📚 Pre-U CAIE Geography: Formula and Theorem Quick Reference Handbook | Pre-U CAIE 地理：公式定理速查手册
This handbook summarises the essential formulae, laws, indices, and conceptual frameworks that underpin the Pre-U CAIE Geography syllabus. From population dynamics to fluvial hydrology and urban systems, each section presents definitions and equations using simple Unicode notation, helping you consolidate key quantitative skills for the examination.
本手册汇总了支撑 Pre-U CAIE 地理课程的核心公式、定律、指数和概念框架。从人口动态到河流水文再到城市体系，每一节都以简洁的 Unicode 符号呈现定义与方程，帮助你巩固考试所需的关键量化技能。
1. Demographic Transition Model (DTM) | 人口转变模型
The DTM describes the historical shift of birth and death rates from high to low levels in a country’s development. It consists of five stages, each with characteristic rates of natural increase (RNI = CBR – CDR). No single formula governs the model, but the relationship RNI = (Births – Deaths) / Total Population × 1000 is used to express per 1000 per year.
人口转变模型描述一个国家在发展过程中出生率和死亡率从高水平向低水平的历史性转变。它包括五个阶段，每个阶段都有特定的自然增长率 (RNI = CBR − CDR)。该模型没有单一公式，但常用 RNI = (出生人数 − 死亡人数) / 总人口 × 1000 来表示每年每千人的自然增长。
The crude birth rate (CBR) is CBR = (Live births / Mid-year population) × 1000, and the crude death rate (CDR) is CDR = (Deaths / Mid-year population) × 1000.
粗出生率 CBR = (活产数 / 年中人口) × 1000，粗死亡率 CDR = (死亡数 / 年中人口) × 1000。
The fertility replacement level is commonly taken as a total fertility rate (TFR) of about 2.1 children per woman to maintain population stability without migration.
生育更替水平通常以总和生育率 (TFR) 约 2.1 个孩子/妇女来计算，保证在没有迁移的情况下人口稳定。
2. Dependency Ratio and Age Structure | 抚养比与年龄结构
The total dependency ratio measures the pressure on the productive population: Dependency Ratio = [(Population aged 0–14 + Population aged 65+) / Population aged 15–64] × 100. The youth dependency ratio and old-age dependency ratio are calculated separately by substituting only the relevant age group in the numerator.
总抚养比衡量生产年龄人口所承受的压力：抚养比 = [(0–14 岁人口 + 65 岁以上人口) / 15–64 岁人口] × 100。少儿抚养比和老年抚养比只需将分子替换为相应年龄组分别计算。
The ageing index is another structural measure: Ageing Index = (Population aged 65+ / Population aged 0–14) × 100. Values above 100 indicate more elderly than children.
老龄化指数是另一个结构性指标：老龄化指数 = (65 岁以上人口 / 0–14 岁人口) × 100。超过 100 表示老年人口多于少儿。
3. Migration and the Gravity Model | 迁移与重力模型
The gravity model predicts the volume of movement between two places based on their population sizes and distance: Mᵢⱼ = k × (Pᵢ × Pⱼ) / dᵢⱼ², where Mᵢⱼ is migration flow from i to j, Pᵢ and Pⱼ are populations, dᵢⱼ is distance, and k is a constant. The exponent on distance may vary from 1 to 2 depending on friction of distance.
重力模型根据人口规模和距离预测两地之间的流动量：Mᵢⱼ = k × (Pᵢ × Pⱼ) / dᵢⱼ²，其中 Mᵢⱼ 是从 i 到 j 的迁移流量，Pᵢ 和 Pⱼ 为人口数，dᵢⱼ 为距离，k 为常数。距离的指数可根据距离摩擦在 1 到 2 之间变化。
Net migration is the difference between in-migrants and out-migrants: Net Migration = I – O. The net migration rate is (I – O) / Mid-year population × 1000.
净迁移是迁入者与迁出者之差：净迁移 = I − O。净迁移率为 (I − O) / 年中人口 × 1000。
4. Rank-Size Rule | 位序-规模法则
The rank-size rule states that the population of a city is inversely proportional to its rank in the urban hierarchy: Pₙ = P₁ / n, where Pₙ is the population of the nth largest city, P₁ is the population of the largest city, and n is the rank. In a perfectly rank-size distribution, the second city has half the population of the first, the third has one-third, and so on.
位序-规模法则指出，城市人口与其在城市等级体系中的位序成反比：Pₙ = P₁ / n，其中 Pₙ 是第 n 大城市的人口，P₁ 是最大城市的人口，n 为位序。在完全符合位序-规模的分布中，第二大城市人口为最大城市的一半，第三大为三分之一，以此类推。
The degree of primacy can be expressed as: Primacy Index = P₁ / P₂, where P₂ is the population of the second city. A value above 2 suggests a primate city pattern.
