📚 Year 12 CIE Geography: Formula & Theorem Quick Reference Handbook | CIE 地理公式定理速查手册
This quick reference handbook presents essential formulae, equations, and key theorems examined in the Cambridge International AS Geography (Year 12) syllabus. From hydrological balances to population dynamics, these quantitative tools enable you to analyse spatial patterns and geographical processes with precision.
本速查手册汇总了剑桥国际 AS 地理（Year 12）课程中考查的核心公式、方程与重要定理。从水文平衡到人口动态，这些定量工具帮助你精准分析空间格局与地理过程。
1. Drainage Basin Water Balance | 流域水分平衡方程
cipitation, Q is runoff (streamflow), E is evapotranspiration, and ΔS is change in soil and groundwater storage over a given time period. This equation underscores the hydrological cycle's continuity.
联系流域降水、径流、蒸散发和储水量变化的基本方程为：P = Q + E + ΔS，式中 P 为降水量，Q 为径流量（河道流量），E 为蒸散发量，ΔS 为特定时段内土壤与地下水储量的变化量。该方程体现了水文循环的连续性。
P = Q + E + ΔS
he equation can be rearranged to estimate missing components, for instance ΔS = P - Q - E.
单位常为时段内的毫米数，可变形以估算缺失成分，如 ΔS = P - Q - E。
2. River Discharge Formula | 河流流量公式
Discharge (Q) is the volume of water passing a given cross‑section per unit time, calculated as Q = A × V, where A is the cross‑ ow velocity (m s⁻¹). Discharge is expressed in cubic metres per second (m³ s⁻¹) or cumecs.
流量 (Q) 是单位时间内通过某一断面的水体体积，计算式为 Q = A × V，其中 A 为河道横截面积（m²），V 为平均流速（m s⁻¹）。流量以立方米每秒（m³ s⁻¹）或“秒立方”为计量单位。
Q = A × V
Velocity can be measured with a flow meter; cross‑sectional area is calculated from channel width multiplied by mean depth.
流速可用流速仪测量；横截面积由河面宽度乘平均水深求得。
3. Stream Gradient (Slope) | 河流纵比降（坡度）
fall in elevation (m) and L is the length of the channel reach (m). It is often given as a dimensionless ratio (e.g., 0.002) or in metres per kilometre.
河流纵比降是河段高差与其水平距离之比，表示为 S = H / L，其中 H 为高程落差（m），L 为河段长度（m）。常用无因次比（如 0.002）或每公里下降米数表示。
S = H / L
Gradient influences flow velocity, sediment transport capacity, and the development of channel morphology.
纵比降影响流速、输沙能力及河床形态的发育。
4. Sinuosity Ratio | 蜿蜒度指数
= Lc / Lv, where Lc is the actual channel length between two points and Lv is the straight‑line valley distance. A ratio greater than 1.5 typically indicates a meandering channel.
蜿蜒度衡量河道弯曲程度，定义为 Sinuosity = Lc / Lv，式中 Lc 为两点间实际河道长度，Lv 为河谷直线距离。比值大于 1.5 通常指示曲流河道。
Sinuosity = Lc / Lv
Values close to 1 indicate a straight channel; values above 2.0 imply high sinuosity with pronounced meanders.
比值接近 1 为顺直河道；大于 2.0 为高蜿蜒度，发育典型河曲。
5. Drainage Density | 河网密度
here ΣL is the sum of channel lengths (km) and A is the drainage basin area (km²). High drainage density indicates efficient runoff and rapid flood response.
河网密度 (Dd) 为流域内所有河道总长度与流域面积之比：Dd = ΣL / A，式中 ΣL 为河道总长度（km），A 为流域面积（km²）。高河网密度表明产汇流高效，洪水响应迅速。
Dd = ΣL / A
Typical values range from very low (2–4 km⁻¹) in permeable catchment areas to very high ( >12 km⁻¹) on impermeable surfaces with sparse vegetation.
典型值在透水性好的流域极低（2–4 km⁻¹），在植被稀少的不透水地表极高（>12 km⁻¹）。
6. Stream Order and Bifurcation Ratio | 河流分级与分叉比
Strahler' ifurcation ratio (Rb) is Rb = Nu / N(u+1), where Nu is the number of streams of order u and N(u+1) is the number of the next higher order. Rb typically falls between 3 and 5.
斯特拉勒河流分级法将源头无支流水系定为1级；同级河流汇合后级别加1。分叉比 (Rb) 为 Rb = Nu / N(u+1)，其中 Nu 为 u 级河流的数量，N(u+1) 为下一高级别河流的数量。Rb 通常介于3至5之间。
Rb = Nu / N(u+1)
A low bifurcation ratio indicates a well‑drained basin with rapid flood peaks; a high ratio suggests elongated basin shapes with delayed flood response.
