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CSCA STEM Formula Knowledge Base

All 58 core equations required for the China Scholastic Competency Assessment. Every entry includes symbolic derivations, examiner trap alerts, and CBT question mappings.

Mathematics

数学公式库

Sets, inequalities, differential calculus, analytic geometry, vectors, and combinatorics.

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Physics

物理公式库

Newtonian mechanics, momentum, thermodynamics, electromagnetism, optics, and quantum physics.

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Chemistry

化学定律与公式

Stoichiometry, gas laws, chemical equilibrium, electrochemistry, and reaction kinetics.

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Complete Formula Index (58)

Verified LaTeX & SymPy Proofs

Mathematics (30)

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Quadratic FormulaSets & Inequalities

一元二次方程求根公式

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

Union of SetsSets & Inequalities

集合的并集

$\[ A \cup B = \{ x \mid x \in A \text{ or } x \in B \} \]$

Intersection of SetsSets & Inequalities

集合的交集

$\[ A \cap B = \{ x \mid x \in A \text{ and } x \in B \} \]$

Complement of a SetSets & Inequalities

集合的补集

$\[ \complement_{U} A = \{ x \in U \mid x \notin A \} \]$

Rational Inequality EquivalenceSets & Inequalities

分式不等式的等价转化

$\[ \frac{ax + b}{cx + d} \le 0 \iff (ax + b)(cx + d) \le 0 \quad \text{and} \quad cx + d \ne 0 \]$

AM-GM InequalitySets & Inequalities

基本均值不等式

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

Change of Base FormulaFunctions

对数的换底公式

$\[ \log_a b = \frac{\log_c b}{\log_c a} \quad (a, b, c > 0, \, a, c \ne 1) \]$

Logarithmic PropertiesFunctions

对数的基本运算性质

$\[ \log_a (xy) = \log_a x + \log_a y \quad \text{and} \quad \log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y \]$

Trigonometric Pythagorean IdentityFunctions

同角三角关系（平方关系）

$\[ \sin^2 \theta + \cos^2 \theta = 1 \]$

Double-Angle Formula (Sine)Functions

正弦的二倍角公式

$\[ \sin 2\theta = 2 \sin \theta \cos \theta \]$

Double-Angle Formula (Cosine)Functions

余弦的二倍角公式

$\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \]$

Function Parity (Odd & Even)Functions

函数的奇偶性

$\[ \text{Even: } f(-x) = f(x), \quad \text{Odd: } f(-x) = -f(x) \]$

Arithmetic Sequence Nth TermSequences

等差数列通项公式

$\[ a_n = a_1 + (n - 1)d \]$

Arithmetic Sequence SumSequences

等差数列前n项和公式

$\[ S_n = \frac{n(a_1 + a_n)}{2} \]$

Geometric Sequence Nth TermSequences

等比数列通项公式

$\[ a_n = a_1 q^{n-1} \]$

Geometric Sequence SumSequences

等比数列前n项和公式

$\[ S_n = \frac{a_1(1 - q^n)}{1 - q} \quad (q \ne 1) \]$

Power Rule for DerivativesDerivatives

导数的幂函数求导法则

$\[ \frac{d}{dx} x^n = n x^{n-1} \]$

Tangent Line EquationDerivatives

曲线切线方程公式

$\[ y - y_0 = f'(x_0)(x - x_0) \]$

Distance FormulaPlane Analytic Geometry

两点间距离公式

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

Equation of a CirclePlane Analytic Geometry

圆的标准方程

$\[ (x - a)^2 + (y - b)^2 = R^2 \]$

Equation of an EllipsePlane Analytic Geometry

椭圆的标准方程

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

Equation of a HyperbolaPlane Analytic Geometry

双曲线的标准方程

$\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad (a > 0, \, b > 0) \]$

Equation of a ParabolaPlane Analytic Geometry

抛物线的标准方程

$\[ y^2 = 2px \quad (p > 0) \]$

Equation of a Line (Slope-Intercept & General Form)Plane Analytic Geometry

直线的方程（斜截式与一般式）

$\[ y = kx + b \quad \text{and} \quad Ax + By + C = 0 \]$

Vector Dot ProductVectors & Complex Numbers

平面向量的数量积（点积）

$\[ \mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y = |\mathbf{a}| |\mathbf{b}| \cos \theta \]$

