tfrac {4\pi }{c}}I(\nu ,T)={\tfrac {8\pi h\nu ^{3}}{c^{3}}}{\tfrac {1}{e^{h\nu /k_{\mathrm {B} }T}-1}}$
4. $u'(\lambda ,T)={\tfrac {4\pi }{c}}I'(\lambda ,T)={\tfrac {8\pi hc}{\lambda ^{5}}}{\tfrac {1}{e^{hc/\lambda k_{\mathrm {B} }T}-1}}$

モル分率や分圧に関する法則。

1. ラウールの法則 $P_{1}=x_{1}P_{1}^{*}$
2. ヘンリーの法則 $P_{2}=Kx_{2}=K'm$
3. 蒸気圧降下 $\Delta {P}=x_{2}P_{1}^{*}$

• 凝固点降下

1. $\Delta {T}=K_{f}m$
2. $K_{f}={\tfrac {MRT_{f}^{2}}{\Delta {H_{f}}}}$

• 沸点上昇

1. $\Delta {T}=K_{b}m$
2. $K_{b}={\tfrac {MRT_{b}^{2}}{\Delta {H_{b}}}}$

コールラウシュの法則及びデバイ-ヒュッケルの式。

1. セル定数 $k=R_{x}\,\kappa$
2. 当量伝導度 $\Lambda ={\tfrac {1000\kappa }{C}}=\Lambda ^{\infty }-A{\sqrt {C}}$
3. モル伝導度 $\Lambda ^{\infty }=n\lambda ^{+}+m\lambda ^{-}$
4. 解離度 $\alpha ={\tfrac {\Lambda }{\Lambda ^{\infty }}}$
5. 見かけの解離定数 $K_{C}={\tfrac {\alpha ^{2}C}{1-\alpha }}={\tfrac {\Lambda ^{2}C}{\Lambda ^{\infty }(\Lambda ^{\infty }-\Lambda )}}$
6. イオン強度 $I={\tfrac {1}{2}}\sum _{i}^{n}m_{i}z_{i}^{2}$
7. 活量 $a_{i}\equiv \gamma _{i}\,x_{i}$
8. イオンの平均活量係数 ${\begin{aligned}\log {\gamma _{\pm }}&=-{\tfrac {1.824\times 10^{6}}{\sqrt {\left(\varepsilon T\right)^{3}}}}\left|z_{+}z_{-}\right|{\sqrt {I}}\approx -0.5094\left|z_{+}z_{-}\right|{\sqrt {I}}\\&=-{\tfrac {1.824\times 10^{6}}{\sqrt {\left(\varepsilon T\right)^{3}}}}{\sqrt {\alpha C}}\approx -0.5094{\sqrt {\alpha C}}\end{aligned}}$
9. $K_{\infty }=K_{C}\,\gamma _{\pm }^{2}$

部分モル体積。

1. ${\begin{aligned}V&=n_{1}\left({\tfrac {\partial V}{\partial n_{1}}}\right)_{n_{2},T,P}+n_{2}\left({\tfrac {\partial V}{\partial n_{2}}}\right)_{n_{1},T,P}\\&=n_{1}{\bar {V_{1}}}+n_{2}{\bar {V_{2}}}=n_{1}V_{1}^{*}+n_{2}\phi _{V}\end{aligned}}$
2. $dV=\left({\tfrac {\partial V}{\partial n_{1}}}\right)_{n_{2},T,P}dn_{1}+\left({\tfrac {\partial V}{\partial n_{2}}}\right)_{n_{1},T,P}dn_{2}={\bar {V_{1}}}dn_{1}+{\bar {V_{2}}}dn_{2}$
3. $\phi _{V}=\phi _{V}^{\circ }+S_{V}{\sqrt {m}}={\tfrac {V-n_{1}V_{1}^{*}}{n_{2}}}={\tfrac {1}{m}}\left[{\tfrac {1000+mM}{d}}-{\tfrac {1000}{d_{0}}}\right]$
4. ${\begin{aligned}{\bar {V_{2}}}&=\left({\tfrac {\partial V}{\partial n_{2}}}\right)_{n_{1},T,P}=\phi _{V}+n_{2}\left({\tfrac {\partial V}{\partial n_{2}}}\right)_{n_{1},T,P}\\&=\phi _{V}+{\tfrac {\sqrt {m}}{2}}\left({\tfrac {\partial \phi _{V}}{\partial {\sqrt {m}}}}\right)=\phi _{V}+{\tfrac {\sqrt {m}}{2}}S_{V}\end{aligned}}$

