KaTeX{\KaTeX}

1. ​​多重积分与分式​

∭V(∂2P∂x2+∂2Q∂y2+∂2R∂z2) dV=∮∂V(P dy∧dz+Q dz∧dx+R dx∧dy)\iiint\limits_{V} \left( \frac{\partial^2 P}{\partial x^2} + \frac{\partial^2 Q}{\partial y^2} + \frac{\partial^2 R}{\partial z^2} \right) \, dV = \oint_{\partial V} \left( P\, dy \wedge dz + Q\, dz \wedge dx + R\, dx \wedge dy \right)

2. ​​矩阵与行列式​

det⁡(a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮an1an2⋯ann)=∑σ∈Snsgn(σ)∏i=1nai,σ(i)\det \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix} = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod_{i=1}^n a_{i,\sigma(i)}

3. ​​级数展开与极限​

lim⁡n→∞(∑k=1n1k2)=π26且eix=cos⁡x+isin⁡x=∑n=0∞(ix)nn!\lim_{n \to \infty} \left( \sum_{k=1}^n \frac{1}{k^2} \right) = \frac{\pi^2}{6} \quad \text{且} \quad e^{ix} = \cos x + i\sin x = \sum_{n=0}^\infty \frac{(ix)^n}{n!}

4. ​​微分方程与运算符​

L{f(t)}=∫0∞e−stf(t) dt满足∂2u∂t2=c2∇2u\mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t) \, dt \quad \text{满足} \quad \frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u

5. ​​概率与组合数学​

P(A∣B)=P(B∣A)P(A)P(B)其中P(B)=∑i=1nP(B∣Ai)P(Ai)P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B)} \quad \text{其中} \quad P(B) = \sum_{i=1}^n P(B \mid A_i) P(A_i)

6. ​​向量分析与微分几何

∇×F=(∂Fz∂y−∂Fy∂z)i+(∂Fx∂z−∂Fz∂x)j+(∂Fy∂x−∂Fx∂y)k\nabla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \mathbf{k}

7. ​​多项式与根号嵌套​

x=−b±b2−4ac2a且a+bc+∫01ln⁡(1+x2)x dxx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \quad \text{且} \quad \sqrt{\frac{a + \sqrt{b}}{c}} + \int_0^1 \frac{\ln(1 + x^2)}{\sqrt{x}} \, dx

8. ​​量子力学算符​

H^ψ=(−ℏ22m∇2+V(r))ψ=iℏ∂ψ∂t\hat{H} \psi = \left( -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}) \right) \psi = i\hbar \frac{\partial \psi}{\partial t}

9. ​​分形与递归关系​

zn+1=zn2+c其中z,c∈C且∣zn∣→∞ 或收敛z_{n+1} = z_n^2 + c \quad \text{其中} \quad z, c \in \mathbb{C} \quad \text{且} \quad |z_n| \to \infty \text{ 或收敛}

10. ​​统计力学与热力学​

Z=∑igie−βEi且S=−kB∑iPiln⁡PiZ = \sum_{i} g_i e^{-\beta E_i} \quad \text{且} \quad S = -k_B \sum_{i} P_i \ln P_i

行内P(A∣B)=P(B∣A)P(A)P(B)其中P(B)=∑i=1nP(B∣Ai)P(Ai)P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B)} \quad \text{其中} \quad P(B) = \sum_{i=1}^n P(B \mid A_i) P(A_i)