Modulus of a Complex NumberVectors & Complex Numbers

复数的模

$\[ |z| = \sqrt{a^2 + b^2} \quad (z = a + bi) \]$

Volume and Surface Area of a SphereSolid Geometry

球的体积与表面积公式

$\[ V = \frac{4}{3}\pi R^3 \quad \text{and} \quad S = 4\pi R^2 \]$

Classical Probability FormulaProbability & Statistics

古典概型概率计算公式

$\[ P(A) = \frac{N(A)}{N} \]$

Variance and Standard DeviationProbability & Statistics

样本方差与标准差公式

$\[ s^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]$

Normal Distribution (3-Sigma Rule)Probability & Statistics

正态分布（3σ原则 / 68-95-99.7准则）

$\[ P(\mu - \sigma \le X \le \mu + \sigma) \approx 0.6826 \quad \text{and} \quad P(\mu - 2\sigma \le X \le \mu + 2\sigma) \approx 0.9544 \]$

Physics (17)

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Kinematics Equations under Constant AccelerationMechanics

匀变速直线运动规律与位移公式

$\[ v = v_0 + at, \quad s = v_0 t + \frac{1}{2} a t^2, \quad v^2 - v_0^2 = 2as \]$

Newton's Second Law of MotionMechanics

牛顿第二运动定律

$\[ \sum \mathbf{F} = m\mathbf{a} \]$

Momentum, Impulse & Law of Conservation of MomentumMechanics

动量定理与动量守恒定律

$\[ \mathbf{I} = \mathbf{F}\Delta t = \Delta \mathbf{p}, \quad m_1 \mathbf{v}_1 + m_2 \mathbf{v}_2 = m_1 \mathbf{v}_1' + m_2 \mathbf{v}_2' \]$

Work-Energy Theorem & Conservation of Mechanical EnergyMechanics

动能定理与机械能守恒定律

$\[ W_{\text{net}} = \Delta E_k = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2, \quad E_{k1} + E_{p1} = E_{k2} + E_{p2} \]$

Uniform Circular Motion & Law of Universal GravitationMechanics

匀速圆周运动与万有引力定律

$\[ F_c = m\frac{v^2}{r} = m\omega^2 r, \quad F_g = G\frac{Mm}{r^2} = m\frac{v^2}{r} \implies v = \sqrt{\frac{GM}{r}} \]$

Simple Harmonic Motion & Mechanical Wave SpeedMechanics

简谐运动周期与机械波传播公式

$\[ T = 2\pi \sqrt{\frac{L}{g}}, \quad v = \frac{\lambda}{T} = \lambda f \]$

Coulomb's Law & Electric Field StrengthElectromagnetism

库仑定律与电场强度公式

$\[ F = k_e \frac{|q_1 q_2|}{r^2}, \quad E = \frac{F}{q} = k_e \frac{Q}{r^2}, \quad U = Ed \]$

Ohm's Law & Closed Circuit EMF EquationElectromagnetism

闭合电路欧姆定律与电功率公式

$\[ I = \frac{E}{R + r}, \quad U = E - Ir, \quad P = UI = I^2 R \]$

Lorentz Force & Magnetic Ampere ForceElectromagnetism

安培力与洛伦兹力公式

$\[ F_{\text{Ampere}} = BIL\sin\theta, \quad F_{\text{Lorentz}} = qvB\sin\theta, \quad r = \frac{mv}{qB} \]$

Faraday's Law of Induction & Motional EMFElectromagnetism

法拉第电磁感应定律与动生电动势

$\[ E = n\frac{\Delta \Phi}{\Delta t}, \quad E = BLv\sin\theta \]$

AC Series RLC Circuit Impedance & PhasorElectromagnetism

交变电流RLC串联电路阻抗与相量公式

$\[ Z = \sqrt{R^2 + (X_L - X_C)^2}, \quad \tan\theta = \frac{X_L - X_C}{R} \]$

AC Active Power & Power Factor FormulaElectromagnetism

交流电路有功功率与功率因数公式

$\[ P = UI\cos\varphi, \quad \cos\varphi = \frac{R}{Z} = \frac{R}{\sqrt{R^2 + (X_L - X_C)^2}} \]$

