全学科 LaTeX 公式大全

$$a+b=b+a$$

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

$$a\times b=b\times a$$

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

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

$$a÷b=\frac{a}{b}(b\ne 0)$$

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

$$a-b-c=a-(b+c)$$

$$a÷b=(a\times k)÷(b\times k)(b\ne 0,k\ne 0)$$

$$\frac{a}{b}=\frac{a\times k}{b\times k}(b\ne 0,k\ne 0)$$

$$\frac{a}{c}\pm \frac{b}{c}=\frac{a\pm b}{c}$$

$$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$$

$$\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}$$

$$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}$$

$$a\mathrm{\%}=\frac{a}{100}$$

$$\overline{x}=\frac{x_1+x_2+\cdots +x_n}{n}$$

$$s=vt$$

$$\text{总价}=\text{单价}\times \text{数量}$$

$$\text{工作总量}=\text{工作效率}\times \text{工作时间}$$

$$\text{利息}=\text{本金}\times \text{利率}\times \text{时间}$$

$$ C = 4a $$

$$ S = a^2 $$

$$ C = 2(a + b) $$

$$ S = ab $$

$$ S = \frac{1}{2}ah $$

$$ S = ah $$

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

$$ C = \pi d = 2\pi r $$

$$ S = \pi r^2 $$

$$ C = a + b + c $$

$$ \angle A + \angle B + \angle C = 180^\circ $$

$$ d = \sqrt{2}\,a $$

$$ d = 2r $$

$$ C = \pi r + 2r $$

$$ S = \pi(R^2 - r^2) $$

$$ V = a^3 $$

$$ V = abc = Sh $$

$$ V = Sh = \pi r^2 h $$

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

$$ S = 6a^2 $$

$$ S = 2(ab + bc + ac) $$

$$ S_{\text{侧}} = 2\pi rh $$

$$ S = 2\pi r^2 + 2\pi rh $$

$$ S_{\text{侧}} = \pi rl $$

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

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

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

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

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

$$ \begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases} $$

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

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

$$ \Delta = b^2 - 4ac $$

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

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

$$ (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca $$

$$ x^2 + (p+q)x + pq = (x+p)(x+q) $$

$$ ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + \frac{4ac - b^2}{4a} $$

$$ a^m \cdot a^n = a^{m+n} $$

$$ a^m \div a^n = a^{m-n} \quad (a \neq 0) $$

$$ (a^m)^n = a^{mn} $$

$$ (ab)^n = a^n b^n $$

$$ a^0 = 1, \quad a^{-n} = \frac{1}{a^n} \quad (a \neq 0) $$

$$ \sqrt{a^2} = |a| $$

$$ \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \quad (a \ge 0,\ b \ge 0) $$

$$ N = a \times 10^n \quad (1 \le |a| < 10) $$

$$ \frac{a}{x} = \frac{b}{y} \iff ay = bx \quad (xy \neq 0) $$

$$ y = kx \quad (k \neq 0) $$

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

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

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

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

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

$$ y = a(x - x_1)(x - x_2) $$

$$ x = -\frac{b}{2a} $$

$$ k = \frac{y_2 - y_1}{x_2 - x_1} \quad (x_1 \neq x_2) $$

$$ S = |k| $$

$$ y = f(x - h) + k $$

$$ \Delta = b^2 - 4ac > 0 \iff \text{与 } x \text{ 轴有两个交点} $$

$$ a^2 + b^2 = c^2 $$

$$ (n-2) \times 180^\circ $$

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

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

$$ \angle ACD = \angle A + \angle B $$

$$ \frac{(n-2) \times 180^\circ}{n} $$

$$ MN = \frac{1}{2}BC $$

$$ \frac{S_1}{S_2} = k^2 $$

$$ \angle ACB = \frac{1}{2}\angle AOB $$

$$ OC \perp AB \Rightarrow AD = DB $$

$$ PA \cdot PB = PC \cdot PD $$

$$ PT^2 = PA \cdot PB $$

$$ S = \sqrt{p(p-a)(p-b)(p-c)}, \quad p = \frac{a+b+c}{2} $$

$$ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} $$

$$ M\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) $$

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

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

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

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

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

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

$$ |2^A| = 2^n \quad (|A| = n) $$

$$ |A \cup B| = |A| + |B| - |A \cap B| $$

$$ \lnot(\forall x \in M,\ p(x)) \iff \exists x \in M,\ \lnot p(x) $$

$$ \lnot(\exists x \in M,\ p(x)) \iff \forall x \in M,\ \lnot p(x) $$

$$ (p \Rightarrow q) \iff (\lnot q \Rightarrow \lnot p) $$

$$ (p \Rightarrow q) \land (q \Rightarrow p) \iff p \Leftrightarrow q $$

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

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

$$ \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta $$

$$ \cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta $$

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

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

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

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

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

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

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

$$ \sin(\pi - \alpha) = \sin\alpha, \quad \cos(\pi - \alpha) = -\cos\alpha $$

$$ \sin\left(\frac{\pi}{2} - \alpha\right) = \cos\alpha, \quad \cos\left(\frac{\pi}{2} - \alpha\right) = \sin\alpha $$

