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a + b = b + a

(a + b) + c = a + (b + c)

a \times b = b \times a

(a \times b) \times c = a \times (b \times c)

a \times (b + c) = a \times b + a \times c

a \div b = \frac{a}{b} \quad (b \neq 0)

a + b \times c = a + (b \times c)

C = 4a

S = a^2

C = 2(a + b)

S = ab

S = \frac{1}{2}ah

S = ah

S = \frac{(a+b)h}{2}

C = \pi d = 2\pi r

S = \pi r^2

V = a^3

V = abc = Sh

V = Sh = \pi r^2 h

V = \frac{1}{3}Sh = \frac{1}{3}\pi r^2 h

a^2 - b^2 = (a+b)(a-b)

(a \pm b)^2 = a^2 \pm 2ab + b^2

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

ax + b = 0 \Rightarrow x = -\frac{b}{a}

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

x_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}

\Delta = b^2 - 4ac

y = kx \quad (k \neq 0)

y = \frac{k}{x} \quad (k \neq 0)

y = kx + b \quad (k \neq 0)

y = ax^2 + bx + c \quad (a \neq 0)

y = a(x-h)^2 + k

\left(-\frac{b}{2a}, \frac{4ac-b^2}{4a}\right)

a^2 + b^2 = c^2

(n-2) \times 180^\circ

l = \frac{n\pi r}{180}

S = \frac{n\pi r^2}{360} = \frac{1}{2}lr

(x-a)^2 + (y-b)^2 = r^2

A \subseteq B \iff \forall x(x \in A \implies x \in B)

A \cap B = \{x | x \in A \land x \in B\}

A \cup B = \{x | x \in A \lor x \in B\}

\complement_U A = \{x | x \in U \land x \notin A\}

\complement_U(A \cup B) = (\complement_U A) \cap (\complement_U B)

\sin^2\alpha + \cos^2\alpha = 1

\tan\alpha = \frac{\sin\alpha}{\cos\alpha}

\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

\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}

\sin 2\alpha = 2\sin\alpha\cos\alpha

\cos 2\alpha = \cos^2\alpha - \sin^2\alpha = 2\cos^2\alpha - 1

\sin^2\frac{\alpha}{2} = \frac{1-\cos\alpha}{2}

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

c^2 = a^2 + b^2 - 2ab\cos C

S = \frac{1}{2}ab\sin C

a_n = a_1 + (n-1)d

S_n = \frac{n(a_1+a_n)}{2} = na_1 + \frac{n(n-1)}{2}d

a_n = a_1 q^{n-1} \quad (q \neq 0)

S_n = \frac{a_1(1-q^n)}{1-q}

S_n = na_1

|\mathbf{a}| = \sqrt{x^2 + y^2}

\mathbf{a} \cdot \mathbf{b} = x_1x_2 + y_1y_2

\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos\theta

\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}

\mathbf{a} \perp \mathbf{b} \iff \mathbf{a} \cdot \mathbf{b} = 0

\mathbf{a} \parallel \mathbf{b} \iff x_1y_2 - x_2y_1 = 0

\frac{a+b}{2} \ge \sqrt{ab} \quad (a,b > 0)

||a| - |b|| \le |a+b| \le |a| + |b|

(a^2+b^2)(c^2+d^2) \ge (ac+bd)^2

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a>b>0)

e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}}

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

y = \pm \frac{b}{a}x

y^2 = 2px

z = a + bi

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

e^{ix} = \cos x + i\sin x

(a+bi)(c+di) = (ac-bd) + (ad+bc)i

A_n^m = \frac{n!}{(n-m)!}

C_n^m = \frac{n!}{m!(n-m)!}

C_n^m = C_n^{n-m}

(a+b)^n = \sum_{k=0}^n C_n^k a^{n-k}b^k

P(B|A) = \frac{P(AB)}{P(A)}

\forall \epsilon > 0, \exists \delta > 0, 0 < |x - x_0| < \delta \implies |f(x) - A| < \epsilon