First Law of Thermodynamics & Ideal Gas LawThermodynamics

热力学第一定律与理想气体状态方程

$\[ \Delta U = Q + W, \quad \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]$

Snell's Law of Refraction & Total Internal ReflectionOptics

光的折射定律与全反射临界角公式

$\[ n = \frac{c}{v} = \frac{\sin\theta_1}{\sin\theta_2}, \quad \sin C = \frac{1}{n} \]$

Young's Double-Slit Interference Fringe SpacingOptics

双缝干涉条纹间距公式

$\[ \Delta x = \frac{L}{d}\lambda \]$

Einstein's Photoelectric Effect & Photon EnergyModern Physics

光电效应方程与光子能量公式

$\[ E_{\text{photon}} = h\nu, \quad E_{k\max} = h\nu - W_0 = eU_c \]$

Bohr Hydrogen Energy Levels & Mass-Energy EquivalenceModern Physics

玻尔氢原子能级跃迁与质能方程

$\[ E_n = \frac{E_1}{n^2} \quad (E_1 = -13.6\text{ eV}), \quad h\nu = E_m - E_n, \quad \Delta E = \Delta m c^2 \]$

Chemistry (11)

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Amount of Substance, Mass & Particle Count RelationsBasic Chemical Concepts and Calculations

物质的量、微粒数与质量换算公式

$\[ n = \frac{m}{M} = \frac{N}{N_A} \]$

Gas Volume and Moles at STPBasic Chemical Concepts and Calculations

气体摩尔体积与标况状态公式

$\[ n = \frac{V}{V_m} \quad (V_m = 22.4\text{ L/mol at } 0^\circ\text{C, } 101.3\text{ kPa}) \]$

Molar Concentration of Solution & Dilution LawBasic Chemical Concepts and Calculations

物质的量浓度公式与稀释定律

$\[ c = \frac{n}{V}, \quad c_1 V_1 = c_2 V_2 \]$

Conversion between Mass Fraction and Molar ConcentrationBasic Chemical Concepts and Calculations

质量分数与物质的量浓度的换算公式

$\[ c = \frac{1000 \rho w}{M} \]$

Ideal Gas Law Equation of StateBasic Chemical Concepts and Calculations

理想气体状态方程（克拉珀龙方程）

$\[ PV = nRT = \frac{m}{M}RT \implies PM = \rho RT \]$

Chemical Reaction Rate & Rate Ratio LawChemical Theories and Laws

化学反应速率与化学计量数比值规律

$\[ v = \frac{\Delta c}{\Delta t}, \quad \frac{v(\text{A})}{a} = \frac{v(\text{B})}{b} = \frac{v(\text{C})}{c} \]$

Chemical Equilibrium Constant ExpressionChemical Theories and Laws

化学平衡常数表达式与转化率公式

$\[ K = \frac{[\text{C}]^c [\text{D}]^d}{[\text{A}]^a [\text{B}]^b}, \quad \alpha = \frac{\Delta n}{n_0} \times 100\% \]$

pH, pOH & Water Ion Product EquilibriumChemical Theories and Laws

pH定义式与水的离子积常数

$\[ \text{pH} = -\log_{10}[\text{H}^+], \quad K_w = [\text{H}^+][\text{OH}^-] = 1.0 \times 10^{-14} \text{ at } 25^\circ\text{C} \]$

Arrhenius Activation Energy EquationChemical Theories and Laws

阿伦尼乌斯公式与活化能

$\[ k = A e^{-\frac{E_a}{RT}} \implies \ln k = -\frac{E_a}{R}\frac{1}{T} + \ln A \]$

Gibbs Free Energy Spontaneity CriterionChemical Theories and Laws

吉布斯自由能判据与反应自发性

$\[ \Delta G = \Delta H - T\Delta S \quad (\Delta G < 0 \implies \text{Spontaneous}) \]$

Galvanic & Electrolytic Cell Electrode ReactionsChemical Theories and Laws

原电池与电解池电极反应规律

$\[ \text{Anode: Oxidation (loss of } e^-), \quad \text{Cathode: Reduction (gain of } e^-) \]$