Year 13 Cambridge Geography: Formula & Theorem Quick Reference | 十三年级剑桥地理公式定理速查手册

📚 Year 13 Cambridge Geography: Formula & Theorem Quick Reference | 十三年级剑桥地理公式定理速查手册

This rapid reference handbook brings together the essential quantitative relationships, indices and models that appear across the Cambridge International A‑Level Geography (9696) syllabus. From water balance and stream power to urban rank‑size rules and demographic multipliers, each entry is presented with clear notation, typical units and a brief explanation of its application in geographical analysis.

本速查手册汇集了剑桥国际 A‑Level 地理 (9696) 考试中常出现的重要定量关系、指数与模型。从水平衡、河流功率到城市位序‑规模法则和人口乘数，每一个条目都配有清晰的符号、常用单位以及对地理分析应用的简要说明。

1. Hydrological Cycle & Water Balance | 水文循环与水平衡

The water balance equation for a drainage basin over a given time period is expressed as 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.

流域在一定时段内的水平衡方程表达为 P = Q + E + ΔS，其中 P 为降水量，Q 为径流量，E 为蒸散发量，ΔS 为土壤水分和地下水储存量的变化。

• / P (dimensionless). It indicates the proportion of precipitation that becomes streamflow.

  径流系数 C = Q / P（无量纲），反映降水转化为河川径流的比例。

• Soil water surplus or deficit can be identified when P exceeds E or falls short of it; ΔS becomes positive (recharge) or negative (utilization).

  当 P 大于 E 时出现水分盈余，反之出现水分亏缺；ΔS 为正（补给）或为负（消耗）。

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2. Discharge & Flood Estimation | 流量与洪水估算

River discharge Q is calculated as the product of cross‑sectional area A and mean velocity v: Q = A × v, with units m³ s⁻¹. Velocity is often obtained from the Manning equation: v = (1/n) R^2/3 S^1/2, where n is Manning's roughness, R is hydraulic radius (A / wetted perimeter) and S is channel slope.

河流流量 Q 等于过水断面面积 A 与平均流速 v 的乘积：Q = A × v，单位为 m³ s⁻¹。流速常用曼宁公式求得：v = (1/n) R^2/3 S^1/2，其中 n 为曼宁糙率，R 为水力半径（面积/湿周），S 为河道坡度。

• Flood peak estimation often uses the rational formula Q_peak = C × I × A, where I is rainfall intensity and A is catchment area.

  洪峰流量估算常用推理公式 Q_peak = C × I × A，I 为降雨强度，A 为流域面积。

• Re‑ (n + 1) / m, where n is number of years of record and m is the rank of a flood magnitude.

  重现期 T = (n + 1) / m，n 为记录年数，m 为洪水大小的排序名次。

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3. Stream Power & Erosion | 河流功率与侵蚀

Total stream power per unit channel length is given by Ω = ρ g Q S, where ρ is water density (1000 kg m⁻³), g is gravity (9.81 m s⁻²), Q is discharge and S is energy slope. It measures the rate of energy expenditure available for bed and bank erosion.

单位河长的总水流功率为 Ω = ρ g Q S，ρ 为水的密度（1000 kg m⁻³），g 为重力加速度（9.81 m s⁻²），Q 为流量，S 为能面比降。该指标反映了可用于河床和河岸侵蚀的能量消耗速率。

• is channel width. It helps identify thresholds for sediment transport and alluvial channel pattern change.

  单位水流功率 ω = Ω / w，w 为河宽，常用于判别泥沙输移和冲积河型转变的临界值。

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4. Climatic Indices & Aridity | 气候指数与干旱度

De Martonne aridity index is widely used in physical geography: I = P / (T + 10), where P is annual precipitation (mm) and T is mean annual temperature (°C). Values below 20 indicate semi‑arid conditions, below 10 arid.

德马顿干燥指数：I = P / (T + 10)，P 为年降水量 (mm)，T 为年均气温 (°C)。I < 20 为半干旱，I < 10 为干旱。

• Lang’s rain factor: RF = P / T (mm/°C). Used for soil climate classifications.

  朗格雨量因子：RF = P / T (mm/°C)，用于土壤气候分类。

• Potential evapotranspiration (PET) is often estimated by the Thornthwaite formula using monthly temperature and day‑length corrections.

  潜在蒸散发 (PET) 常用桑斯威特公式，依据月均温和日长订正进行估算。

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5. Soil Erosion Models | 土壤侵蚀模型 (USLE)

The Universal Soil Loss Equation estimates annual soil loss: A = R × K × LS × C × P, where A is soil loss (tonnes ha⁻¹ yr⁻¹), R rainfall erosivity, K soil erodibility, LS slope length‑steepness, C cover‑management and P support practice factor.