\lim_{x \to 0} \frac{\sin x}{x} = 1

\lim_{x \to \infty} (1 + \frac{1}{x})^x = e

\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}

f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots

f'(x) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}

(uv)' = u'v + uv'

(\frac{u}{v})' = \frac{u'v - uv'}{v^2}

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

\int_a^b f(x)dx = F(b) - F(a)

\int u dv = uv - \int v du

\int f(g(x))g'(x)dx = \int f(u)du

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

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

\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h}

dz = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y}dy

\nabla f = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z})

\iint_D f(x,y)dxdy = \iint_{D'} f(r\cos\theta, r\sin\theta)r dr d\theta

\oint_L Pdx + Qdy = \iint_D (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})dxdy

\iiint_V \nabla \cdot \mathbf{F} dV = \oiint_{\partial V} \mathbf{F} \cdot \mathbf{n} dS

\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r}

(AB)_{ij} = \sum_{k} A_{ik}B_{kj}

A^{-1} = \frac{1}{|A|} A^*

x_i = \frac{D_i}{D}

A\mathbf{x} = \lambda \mathbf{x}

|\lambda I - A| = 0

\beta_k = \alpha_k - \sum_{i=1}^{k-1} \frac{\langle \alpha_k, \beta_i \rangle}{\langle \beta_i, \beta_i \rangle} \beta_i

P(A|B) = \frac{P(B|A)P(A)}{P(B)}

E(X) = \sum x_k p_k

E(X) = \int_{-\infty}^{+\infty} x f(x) dx

D(X) = E(X^2) - [E(X)]^2

f(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Cov(X,Y) = E[(X-E(X))(Y-E(Y))]

x = v_0t + \frac{1}{2}at^2

F = ma

F = G\frac{Mm}{r^2}

F = -kx

W = \Delta E_k = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2

\frac{1}{2}mv_1^2 + mgh_1 = \frac{1}{2}mv_2^2 + mgh_2

T = 2\pi \sqrt{\frac{L}{g}}

F = k\frac{q_1q_2}{r^2}

E = \frac{F}{q} = k\frac{Q}{r^2}

I = \frac{U}{R}

Q = I^2Rt

f = qvB\sin\theta

F = BIL\sin\theta

E = n\frac{\Delta \Phi}{\Delta t}

\frac{U_1}{U_2} = \frac{n_1}{n_2}

pV = nRT

\Delta U = Q + W

\eta = \frac{W}{Q_1} = 1 - \frac{Q_2}{Q_1}

E_k = h\nu - W_0

E = mc^2

\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

\Delta t' = \gamma \Delta t

L' = \frac{L}{\gamma}

i\hbar\frac{\partial}{\partial t}\Psi = \hat{H}\Psi

\Delta x \Delta p \ge \frac{\hbar}{2}

\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}

\nabla \cdot \mathbf{B} = 0

\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}

\nabla \times \mathbf{B} = \mu_0(\mathbf{J} + \epsilon_0\frac{\partial \mathbf{E}}{\partial t})

\ce{6CO2 + 6H2O ->[\text{光照}][\text{叶绿素}] C6H12O6 + 6O2}

\ce{CaCO3 ->[\text{高温}] CaO + CO2\uparrow}

\ce{Cu + 4HNO3(\text{浓}) -> Cu(NO3)2 + 2NO2\uparrow + 2H2O}

\ce{4NH3 + 5O2 ->[\text{催化剂}][\Delta] 4NO + 6H2O}

\ce{CH3COOH + C2H5OH <=>[加热][浓硫酸] CH3COOC2H5 + H2O}

n = \frac{m}{M} = \frac{N}{N_A} = \frac{V}{V_m}

\text{pH} = -\lg c(\ce{H+})

K_w = c(\ce{H+}) \cdot c(\ce{OH-}) = 10^{-14} \quad (25^\circ\text{C})

\ce{[Ar]3d^6 4s^2}

\Delta G = \Delta H - T\Delta S

k = Ae^{-\frac{E_a}{RT}}

E = E^\theta - \frac{RT}{nF}\ln Q

2d\sin\theta = n\lambda