$$ \tan 2\alpha = \frac{2\tan\alpha}{1 - \tan^2\alpha} $$

$$ \sin 3\alpha = 3\sin\alpha - 4\sin^3\alpha $$

$$ \cos 3\alpha = 4\cos^3\alpha - 3\cos\alpha $$

$$ \cos^2\frac{\alpha}{2} = \frac{1+\cos\alpha}{2} $$

$$ \sin^2\alpha = \frac{1 - \cos 2\alpha}{2}, \quad \cos^2\alpha = \frac{1 + \cos 2\alpha}{2} $$

$$ \sin\alpha = \frac{2t}{1+t^2}, \quad \cos\alpha = \frac{1-t^2}{1+t^2}, \quad t = \tan\frac{\alpha}{2} $$

$$ a\sin x + b\cos x = \sqrt{a^2+b^2}\,\sin(x + \varphi) $$

$$ \sin\alpha\cos\beta = \frac{1}{2}[\sin(\alpha+\beta) + \sin(\alpha-\beta)] $$

$$ \sin\alpha + \sin\beta = 2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2} $$

$$ \cos C = \frac{a^2 + b^2 - c^2}{2ab} $$

$$ \tan\alpha \cdot \cot\alpha = 1 $$

$$ a_n = a_1 + (n-1)d $$

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

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

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

$$ S_n = na_1 $$

$$ 2a_n = a_{n-1} + a_{n+1} $$

$$ a_n^2 = a_{n-1} \cdot a_{n+1} $$

$$ m + n = p + q \Rightarrow a_m + a_n = a_p + a_q $$

$$ a_n = \begin{cases} S_1, & n = 1 \\ S_n - S_{n-1}, & n \ge 2 \end{cases} $$

$$ 1 + 2 + \cdots + n = \frac{n(n+1)}{2} $$

$$ \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6} $$

$$ \sum_{k=1}^n k^3 = \left[\frac{n(n+1)}{2}\right]^2 $$

$$ S = \frac{a_1}{1-q} \quad (|q| < 1) $$

$$ \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} $$

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

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

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

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

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

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

$$ \mathbf{a} + \mathbf{b} = (x_1 + x_2,\ y_1 + y_2) $$

$$ \mathbf{a} - \mathbf{b} = (x_1 - x_2,\ y_1 - y_2) $$

$$ \lambda\mathbf{a} = (\lambda x,\ \lambda y) $$

$$ \mathrm{Prj}_{\mathbf{b}}\mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} $$

$$ \mathbf{a} = \lambda_1\mathbf{e}_1 + \lambda_2\mathbf{e}_2 $$

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

$$ \mathbf{a} \cdot \mathbf{b} = x_1x_2 + y_1y_2 + z_1z_2 $$

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

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

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

$$ \frac{a+b+c}{3} \ge \sqrt[3]{abc} \quad (a,b,c > 0) $$

$$ \frac{2}{\frac{1}{a}+\frac{1}{b}} \le \sqrt{ab} \le \frac{a+b}{2} \le \sqrt{\frac{a^2+b^2}{2}} $$

$$ ax^2 + bx + c > 0 \ (a>0,\ \Delta>0) \iff x < x_1 \ \text{或} \ x > x_2 $$

$$ |x| < a \iff -a < x < a \quad (a > 0) $$

$$ (1+x)^n \ge 1 + nx \quad (x > -1,\ n \ge 1) $$

$$ \sum_{i=1}^n a_i b_{n+1-i} \le \sum_{i=1}^n a_i b_{\sigma(i)} \le \sum_{i=1}^n a_i b_i $$

$$ y - y_0 = k(x - x_0) $$

$$ y = kx + b \quad (k \text{ 为斜率},\ b \text{ 为纵截距}) $$

$$ \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} $$

$$ \frac{x}{a} + \frac{y}{b} = 1 \quad (ab \neq 0) $$

$$ Ax + By + C = 0 \quad (A^2 + B^2 \neq 0) $$

$$ k = \tan\alpha \quad (\alpha \neq 90^\circ) $$

$$ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} $$

$$ d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} $$

$$ l_1 \parallel l_2 \iff A_1B_2 - A_2B_1 = 0 $$

$$ l_1 \perp l_2 \iff A_1A_2 + B_1B_2 = 0 $$

$$ \tan\theta = \left|\frac{k_2 - k_1}{1 + k_1k_2}\right| \quad (k_1k_2 \neq -1) $$

