📚 CIE AS Geography Formula & Theorem Quick Reference Handbook | CIE AS 地理公式定理速查手册
This handbook provides essential formulas, equations and key theoretical principles required for the Cambridge International AS Level Geography syllabus. It covers core topics in physical and human geography, offering a quick-reference tool for revision and exam preparation. Each entry presents the formula or theorem with a concise explanation to reinforce understanding of quantitative and conceptual aspects of the course.
本手册汇总了剑桥国际 AS 地理课程必备的公式、方程和核心理论原理，涵盖自然地理与人文地理的核心主题，为复习与备考提供速查工具。每个条目均给出公式或定理并附简要解释，以加强对课程中定量与概念性内容的理解。
1. Fluvial Discharge and Hydraulic Geometry | 河流流量与水力学几何
Discharge (Q) is the volume of water flowing through a channel cross‑section per unit time. It is calculated using the continuity equation:
流量 (Q) 是单位时间内通过河道横截面的水体体积，由连续性方程计算：
Q = A × V
where A is the cross‑sectional area of the channel (m²) and V is the mean flow velocity (m s⁻¹). Field measurements of width, depth and velocity allow accurate discharge calculation. This fundamental relationship underpins river regime analysis and flood modelling.
其中 A 为河道横截面积（m²），V 为平均流速（m s⁻¹）。通过野外测量河宽、水深和流速即可准确计算流量。这一基本关系是河流情势分析与洪水模拟的基础。
Hydraulic radius (R) measures the efficiency of a channel cross‑section:
水力半径 (R) 衡量河道横截面的输水效率：
R = A ÷ P
where P is the wetted perimeter. A larger hydraulic radius indicates lower frictional resistance relative to the cross‑sectional area, promoting faster flow. It is an essential variable in the Manning and Chezy equations used in advanced hydraulics.
其中 P 为湿周。水力半径越大，相对于横截面积的摩擦阻力越小，流速越快。它是高级水力学中曼宁公式和谢才公式的关键变量。
2. Catchment Water Balance and Runoff Formulas | 流域水量平衡与径流公式
The water balance equation expresses the hydrological function of a drainage basin over a given period:
水量平衡方程 表达了给定时段内流域的水文收支关系：
P = Q + E + ΔS
where P = precipitation, Q = runoff (streamflow), E = evapotranspiration, and ΔS = change in storage (soil moisture, groundwater, surface detention). This equation is crucial for understanding river regimes and water resource availability.
其中 P 为降水量，Q 为径流量（河道流量），E 为蒸散发量，ΔS 为储水量变化（土壤水、地下水、地表滞水）。该方程对于理解河流情势和水资源可用性至关重要。
The runoff coefficient (C) relates total runoff to total rainfall:
径流系数 (C) 将总径流量与总降雨量联系起来：
C = Q / P
Values approaching 1 indicate impermeable surfaces and rapid storm response; lower values reflect higher infiltration and storage. In urban design, the Rational Method estimates peak discharge: Qp = C × I × A, where I is rainfall intensity and A is catchment area.
数值接近 1 表示地表不透水且暴雨响应快；较低值反映渗透强、储水多。在城市设计中，推理公式法 估算峰值流量：Qp = C × I × A，其中 I 为降雨强度，A 为流域面积。
3. Population Change Equations | 人口变动方程
The Crude Birth Rate (CBR) and Crude Death Rate (CDR) are foundational demographic indicators:
粗出生率 (CBR) 和 粗死亡率 (CDR) 是基础的人口统计指标：
CBR = (B / P) × 1000
CDR = (D / P) × 1000
where B = live births in a year, D = deaths in a year, and P = total mid‑year population. Rates are expressed per 1000 people per year.
其中 B 为一年内活产人数，D 为一年内死亡人数，P 为年中总人口。比率以每年每千人为单位表示。
The Rate of Natural Increase (RNI) shows population change excluding migration:
自然增长率 (RNI) 显示不含迁移的人口变动：
RNI = CBR − CDR
It is often converted to a percentage: RNI (%) = (CBR − CDR) / 10. The doubling time estimates the years needed for a population to double at the current growth rate, using the Rule of 70:
通常转化为百分比：RNI (%) = (CBR − CDR) / 10。人口倍增时间 估算在现有增长率下人口翻番所需年数，运用“70 法则”：
Doubling Time (years) = 70 / Growth Rate (%)
A growth rate of 2% yields a doubling time of 35 years. This simple calculation helps highlight demographic momentum.