低分叉比指示排水良好、洪峰陡涨的流域；高分叉比则反映流域形状狭长、洪水响应滞后。
7. Population Change Equation | 人口变动公式
ion change over a period is the result of natural increase and net migration: P₂ = P₁ + (B – D) + (I – ths, I is immigration, and O is emigration. All components are for the same time interval.
某一时期内人口变动是自然增长与净迁移的总和：P₂ = P₁ + (B – D) + (I – O)，式中 P₁ 和 P₂ 分别为期初和期末人口，B 为出生人数，D 为死亡人数，I 为迁入人数，O 为迁出人数。所有成分均对应同一时间区间。
P₂ = P₁ + B − D + I − O
This fundamental demographic identity helps analyse the drivers of population growth or decline in any given area.
这一基本人口恒等式有助于分解任一地区人口增长或下降的驱动力。
8. Crude Birth Rate and Crude Death Rate | 粗出生率与粗死亡率
te (CBR) = (B / P) × 1000, where B is the number of live births in a year and P is the mid‑ R) = (D / P) × 1000, where D is deaths in a year. Rates are expressed per 1000 population.
粗出生率 (CBR) = (B / P) × 1000，B 为年活产婴儿数，P 为年中人口数。粗死亡率 (CDR) = (D / P) × 1000，D 为年死亡人数。比率以每千人为单位。
CBR = (B / P) × 1000
CDR = (D / P) × 1000
nverted to a percentage by dividing by 10 (e.g., NIR = 20 − 5 per 1000 = 15 per 1000 = 1.5%).
自然增长率 (NIR) = CBR − CDR，通常除以10转换成百分数（如 NIR = 20 − 5‰ = 15‰ = 1.5%）。
9. Fertility and Infant Mortality Rates | 生育率与婴儿死亡率
woman would have during her reproductive years, not a simple formula but derived from age‑ where D₀₋₁ is deaths of infants under 1 year in a year and B is live births in the same year.
总和生育率 (TFR) 是育龄妇女平均生育子女数，并非简单公式，而是由年龄别生育率汇总。婴儿死亡率 (IMR) = (D₀₋₁ / B) × 1000，D₀₋₁ 为年内未满1岁婴儿死亡数，B 为同年活产数。
IMR = (D₀₋₁ / B) × 1000
Both indicators are crucial measures of a country's health and development status, with IMR being especially sensitive to medical care and sanitation.
两者均为衡量国家健康与发展水平的关键指标，其中婴儿死亡率对医疗和卫生条件尤为敏感。
10. Net Migration Rate and Migration Efficiency | 净迁移率与迁移效率
Net Migration Rate (NMR) = ((I − O) / P) × 1000, where I is in‑migration, O is out‑migration, and P is the mid‑ I + O) × 100, measuring the effectiveness of gross migration in redistributing population.
净迁移率 (NMR) = ((I − O) / P) × 1000，I 为迁入，O 为迁出，P 为年中人口。迁移效率 = (I − O) / (I + O) × 100，衡量总迁移对人口再分配的有效程度。
NMR = ((I − O) / P) × 1000
Migration Efficiency = ((I − O) / (I + O)) × 100
Positive NMR indicates net in‑migration; high positive efficiency means a large proportion of gross migration is directed into the area.
NMR 为正表示净迁入；高效率正值意味着总迁移中的较大部分流入了该地区。
11. Urbanisation Level and Population Density | 城市化水平与人口密度
Urbanisation Level = (Urban Population / Total Population) × 100% l Density = Total Population / Arable Land Area, highlighting pressure on productive land.
城市化水平 = (城镇人口 / 总人口) × 100%。算术人口密度 = 总人口 / 陆地总面积（人/km²）。生理密度 = 总人口 / 耕地面积，突显生产性土地的压力。
Urbanisation = (Purban / Ptotal) × 100
Arithmetic Density = P / A
Physiological Density = P / Aarable
These indices are essential for describing settlement patterns and resource stress in AS Geography case studies.
这些指数在 AS 地理案例研究中对于描述聚落格局和资源压力至关重要。
12. Key Theorems and Models at a Glance | 重要定理与模型速览
While not purely formulaic, several cornerstone models carry quantitative implications. The Bradshaw model compares how channel variables (e.g., discharge, depth, velocity, sediment size) change downstream, often represented by a trend line. Ravenstein's Laws of Migration state that most migrants move short distances, and each migration flow produces a counterflow. The Gravity Model of migration expresses the interaction between two places as Iij = k × (Pi × Pj) / dij², where P are populations and d is distance. These models aid spatial analysis and must be critically evaluated in exam responses.
虽然并非纯粹公式，若干基石模型具有定量内涵。布拉德肖模型比较了河道变量（如流量、水深、流速、泥沙粒径）沿下游方向的变化趋势，常以趋势线呈现。拉文斯坦迁移法则指出多数迁移距离短、每一迁移流会产生反向流。迁移重力模型将两地相互作用表达为 Iij = k × (Pi × Pj) / dij²，其中 P 为人口，d 为距离。这些模型有助于空间分析，在考试答题中需要批判性评估。
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