気体の赤外吸収スペクトル。

1. 振動数 $\nu =c{\tilde {\nu }}={\tfrac {1}{2\pi }}{\sqrt {\tfrac {k}{\mu }}}$
2. 波数 ${\tilde {\nu }}=2{\tilde {B}}J''+{\tilde {\nu }}_{0}$
3. 換算質量 $\mu ={\tfrac {m_{1}m_{2}}{m_{1}+m_{2}}}$
4. 慣性モーメント $I=\mu r_{e}^{2}$
5. 平均結合距離 $r_{e}={\sqrt {\tfrac {I}{\mu }}}$

光合成 ${\ce {6{CO_{2}}+6{H_{2}O}}}\,{\xrightarrow {h\nu }}\,{\ce {{C_{6}H_{12}O_{6}}+6{O_{2}}}}$

フーリエの法則 ${\boldsymbol {Q}}=-\kappa \,\nabla \,T$

拡散方程式 ${\tfrac {\partial c}{\partial t}}=\left[-D\left({\tfrac {\partial ^{2}c}{\partial x^{2}}}+{\tfrac {\partial ^{2}c}{\partial y^{2}}}+{\tfrac {\partial ^{2}c}{\partial z^{2}}}\right)=-D\,\nabla ^{2}\,c=\right]-D\,\Delta \,c$

ブラウン運動に関するアインシュタインの関係式 $D=\mu k_{\mathrm {B} }T$

物性論におけるアインシュタインの比熱式 $C_{V}=3R\left({\tfrac {h\nu }{k_{\mathrm {B} }T}}\right)^{2}{\tfrac {e^{h\nu /k_{\mathrm {B} }T}}{\left(e^{h\nu /k_{\mathrm {B} }T}-1\right)^{2}}}$

ヴィーデマン＝フランツ則。

1. ${\tfrac {\mathrm {K} }{\sigma }}=LT$
2. $L={\tfrac {\mathrm {K} }{\sigma T}}={\tfrac {(\pi k_{\text{B}})^{2}}{3e^{2}}}$

マクスウェルの方程式。

1. $\nabla \cdot {\vec {\boldsymbol {D}}}={\mathsf {div}}\,{\vec {\boldsymbol {D}}}=\rho _{e}$
2. $\nabla \cdot {\vec {\boldsymbol {B}}}={\mathsf {div}}\,{\vec {\boldsymbol {B}}}=0$
3. $\nabla \times {\vec {\boldsymbol {E}}}+{\tfrac {\partial {\vec {\boldsymbol {B}}}}{\partial t}}={\mathsf {rot}}\,{\vec {\boldsymbol {E}}}+{\tfrac {\partial {\vec {\boldsymbol {B}}}}{\partial t}}=0$
4. $\nabla \times {\vec {\boldsymbol {H}}}={\mathsf {rot}}\,{\vec {\boldsymbol {H}}}={\tfrac {\partial {\vec {\boldsymbol {D}}}}{\partial t}}+{\vec {i}}$