增长率为 2% 时倍增时间为 35 年。这一简易计算有助于凸显人口惯性。
4. Migration Measures | 迁移指标
Net migration rate (NMR) captures the balance between immigration (I) and emigration (E):
净迁移率 (NMR) 反映迁入 (I) 与迁出 (E) 的差额：
NMR = ((I − E) / P) × 1000
A positive NMR indicates net in‑migration, adding to population growth. The total population growth rate is then:
正净迁移率表示净迁入，促进人口增长。于是总人口增长率为：
Population Growth Rate (%) = (RNI + NMR) / 10
These measures are essential for analysing population dynamics in source and destination regions, and for evaluating the demographic impacts of migration policies.
这些指标对于分析迁出地与迁入地的人口动态以及评价迁移政策的人口影响必不可少。
5. Urbanisation and Urban Indicators | 城市化与城市指标
The level of urbanisation quantifies the proportion of a population living in urban areas:
城市化水平 量化居住在城市地区的人口比例：
Urbanisation Level (%) = (Urban Population / Total Population) × 100
This figure is used to compare countries and regions, and to track changes over time. The rate of urban population growth considers both natural increase and net rural‑to‑urban migration.
该指标用以比较国家与地区并追踪时间变化。城市人口增长率 则同时考虑自然增长和城乡净迁移。
Urban population density is simply:
城市人口密度 定义为：
Density = Urban Population / Urban Area (km²)
High densities often characterise compact cities and can relate to land‑use intensity, infrastructure demand and quality of life assessments.
高密度常为紧凑城市的特征，并与土地利用强度、基础设施需求和生活质量评估相关。
6. Rank‑Size Rule and Primacy | 位序规模法则与首位度
The rank‑size rule (Zipf’s law) describes a regular pattern in the size distribution of cities within a country:
位序规模法则（齐普夫定律）描述了一国内部城市规模分布的规律模式：
Pn = P1 / n
where Pn is the population of the n‑th largest city and P1 is the population of the largest city. According to the rule, the second city is half the size of the largest, the third is one‑third, and so on. Deviations suggest primacy or a binary distribution.
其中 Pn 是第 n 大城市的人口，P1 是最大城市的人口。按法则，第二大城市人口为最大城市的一半，第三大城市为三分之一，以此类推。偏离该法则表明首位分布或双核分布。
The primacy index measures the dominance of the largest city:
首位度 衡量最大城市的支配程度：
Two‑City Index = P1 / P2
A value significantly above 2 suggests a primate city pattern. A four‑city index (P1 / (P2 + P3 + P4)) and an eleven‑city index can also be used for more robust assessments.
比值显著大于 2 表明存在首位城市格局。也可采用四城市指数 (P1 / (P2 + P3 + P4)) 和十一城市指数以获得更可靠的评估。
7. Central Place Theory Principles | 中心地理论原理
Christaller’s Central Place Theory explains the size, spacing and hierarchy of settlements based on the provision of goods and services. The spatial organisation is governed by three k‑principles:
克里斯塔勒的中心地理论根据商品和服务的供给解释聚落的规模、间距与等级体系。空间组织由三个 k 原则 支配：
Marketing principle (k = 3): a central place serves a hexagonal market area with three times its own population; settlements are arranged to minimise travel for consumers.
市场原则 (k = 3)：一个中心地服务于六边形市场区域，其服务人口为自身人口的三倍；聚落布局旨在最小化消费者的出行。
Transport principle (k = 4): central places align along transport routes, resulting in a hierarchy where each higher‑order centre dominates four times the population of the next level.
交通原则 (k = 4)：中心地沿交通线分布，形成的等级体系中每个高级中心地支配四倍于下一等级的人口。
Administrative principle (k = 7): the market area is nested entirely within the higher‑order territory, emphasising political or administrative control; the factor is 7.
行政原则 (k = 7)：市场区完全嵌套在高级政区之内，强调政治或行政控制；倍数为 7。
These k‑values define the ratio of the number of lower‑order centres served by one higher‑order centre, including the higher‑order centre itself.
这些 k 值定义了一个高级中心地所服务的低级中心地数量之比（含自身）。
8. Gravity Model and Spatial Interaction | 引力模型与空间相互作用
The gravity model predicts the interaction (e.g. migration, trade, commuting) between two places based on their size and distance:
引力模型 根据规模与距离预测两地间的相互作用（如迁移、贸易、通勤）：
Iij = (Pi × Pj) / dij²
where Iij is the interaction between places i and j, Pi and Pj are their populations (or economic mass), and dij is the distance between them. The squared distance emphasises the friction of distance: interaction declines sharply with increasing distance. In applied studies, distance may be replaced by travel time or cost, and the exponent can be calibrated empirically.
其中 Iij 为地点 i 与 j 之间的相互作用，Pi 和 Pj 为其人口（或经济规模），dij 为两地距离。距离平方强调了距离衰减效应：相互作用随距离增加急剧减弱。在应用研究中，距离可替换为旅行时间或成本，指数也可通过经验数据校准。
This model underpins central place theory and helps explain retail catchment areas and commuter flows.