通用土壤流失方程估算年均土壤流失量：A = R × K × LS × C × P，A 为土壤流失量 (t ha⁻¹ yr⁻¹)，R 为降雨侵蚀力，K 为土壤可蚀性，LS 为坡长坡度因子，C 为覆盖管理因子，P 为水土保持措施因子。

• R factor is computed from the kinetic energy of storms times maximum 30‑minute intensity; higher values indicate greater erosion potential.

  R 因子由暴雨动能乘以最大 30 分钟雨强计算；值越大侵蚀潜力越大。

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6. Population Growth Models | 人口增长模型

The basic exponential growth model is P_t = P₀ e^rt, where P₀ is initial population, r is growth rate (decimals), t is time and e is Euler's number. For short intervals, a geometric model P_t = P₀ (1 + r)^t is used.

基本指数增长模型为 P_t = P₀ e^rt，P₀ 为起始人口，r 为增长率（小数），t 为时间，e 为自然常数。短时段可用几何增长模型 P_t = P₀ (1 + r)^t。

• Doubling time: T_d = 70 / r(%) (approx.), a quick rule very common in human geography.

  人口倍增时间：T_d = 70 / r(%)（近似），是人文地理中常用的简便算法。

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7. Demographic Indicators | 人口指标

th rate CDR = (D / P) × 1000, where B and D are live births and deaths in a year, P is mid‑year population. Natural increase rate = (CBR – CDR) / 10, expressed as a percentage.

粗出生率 CBR = (B / P) × 1000，粗死亡率 CDR = (D / P) × 1000，B、D 为年度活产和死亡数，P 为年中人口。自然增长率 = (CBR – CDR)/10，以百分比表示。

• ar / live births) × 1000. Total fertility rate TFR = average number of children per woman.

  婴儿死亡率 IMR = (1 岁以下死亡数/活产数) × 1000。总和生育率 TFR = 每名妇女平均生育子女数。

• Dependency ratio = [(population 0–14 + population 65+) / population 15–64] × 100.

  抚养比 = [(0–14 岁人口 + 65 岁以上人口) / 15–64 岁人口] × 100。

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8. Urban Models & Rank‑Size Rule | 城市模型与位序‑规模法则

Zipf’s rank‑size rule states that the population of a settlement is inversely proportional to its rank in the national hierarchy: P_n = P₁ / n, where P₁ is the population of the largest city and n is the rank. A more general form is P_n = P₁ / n^q, where q indicates the degree of primacy.

齐夫位序‑规模法则：城市人口与其在**城市体系中的位序成反比：P_n = P₁ / n，P₁ 为首位城市人口，n 为位序。更一般的形式为 P_n = P₁ / n^q，q 值反映首位度强弱。

• Primacy index: ratio of the largest city's population to the sum of populations of the next three cities. A value above 1 signals strong primacy.

  首位度指数：最大城市人口与第二、三、四大城市人口之和的比值；大于 1 表示强首位分布。

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9. Location Quotient & Multiplier | 区位商与乘数效应

The location quotient measures the regional concentration of an industry relative to the national average: LQ = (e_i / E) / (E_i / E_nat), where e_i is regional employment in sector i, E is total regional employment, E_i is national employment in sector i, E_nat is total national employment. LQ > 1 indicates export orientation.

区位商衡量某一产业在区域中的集聚程度：LQ = (e_i / E) / (E_i / E_nat)，e_i 为区域 i 产业就业人数，E 为区域总就业，E_i 为全国 i 产业就业，E_nat 为全国总就业。LQ > 1 表示该产业为外向型。

• Local multiplier k = 1 / (1 – me locally. It estimates how an injection of spending circulates within the local economy.

  地方乘数 k = 1 / (1 – MPC_local)，MPC_local 为本地边际消费倾向，用于估算一笔初始支出在地方经济中的循环放大效应。

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10. Gravity Model & Interaction | 重力模型与相互作用

The gravity model predicts the flow of people, goods or information between two places: I_ij = k (P_i × P_j) / d_ij^b, where I is interaction, P_i and P_j are population masses, d_ij is distance, k is a constant and b is the distance exponent (often calibrated).

重力模型用于预测两地之间的人员、货物或信息流动：I_ij = k (P_i × P_j) / d_ij^b，I 为相互作用量，P_i、P_j 为人口规模，d_ij 为距离，k 为常数，b 为距离衰减指数（需通过实际数据校准）。

• Re‑formulated for retail, Reilly’s law of retail gravitation calculates the breaking point between two centres: d_A = d_AB / (1 + √P_B / P_A), where d_A is distance from the larger centre.

  在零售地理中，赖利零售引力定律计算两个商业中心之间的断裂点：d_A = d_AB / (1 + √P_B / P_A)，d_A 为离较大中心的距离。

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