電気に関する法則。

1. 電束密度 $\mathbf {D} =\varepsilon _{0}\mathbf {E}$
2. ジュールの法則 $Q=RI^{2}t$

• クーロンの法則

1. クーロン力 $\mathbf {F} ={\tfrac {1}{4\pi \varepsilon _{0}}}{\tfrac {q_{1}q_{2}}{r^{2}}}\mathbf {\hat {r}} =q\mathbf {E}$
2. 電場 $\mathbf {E} (r)={\tfrac {1}{4\pi \varepsilon _{0}}}{\tfrac {q}{r^{2}}}\mathbf {\hat {r}} =-{\tfrac {1}{4\pi \varepsilon _{0}}}\,{\mathsf {grad}}\,{\tfrac {q}{r}}=-{\tfrac {1}{4\pi \varepsilon _{0}}}\nabla \,{\tfrac {q}{r}}=-{\mathsf {grad}}\,\phi =-\nabla \,\phi$
3. 電位 $\phi (r)={\tfrac {1}{4\pi \varepsilon _{0}}}{\tfrac {q}{r}}$
4. $\mathbf {\hat {r}} ={\tfrac {\mathbf {r} }{|\mathbf {r} |}}={\tfrac {\mathbf {r} }{r}}$

• オームの法則

1. 電圧 $V=RI$
2. 電気抵抗 $R=\rho {\tfrac {L}{S}}$
3. 電流密度 $\mathbf {j} =\sigma \mathbf {E}$
4. 電場 $\mathbf {E} =\rho \mathbf {j}$
5. 電力 $P=VI=RI^{2}={\tfrac {V^{2}}{R}}$

• コンデンサ

1. 電荷 $Q=CV$
2. 静電容量 $C={\tfrac {Q}{V}}=\varepsilon {\tfrac {S}{d}}$
3. 電力量 $W={\tfrac {1}{2}}CV^{2}={\tfrac {1}{2}}QV={\tfrac {Q^{2}}{2C}}$

磁気や電磁気に関する法則。

1. 磁場 $\mathbf {H} =-{\tfrac {1}{4\pi \mu _{0}}}\,{\mathsf {grad}}\,{\tfrac {\mathbf {m} \cdot \mathbf {r} }{r^{3}}}=-{\tfrac {1}{4\pi \mu _{0}}}\,\nabla \,{\tfrac {\mathbf {m} \cdot \mathbf {r} }{r^{3}}}=-{\mathsf {grad}}\,\phi _{m}=-\nabla \,\phi _{m}$
2. ポテンシャル $\phi _{m}={\tfrac {1}{4\pi \mu _{0}}}{\tfrac {\mathbf {m} \cdot \mathbf {r} }{r^{3}}}$
3. 磁束密度 $\mathbf {B} =\mu _{0}\mathbf {H}$
4. 磁化 $\mathbf {M} =\chi _{m}\mathbf {B} =\chi _{m}\mu _{0}{H}$
5. アンペールの力 $\mathbf {F} =I\mathbf {s} \times \mathbf {B}$
6. ローレンツ力 $\mathbf {F} =q\mathbf {E} +q\mathbf {v} \times \mathbf {B}$
7. ビオ・サバールの法則 $d\mathbf {B} ={\tfrac {\mu _{0}}{4\pi }}{\tfrac {Id\mathbf {s} \times \mathbf {r} }{r^{3}}}={\tfrac {\mu _{0}}{4\pi }}{\tfrac {I\sin \theta }{r^{2}}}\,d\mathbf {s}$

• クーロンの法則

1. クーロン力 $\mathbf {F} ={\tfrac {1}{4\pi \mu _{0}}}{\tfrac {q_{m1}q_{m2}}{r^{2}}}\mathbf {\hat {r}} =q_{m}\mathbf {H}$
2. 磁場 $\mathbf {H} (r)={\tfrac {1}{4\pi \mu _{0}}}{\tfrac {q_{m}}{r^{2}}}\mathbf {\hat {r}} =-{\tfrac {1}{4\pi \mu _{0}}}\,{\mathsf {grad}}\,{\tfrac {q_{m}}{r}}=-{\tfrac {1}{4\pi \mu _{0}}}\nabla \,{\tfrac {q_{m}}{r}}=-{\mathsf {grad}}\,\phi _{m}=-\nabla \,\phi _{m}$
3. 磁場ポテンシャル $\phi _{m}(r)={\tfrac {1}{4\pi \mu _{0}}}{\tfrac {q_{m}}{r}}$
4. $\mathbf {\hat {r}} ={\tfrac {\mathbf {r} }{|\mathbf {r} |}}={\tfrac {\mathbf {r} }{r}}$