首位度可以用首位度指数表示为：首位度指数 = P₁ / P₂，其中 P₂ 为第二大城市的人口。大于 2 的值表明存在首位城市模式。
5. Nearest Neighbour Analysis | 最近邻分析
Nearest neighbour analysis quantifies the spatial distribution of points (e.g. settlements) as clustered, random or regular. The R statistic is: R = 2 × d̄ × √(N/A), where d̄ is the observed mean nearest neighbour distance, N is the number of points, and A is the area. The null hypothesis of randomness gives R = 1; R < 1 indicates clustering, R > 1 indicates regularity.
最近邻分析将点状地物（如聚落）的空间分布量化为集聚、随机或均匀分布。R 统计量为：R = 2 × d̄ × √(N/A)，其中 d̄ 为观测平均最近邻距离，N 为点数，A 为面积。随机分布的零假设下 R = 1；R < 1 表示集聚，R > 1 表示均匀。
The expected mean distance under randomness is E(d̄) = 1 / (2 × √(N/A)). The significance test uses Z = (d̄ − E(d̄)) / SE, but for CAIE, interpretation of R is usually sufficient.
随机分布下的期望平均距离为 E(d̄) = 1 / (2 × √(N/A))。显著性检验使用 Z = (d̄ − E(d̄)) / SE，但 CAIE 通常只需解读 R 值。
6. Hydrological Cycle and Water Balance | 水文循环与水平衡
The water balance equation for a drainage basin is: P = Q + E + ΔS, where P is precipitation, Q is runoff, E is evapotranspiration, and ΔS is the change in soil moisture and groundwater storage. This is a central concept in catchment hydrology.
流域的水平衡方程为：P = Q + E + ΔS，其中 P 为降水量，Q 为径流量，E 为蒸发蒸腾量，ΔS 为土壤水和地下水储存的变化量。这是流域水文学的核心概念。
Evapotranspiration can be estimated using the Thornthwaite formula, but for exam purposes, potential evapotranspiration (PE) is often compared with precipitation to assess soil moisture deficit: Deficit = PE − P when PE > P.
蒸发蒸腾量可以用 Thornthwaite 公式估算，但在考试中，通常将潜在蒸发蒸腾量 (PE) 与降水量比较，以评估土壤水分亏缺：亏缺 = PE − P（当 PE > P 时）。
7. Storm Hydrographs | 暴雨流量过程线
A storm hydrograph plots river discharge (Q) against time. Key indices include lag time (time between peak rainfall and peak discharge), rising limb gradient, and peak discharge. The runoff coefficient is the proportion of rainfall that becomes direct runoff: Runoff coefficient = Direct runoff / Total rainfall.
暴雨流量过程线描绘河流流量 (Q) 随时间的变化。关键指标包括滞时（雨峰与洪峰之间的时间）、涨水段坡度和洪峰流量。径流系数是降水转化为直接径流的比例：径流系数 = 直接径流 / 总降水量。
Discharge is calculated as Q = V × A × v̄, where A is cross-sectional area and v̄ is mean velocity. In practice, velocity is measured using current meters and the cross-section is surveyed.
流量计算公式为 Q = V × A × v̄，其中 A 为过水断面面积，v̄ 为平均流速。实际测量中，流速用流速仪测定，断面通过测量获得。
8. River Discharge and Manning’s Equation | 河流流量与曼宁公式
Manning’s equation estimates mean flow velocity in an open channel: v̄ = (R²/³ × S¹/²) / n, where R is the hydraulic radius (cross-sectional area/wetted perimeter), S is the water surface slope, and n is Manning’s roughness coefficient. It links channel form to discharge capacity.
曼宁公式估算明渠的平均流速：v̄ = (R²/³ × S¹/²) / n，其中 R 为水力半径（过水面积/湿周），S 为水面比降，n 为曼宁粗糙系数。它将河道形态与过水能力联系起来。
Darcy’s Law for groundwater flow through porous media is: Q = k × A × (Δh / L), where Q is discharge, k is hydraulic conductivity, A is cross-sectional area, and Δh / L is the hydraulic gradient. This is relevant to spring line settlements and aquifer studies.
达西定律描述地下水通过多孔介质的流动：Q = k × A × (Δh / L)，其中 Q 为流量，k 为渗透系数，A 为过水面积，Δh / L 为水力梯度。这在泉线聚落和含水层研究中有所应用。
9. Coastal Processes: Wave Energy and Longshore Drift | 海岸过程：波能与沿岸漂移
Wave power per unit crest length is approximately P = (ρg² / 64π) × T × H², where ρ is water density, g is gravity, T is wave period, and H is wave height. This helps explain erosion potential on high-energy coasts.
单位波峰线长度的波浪功率约为 P = (ρg² / 64π) × T × H²，其中 ρ 为水密度，g 为重力加速度，T 为波周期，H 为波高。这有助于解释高能海岸的侵蚀潜力。
Longshore drift volume Ql depends on wave approach angle and breaker height. A simplified expression is Ql ∝ Hb² × sin(2αb), where Hb is breaker height and αb is angle between breaking wave crest and shoreline.