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

$$ x^2 + y^2 + Dx + Ey + F = 0 \quad (D^2 + E^2 - 4F > 0) $$

$$ \left(-\frac{D}{2},\ -\frac{E}{2}\right), \quad r = \frac{\sqrt{D^2 + E^2 - 4F}}{2} $$

$$ d < r \iff \text{相交}, \quad d = r \iff \text{相切}, \quad d > r \iff \text{相离} $$

$$ |AB| = 2\sqrt{r^2 - d^2} $$

$$ |PT| = \sqrt{(x_0-a)^2 + (y_0-b)^2 - r^2} $$

$$ |r_1 - r_2| < d < r_1 + r_2 $$

$$ V = Sh $$

$$ S_{\text{侧}} = ch \quad (c \text{ 为底面周长}) $$

$$ V = \frac{1}{3}Sh $$

$$ S_{\text{侧}} = \frac{1}{2}ch' \quad (h' \text{ 为斜高}) $$

$$ V = \frac{1}{3}h\left(S_1 + \sqrt{S_1S_2} + S_2\right) $$

$$ V = \frac{1}{3}\pi h(R^2 + Rr + r^2) $$

$$ V = \frac{4}{3}\pi r^3 $$

$$ S = 4\pi r^2 $$

$$ r = \sqrt{R^2 - d^2} $$

$$ V = \frac{\sqrt{2}}{12}a^3 $$

$$ V - E + F = 2 $$

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

$$ \sin\theta = \frac{|\mathbf{a} \cdot \mathbf{n}|}{|\mathbf{a}||\mathbf{n}|} $$

$$ \cos\theta = \frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{|\mathbf{n}_1||\mathbf{n}_2|} $$

$$ d = \frac{|\overrightarrow{AP} \cdot \mathbf{n}|}{|\mathbf{n}|} $$

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

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

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

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

$$ y^2 = 2px $$

$$ a^2 = b^2 + c^2 $$

$$ c^2 = a^2 + b^2 $$

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

$$ \frac{2b^2}{a} $$

$$ x = -\frac{p}{2} $$

$$ |PF| = x_0 + \frac{p}{2} $$

$$ C = 2a + 2c $$

$$ z = a + bi $$

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

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

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

$$ (a+bi) \pm (c+di) = (a \pm c) + (b \pm d)i $$

$$ \frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{c^2+d^2} $$

$$ \bar{z} = a - bi $$

$$ z\bar{z} = |z|^2 $$

$$ z = r(\cos\theta + i\sin\theta) $$

$$ [r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta) $$

$$ i^2 = -1, \quad i^3 = -i, \quad i^4 = 1 $$

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

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

$$ C_n^m = C_n^{n-m} $$

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

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

$$ T_{k+1} = C_n^k a^{n-k} b^k $$

$$ \sum_{k=0}^n C_n^k = 2^n $$

$$ C_{n+1}^m = C_n^m + C_n^{m-1} $$

$$ P(A \cup B) = P(A) + P(B) $$

$$ P(\bar{A}) = 1 - P(A) $$

$$ P(AB) = P(A)P(B) $$

$$ P(B) = \sum_{i=1}^n P(A_i)P(B|A_i) $$

$$ P_n(k) = C_n^k p^k (1-p)^{n-k} $$

$$ E(X) = np, \quad D(X) = np(1-p) $$

$$ P(X = k) = \frac{C_M^k C_{N-M}^{n-k}}{C_N^n} $$

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

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

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

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

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

$$ \lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e $$

$$ g(x) \le f(x) \le h(x),\ \lim g(x) = \lim h(x) = A \Rightarrow \lim f(x) = A $$

$$ \{a_n\} \text{ 单调有界} \Rightarrow \lim_{n \to \infty} a_n \text{ 存在} $$

$$ \sin x \sim x, \quad \tan x \sim x, \quad 1 - \cos x \sim \frac{x^2}{2} \quad (x \to 0) $$