• 磁気モーメント

1. $\mathbf {m} =q_{m}\mathbf {r}$
2. $m=\mu _{0}IS$

• 磁気双極子モーメント

1. ポテンシャル $U_{m}=-{\tfrac {1}{4\pi \mu _{0}}}\left[{\tfrac {3(\mathbf {m_{1}} \cdot \mathbf {r} )(\mathbf {m_{2}} \cdot \mathbf {r} )}{r^{5}}}-{\tfrac {\mathbf {m_{1}} \cdot \mathbf {m_{2}} }{r^{3}}}\right]$

• キュリーの法則

1. $\mathbf {M} =C{\tfrac {\mathbf {B} }{T}}$
2. $\chi _{m}={\tfrac {C}{T}}$

• キュリー・ワイスの法則

1. $\mathbf {M} =C{\tfrac {\mathbf {B} }{T-\theta _{p}}}$
2. $\chi _{m}={\tfrac {C}{T-\theta _{p}}}$

デバイの長さ。

1. プラズマの場合 $\lambda _{D}={\sqrt {\tfrac {\varepsilon _{0}k_{\mathrm {B} }T}{n_{\mathrm {e} }e^{2}}}}$
2. 強電解質の場合 $\varkappa ^{-1}={\sqrt {\tfrac {\varepsilon k_{\mathrm {B} }T}{2N_{\mathrm {A} }e^{2}I}}}$

プラズマ振動数 $\omega _{pe}={\sqrt {\tfrac {n_{\mathrm {e} }e^{2}}{m_{\mathrm {e} }\varepsilon _{0}}}}$

波動方程式。

1. ${\tfrac {1}{v^{2}}}{\tfrac {\partial ^{2}u}{\partial t^{2}}}={\tfrac {\partial ^{2}u}{\partial x^{2}}}+{\tfrac {\partial ^{2}u}{\partial y^{2}}}+{\tfrac {\partial ^{2}u}{\partial z^{2}}}$
2. $u=u(x,y,z,t)$
3. $\Delta \,u={\tfrac {1}{v^{2}}}{\tfrac {\partial ^{2}u}{\partial t^{2}}}$
4. $\square \,u=0$
5. $v^{2}\,\Delta \,\psi ={\tfrac {\partial ^{2}\psi }{\partial t^{2}}}$

シュレーディンガー方程式。

1. ハミルトニアン ${\hat {H}}=-{\tfrac {\hbar ^{2}}{2m}}\Delta +V$

• 時間に依存する場合

1. ${\hat {H}}\psi (t)=i\hbar \left({\tfrac {\partial \psi }{\partial t}}\right)$
2. ${\tfrac {\partial ^{2}\psi }{\partial x^{2}}}+{\tfrac {\partial ^{2}\psi }{\partial y^{2}}}+{\tfrac {\partial ^{2}\psi }{\partial z^{2}}}+{\tfrac {\partial ^{2}\psi }{\partial t^{2}}}+{\tfrac {4\pi mi}{h}}\left({\tfrac {\partial \psi }{\partial t}}\right)-{\tfrac {8\pi ^{2}m}{h^{2}}}V(x,y,z,t)\psi (x,y,z,t)=0$

• 時間に依存しない場合

1. ${\hat {H}}\psi (x,y,z)=E\psi (x,y,z)$
2. ${\tfrac {\partial ^{2}\psi }{\partial {x}^{2}}}+{\tfrac {\partial ^{2}\psi }{\partial {y}^{2}}}+{\tfrac {\partial ^{2}\psi }{\partial {z}^{2}}}+{\tfrac {8\pi ^{2}m}{h^{2}}}\left[E-V(x,y,z)\right]\psi (x,y,z)=0$

波動関数。

1. $\psi ({\boldsymbol {r}},t)=A\cdot e^{i({\boldsymbol {k}}\cdot {\boldsymbol {r}}-\omega t)}$
2. 波数 $k={\tfrac {\sqrt {2mE}}{\hbar }}$
3. 角周波数 $\omega ={\tfrac {E}{\hbar }}$