沿岸漂移量 Ql 取决于波浪入射角与破波波高。简化表达式为 Ql ∝ Hb² × sin(2αb)，其中 Hb 为破波波高，αb 为破波波峰线与岸线的夹角。
Beach sediment size influences percolation and slope. The fall velocity of a particle (Stokes’ Law) is w = (Δρ g d²) / (18 μ), where Δρ is density difference, d is grain diameter, and μ is viscosity. This is useful for understanding beach morphology.
海滩沉积物粒度影响渗透与坡度。颗粒沉速（斯托克斯定律）为 w = (Δρ g d²) / (18 μ)，其中 Δρ 为密度差，d 为粒径，μ 为粘滞系数。这有助于理解海滩形态。
10. Climate: Lapse Rates | 气候：温度递减率
The environmental lapse rate (ELR) measures the decrease in temperature with altitude in the troposphere, averaging about 6.5 °C per 1000 m. The dry adiabatic lapse rate (DALR) is 9.8 °C km⁻¹, and the saturated adiabatic lapse rate (SALR) varies around 5 °C km⁻¹. These determine atmospheric stability.
环境气温递减率 (ELR) 衡量对流层气温随高度降低的速率，平均约为 6.5 °C/1000 m。干绝热递减率 (DALR) 为 9.8 °C km⁻¹，湿绝热递减率 (SALR) 约 5 °C km⁻¹。这些决定了大气稳定度。
Absolute humidity (AH) = mass of water vapour / volume of air. Relative humidity (RH) = (actual vapour pressure / saturation vapour pressure) × 100%. Saturation vapour pressure increases exponentially with temperature following the Clausius–Clapeyron relation.
绝对湿度 AH = 水汽质量 / 空气体积。相对湿度 RH = (实际水汽压 / 饱和水汽压) × 100%。饱和水汽压随温度呈指数增长，遵循克劳修斯–克拉佩龙关系。
11. Urban Models: Central Place Theory | 城市模型：中心地理论
Christaller’s central place theory predicts the spacing, size and number of settlements using the service range (upper limit) and threshold population. The market area of a central place is circular but transforms into hexagons to avoid overlaps. Key ratios: K=3 (marketing principle), K=4 (transport principle), K=7 (administrative principle), where K is the number of lower-order places served by one higher-order centre.
克里斯泰勒的中心地理论使用服务范围上限和门槛人口预测聚落的间距、规模与数量。中心地的市场区为圆形，但为避免重叠而转化为六边形。关键比率：K=3（市场原则）、K=4（交通原则）、K=7（行政原则），其中 K 表示一个高级中心所服务的低级地方数。
The threshold population (T) is the minimum number of people needed to support a service. It is calculated as T = Total fixed cost / (Price − Variable cost per unit). The range of a good is the maximum distance a consumer will travel; this can be expressed using the gravity model or simply as R = dmax.
门槛人口 (T) 是支撑某服务所需的最低人口数。计算公式为 T = 总固定成本 / (价格 − 单位可变成本)。商品的服务范围是消费者愿意出行的最大距离，可用重力模型表达或简写为 R = dmax。
12. Economic and Development Indices | 经济与发展指数
Gross Domestic Product (GDP) per capita is GDP / Total population. Purchasing power parity (PPP) adjustment accounts for price differences between countries. The Human Development Index (HDI) combines life expectancy, education (mean and expected years), and GNI per capita. HDI = (Ihealth × Ieducation × Iincome)^(1/3), where each index is normalised between 0 and 1.
人均国内生产总值 = GDP / 总人口。购买力平价 (PPP) 调整考虑了各国间的价格差异。人类发展指数 (HDI) 综合了预期寿命、受教育程度（平均与预期教育年限）和人均国民总收入。HDI = (Ihealth × Ieducation × Iincome)^(1/3)，每个指数归一化到 0 至 1。
The Gini coefficient measures inequality on a scale from 0 (perfect equality) to 1 (perfect inequality). It is derived from the Lorenz curve: G = A / (A + B), where A is area between line of equality and Lorenz curve, B is area under Lorenz curve.
基尼系数衡量不平等程度，取值范围 0（完全平等）至 1（完全不平等）。它由洛伦兹曲线导出：G = A / (A + B)，其中 A 为绝对平等线与洛伦兹曲线之间的面积，B 为洛伦兹曲线下方的面积。
The location quotient (LQ) identifies spatial concentration of an industry: LQ = (Regional employment in sector / Total regional employment) / (National employment in sector / Total national employment). LQ > 1 indicates local specialisation.
区位商 (LQ) 用于识别某一产业的空间集聚度：LQ = (区域某行业就业人数 / 区域总就业) / (全国该行业就业人数 / 全国总就业)。LQ > 1 表示地方专业化。
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