$$ e^x - 1 \sim x, \quad \ln(1+x) \sim x \quad (x \to 0) $$

$$ \lim_{x \to x_0} f(x) = f(x_0) $$

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

$$ \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} $$

$$ \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n} $$

$$ \sum_{n=0}^{\infty} q^n = \frac{1}{1-q} \quad (|q| < 1) $$

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

$$ (uv)' = u'v + uv' $$

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

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

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

$$ (a^x)' = a^x \ln a $$

$$ (\log_a x)' = \frac{1}{x \ln a} $$

$$ (\sin x)' = \cos x, \quad (\cos x)' = -\sin x $$

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

$$ \frac{dy}{dx} = -\frac{F_x}{F_y} \quad (F_y \neq 0) $$

$$ \frac{dy}{dx} = \frac{\psi'(t)}{\varphi'(t)} \quad (\varphi'(t) \neq 0) $$

$$ (uv)^{(n)} = \sum_{k=0}^{n} C_n^k u^{(n-k)} v^{(k)} $$

$$ f(a) = f(b) \Rightarrow \exists \xi \in (a,b),\ f'(\xi) = 0 $$

$$ f(b) - f(a) = f'(\xi)(b - a), \quad \xi \in (a,b) $$

$$ \frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(\xi)}{g'(\xi)} $$

$$ dy = f'(x)\,dx $$

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

$$ \int u dv = uv - \int v du $$

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

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

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

$$ \int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) $$

$$ \int \frac{1}{x} dx = \ln|x| + C $$

$$ \int e^x dx = e^x + C $$

$$ \int \sin x \, dx = -\cos x + C $$

$$ \int \frac{dx}{1 + x^2} = \arctan x + C $$

$$ \int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx $$

$$ \int_{-a}^{a} f(x)dx = 0 \quad (f \text{ 为奇函数}) $$

$$ \int_a^b f(x)dx = f(\xi)(b - a), \quad \xi \in [a,b] $$

$$ \int_a^{+\infty} f(x)dx = \lim_{b \to +\infty} \int_a^b f(x)dx $$

$$ A = \int_a^b [f(x) - g(x)]dx $$

$$ S = 2\pi \int_a^b f(x)\sqrt{1 + [f'(x)]^2}\, dx $$

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

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

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

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

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

$$ \iiint_V \nabla \cdot \mathbf{F} \, dV = \iint_{\partial V} \mathbf{F} \cdot \mathbf{n} \, dS $$

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

$$ \frac{\partial f}{\partial \mathbf{l}} = f_x \cos\alpha + f_y \cos\beta $$

$$ \frac{\partial^2 z}{\partial x \partial y} = \frac{\partial^2 z}{\partial y \partial x} $$

$$ \nabla f = \lambda \nabla g $$

$$ \iint_D f(x,y)\,d\sigma = \int_a^b dx \int_{y_1(x)}^{y_2(x)} f(x,y)\,dy $$

$$ \iiint_\Omega f \, dV = \iiint_{\Omega'} f \, r \, dr \, d\theta \, dz $$

$$ \iiint_\Omega f \, dV = \iiint_{\Omega'} f \, r^2 \sin\varphi \, dr \, d\varphi \, d\theta $$

$$ J = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} x_u & x_v \\ y_u & y_v \end{vmatrix} $$

$$ \int_L f(x,y)\,ds $$

$$ du = u_x \, dx + u_y \, dy + u_z \, dz $$

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

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

$$ x_i = \frac{D_i}{D} $$

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

$$ |\lambda I - A| = 0 $$

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

$$ \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc $$

$$ |A| = \sum_{j=1}^n a_{ij} A_{ij} $$

$$ (AB)^T = B^T A^T $$

$$ (AB)^{-1} = B^{-1} A^{-1} $$

$$ \begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix} $$

$$ r(A + B) \le r(A) + r(B) $$

$$ B = P^{-1} A P $$

$$ A^T A = I $$

$$ f(\mathbf{x}) = \mathbf{x}^T A \mathbf{x} $$

$$ \sum_{i=1}^n \lambda_i = \mathrm{tr}(A), \quad \prod_{i=1}^n \lambda_i = |A| $$