该模型支撑了中心地理论，有助于解释零售业服务范围和通勤流。
9. Universal Soil Loss Equation (USLE) | 通用土壤流失方程
The USLE estimates long‑term average annual soil loss from sheet and rill erosion:
通用土壤流失方程 (USLE) 估算坡面与细沟侵蚀的长期年均土壤流失量：
A = R × K × LS × C × P
where:
其中：
- A = average annual soil loss (t ha⁻¹ yr⁻¹)
- R = rainfall erosivity factor
- K = soil erodibility factor
- LS = slope length‑steepness factor
- C = cover‑management factor
- P = support practice factor (e.g. contour farming, terracing)
This multiplicative model allows planners to assess erosion risk and compare the effectiveness of conservation practices. Although developed for agricultural land, the framework is widely applied in physical geography to understand landscape denudation.
这一乘法模型使规划者可评估侵蚀风险并比较保护措施的有效性。虽然最初为农地开发，但其框架在自然地理解释地表剥蚀中得到广泛应用。
10. Weathering Chemical Reactions | 风化化学反应
Chemical weathering transforms primary minerals into secondary minerals and solutes. Key reactions include:
化学风化将原生矿物转化为次生矿物和溶解物质。关键反应包括：
Carbonation of limestone, critical in karst landscapes:
CaCO₃ + H₂O + CO₂ → Ca(HCO₃)₂
Calcium carbonate reacts with water and dissolved carbon dioxide to form soluble calcium bicarbonate, which can be removed in solution, enlarging joints and caverns.
碳酸钙与水及溶解二氧化碳反应，生成可溶的碳酸氢钙，通过溶液被带走，扩大裂隙和洞穴。
Hydrolysis of feldspar, dominant in granite weathering:
2KAlSi₃O₈ + 2H₂O + CO₂ → Al₂Si₂O₅(OH)₄ + 4SiO₂ + K₂CO₃
Potassium feldspar reacts with water and carbon dioxide to form kaolinite (clay), dissolved silica and potassium carbonate. This process weakens granite and produces clay‑rich regolith.
钾长石与水及二氧化碳反应生成高岭石（黏土）、溶解二氧化硅和碳酸钾。此过程削弱花岗岩并形成富含黏土的风化层。
Oxidation in iron‑bearing minerals: e.g. conversion of ferrous iron to ferric oxides, giving reddish‑brown stains in weathered rock profiles.
含铁矿物的氧化作用：例如亚铁转变为氧化铁，在风化岩剖面上形成红褐色锈斑。
11. Climate Data Calculations | 气候数据计算
Basic climatic analysis requires computing key descriptive statistics from temperature and precipitation data:
基础气候分析需要根据气温和降水数据计算描述性统计量：
Mean Daily Temperature = (Tmax + Tmin) / 2
日平均气温 = (最高气温 + 最低气温) / 2
Diurnal Temperature Range = Tmax − Tmin
日较差 = 最高气温 − 最低气温
Mean Monthly Temperature = sum of mean daily temperatures / number of days in month
月平均气温 = 全月日平均气温总和 / 该月天数
Annual Temperature Range = warmest monthly mean − coldest monthly mean
年较差 = 最热月平均气温 − 最冷月平均气温
Total Annual Precipitation is the sum of monthly precipitation. These quantities are used to construct climate graphs and calculate aridity indices such as the De Martonne index: I = P / (T + 10), where P is annual precipitation (mm) and T is mean annual temperature (°C).
年总降水量为各月降水量之和。这些数据用于绘制气候图并计算干燥指数，如德马通指数：I = P / (T + 10)，其中 P 为年降水量 (mm)，T 为年均温 (°C)。
12. Population Structure Indicators | 人口结构指标
Age–sex pyramids are complemented by several dependency and support ratios:
年龄性别金字塔可辅以若干抚养比与扶养比：
Total Dependency Ratio = ((Population aged 0–14 + Population aged 65+) / Population aged 15–64) × 100
总抚养比 = ((0–14 岁人口 + 65 岁及以上人口) / 15–64 岁人口) × 100
Child Dependency Ratio = (Population aged 0–14 / Population aged 15–64) × 100
少儿抚养比 = (0–14 岁人口 / 15–64 岁人口) × 100
Old‑age Dependency Ratio = (Population aged 65+ / Population aged 15–64) × 100
老年抚养比 = (65 岁及以上人口 / 15–64 岁人口) × 100
Sex Ratio = (Number of males / Number of females) × 100
性别比 = (男性人数 / 女性人数) × 100
These indicators inform social and economic planning, highlighting pressures on education, healthcare and pension systems.
这些指标为社会与经济规划提供依据，凸显教育、医疗和养老金系统所面临的压力。
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