$$ A\mathbf{x} = \mathbf{0} \ \text{有非零解} \iff |A| = 0 $$

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

$$ E(X) = \sum x_k p_k $$

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

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

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

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

$$ F(x) = P(X \le x) $$

$$ \int_{-\infty}^{+\infty} f(x)\,dx = 1 $$

$$ P(X = k) = C_n^k p^k (1-p)^{n-k} $$

$$ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} $$

$$ f(x) = \frac{1}{b-a}, \quad a \le x \le b $$

$$ f(x) = \lambda e^{-\lambda x}, \quad x > 0 $$

$$ E(aX + bY) = aE(X) + bE(Y) $$

$$ D(aX + b) = a^2 D(X) $$

$$ P(|X - \mu| \ge \varepsilon) \le \frac{\sigma^2}{\varepsilon^2} $$

$$ \lim_{n \to \infty} P\left(\frac{\sum_{i=1}^n X_i - n\mu}{\sigma\sqrt{n}} \le x\right) = \Phi(x) $$

$$ \rho_{XY} = \frac{Cov(X,Y)}{\sqrt{D(X)}\sqrt{D(Y)}} $$

$$ \bar{X} = \frac{1}{n}\sum_{i=1}^n X_i $$

$$ S^2 = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2 $$

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

$$ F = ma $$

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

$$ F = -kx $$

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

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

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

$$ v = v_0 + at $$

$$ v^2 - v_0^2 = 2ax $$

$$ \bar{v} = \frac{x}{t} = \frac{v_0 + v}{2} $$

$$ h = \frac{1}{2}gt^2 $$

$$ x = v_0 t, \quad y = \frac{1}{2}gt^2 $$

$$ v = \omega r = \frac{2\pi r}{T} $$

$$ a_n = \frac{v^2}{r} = \omega^2 r $$

$$ F_n = m\frac{v^2}{r} = m\omega^2 r $$

$$ Ft = mv_2 - mv_1 $$

$$ m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2' $$

$$ W = Fs\cos\theta $$

$$ P = \frac{W}{t} = Fv\cos\theta $$

$$ E_k = \frac{1}{2}mv^2 $$

$$ E_p = mgh $$

$$ E_p = \frac{1}{2}kx^2 $$

$$ F_{\text{浮}} = \rho g V_{\text{排}} $$

$$ p = \frac{F}{S} $$

$$ p = \rho g h $$

$$ v = \sqrt{\frac{GM}{R}} $$

$$ \frac{a^3}{T^2} = k $$

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

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

$$ I = \frac{U}{R} $$

$$ Q = I^2Rt $$

$$ f = qvB\sin\theta $$

$$ F = BIL\sin\theta $$

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

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

$$ I = \frac{q}{t} $$

$$ W = UIt $$

$$ P = UI = I^2 R = \frac{U^2}{R} $$

$$ R = R_1 + R_2 + \cdots + R_n $$

$$ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n} $$

$$ R = \rho\frac{L}{S} $$

$$ I = \frac{\mathcal{E}}{R + r} $$

$$ C = \frac{Q}{U} $$

$$ C = \frac{\varepsilon_r S}{4\pi k d} $$

$$ \Phi = BS $$

$$ \mathcal{E} = BLv $$

$$ W = qU $$

$$ U_{AB} = \frac{W_{AB}}{q} $$

$$ E_p = q\varphi $$

$$ U = \frac{U_m}{\sqrt{2}} $$

$$ c = \lambda f $$

$$ pV = nRT $$

$$ \Delta U = Q + W $$

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

$$ p_1 V_1 = p_2 V_2 $$

$$ \frac{p_1}{T_1} = \frac{p_2}{T_2} $$

$$ \frac{V_1}{T_1} = \frac{V_2}{T_2} $$

$$ \frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2} $$

$$ Q = cm\Delta t $$

$$ N = nN_A = \frac{m}{M}N_A $$

$$ U = \frac{i}{2}nRT $$

$$ \eta = 1 - \frac{T_2}{T_1} $$

$$ \Delta S \ge 0 $$

$$ T = t + 273.15\ \mathrm{K} $$

$$ E_k = h\nu - W_0 $$

$$ E = mc^2 $$

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

$$ \Delta t' = \gamma \Delta t $$

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

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

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

$$ \lambda = \frac{h}{p} = \frac{h}{mv} $$

$$ E_n = \frac{E_1}{n^2}, \quad E_1 = -13.6\ \mathrm{eV} $$

$$ h\nu = E_m - E_n $$

$$ N = N_0\left(\frac{1}{2}\right)^{\frac{t}{T}} $$

$$ \Delta E = \Delta m c^2 $$

$$ m = \frac{m_0}{\sqrt{1 - v^2/c^2}} $$

$$ E_k = (\gamma - 1)m_0 c^2 $$

$$ \Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta) $$