Year 12 CAIE Geography: Key Formulas and Theorems Quick Reference Handbook | Year 12 CAIE 地理：公式定理速查手册

📚 Year 12 CAIE Geography: Key Formulas and Theorems Quick Reference Handbook | Year 12 CAIE 地理：公式定理速查手册

This quick reference handbook compiles the essential formulas, theoretical constants and key theorems that Year 12 CAIE Geography students must master for both physical and human geography topics. From stream discharge and lapse rates to demographic rates and urban rank-size rule, each entry is presented with a clear explanation, units and practical context.

本速查手册汇集了 Year 12 CAIE 地理课程中学生必须掌握的核心公式、理论常量和关键定理，涵盖自然地理与人文地理两大领域。从河流流量、气温递减率到人口统计指标和城市位序—规模法则，每个条目都配有清晰的说明、单位和实际应用背景。

1. Stream Discharge and Velocity | 河流流量与流速

Q = A × V

Stream discharge (Q) represents the volume of water passing a given cross-section per unit time, typically measured in cubic metres per second (m³ s⁻¹) or cumecs. It is calculated by multiplying the cross-sectional area (A, in m²) of the channel by the mean flow velocity (V, in m s⁻¹). Discharge increases downstream as tributaries join the main channel, and it is a fundamental variable for constructing flood hydrographs and assessing flood risk.

河流流量 (Q) 表示单位时间内通过某一断面的水体积，通常以立方米每秒 (m³ s⁻¹) 为单位。它的计算公式为河道横截面积 (A, 以 m² 计) 乘以平均流速 (V, 以 m s⁻¹ 计)。下游随着支流汇入，流量逐渐增加。流量是绘制洪水过程线和评估洪水风险的基础变量。

2. Hydraulic Radius and Channel Efficiency | 水力半径与河道效率

R = A / P

Hydraulic radius (R) is a measure of channel efficiency, defined as the ratio of the cross-sectional area (A) to the wetted perimeter (P), the length of the channel bed and banks in contact with water. A larger hydraulic radius means less friction per unit volume of water, allowing higher flow velocity for the same gradient. Deep, semicircular channels tend to be more efficient than wide, shallow ones.

水力半径 (R) 是衡量河道输水效率的指标，定义为过水断面面积 (A) 与湿周 (P, 即水流与河道边界接触的长度) 之比。水力半径越大，单位水体所受的摩擦阻力越小，在相同比降下流速就越高。深而呈半圆形的河道通常比宽浅型河道更为高效。

3. Drainage Density | 河网密度

D = ΣL / A

Drainage density (D) is the total length of all stream channels (ΣL) within a drainage basin divided by the total area of the basin (A). It is usually expressed in kilometres per square kilometre (km km⁻² or km⁻¹). High drainage density indicates a well-integrated stream network, often found in areas with impermeable rocks, steep slopes and sparse vegetation, which promote rapid surface runoff and flashy hydrographs.

河网密度 (D) 是流域内所有河道总长度 (ΣL) 与流域总面积 (A) 的比值，通常以每平方公里的河道长度 (km km⁻² 或 km⁻¹) 表示。高河网密度说明水系发育完善、整合度高，常见于不透水岩层、陡坡和植被稀疏的地区，这些条件有利于快速地表径流并形成暴涨暴落的洪水过程线。

4. Relative Humidity | 相对湿度

RH = (eₐ / eₛ) × 100%

Relative humidity (RH) expresses the amount of water vapour present in the air as a percentage of the amount needed for saturation at the same temperature. eₐ is the actual vapour pressure and eₛ is the saturation vapour pressure. When RH reaches 100%, the air is saturated and condensation may occur. RH plays a critical role in cloud formation, precipitation and the assessment of atmospheric stability.

相对湿度 (RH) 表示空气中实际水汽含量与同温度下饱和水汽含量的百分比。eₐ 为实际水汽压，eₛ 为饱和水汽压。当相对湿度达到 100% 时，空气达到饱和，可能发生凝结。相对湿度对云的形成、降水以及大气稳定度的判断具有关键作用。

5. Adiabatic Lapse Rates | 绝热递减率

DALR = 9.8 °C km⁻¹

The Dry Adiabatic Lapse Rate (DALR) is the rate at which a rising, unsaturated parcel of air cools due to expansion. It is a constant 9.8 °C per 1000 m (9.8 °C km⁻¹). The saturated adiabatic lapse rate (SALR) is lower, averaging about 6 °C km⁻¹, because latent heat is released during condensation, reducing the rate of cooling. The relationship between the environmental lapse rate (ELR), DALR and SALR determines atmospheric stability: absolute stability when ELR < SALR, conditional instability when SALR < ELR < DALR, and absolute instability when ELR > DALR.

干绝热递减率 (DALR) 是未饱和空气块在上升过程中因膨胀而冷却的速率，恒定为每上升 1000 米降温 9.8 °C。湿绝热递减率 (SALR) 则较低，平均约 6 °C km⁻¹，因为水汽凝结释放潜热减缓了降温。环境递减率 (ELR) 与 DALR、SALR 的关系决定了大气稳定度：ELR 小于 SALR 时为绝对稳定，SALR < ELR < DALR 时为条件不稳定，ELR 大于 DALR 时为绝对不稳定。

6. Crude Birth and Death Rates | 粗出生率与粗死亡率

CBR = (B / P) × 1000

CDR = (D / P) × 1000

The crude birth rate (CBR) is the number of live births in a year per 1000 people in the total population. Similarly, the crude death rate (CDR) is the number of deaths per 1000 people per year. B and D represent the total number of live births and deaths respectively, while P is the mid-year population. These crude rates are simple but do not account for the age structure of a population, which is why they are often used alongside age-specific rates.

粗出生率 (CBR) 是某年每 1000 人中活产婴儿的数量，粗死亡率 (CDR) 则是每 1000 人中的死亡人数。B 和 D 分别代表活产数和死亡总数，P 为年中人口。这两个粗率指标计算简单，但未考虑人口的年龄结构，因此常与年龄别生育率和死亡率配合使用。

7. Rate of Natural Increase and Doubling Time | 自然增长率与倍增时间

RNI (%) = (CBR − CDR) / 10

Doubling time (years) = 70 / growth rate (%)

The rate of natural increase (RNI) is the percentage annual growth of a population excluding migration. It is obtained by subtracting CDR from CBR and dividing by 10. The doubling time estimates the number of years required for a population to double in size at a constant annual growth rate, calculated using the rule of 70. For example, at a growth rate of 2% per year, the doubling time is 35 years. Both measures are essential for understanding demographic momentum and future resource demands.

自然增长率 (RNI) 是不考虑迁移的人口年增长百分比，由 CBR 减去 CDR 再除以 10 得出。倍增时间则估算在恒定年增长率下人口翻倍所需的年数，使用“70法则”计算。例如，年增长率为 2% 时，倍增时间为 35 年。这两个指标对于理解人口惯性和未来的资源需求至关重要。

8. Net Migration Rate and Population Density | 净迁移率与人口密度

Net migration rate = ((I − E) / P) × 1000

The net migration rate measures the difference between the number of immigrants (I) and emigrants (E) in a year per 1000 people in the total population (P). A positive figure indicates net in-migration, while a negative value shows net out-migration. It does not capture internal movements but is vital for understanding population change at national or regional scales.

净迁移率衡量的是某年迁入人数 (I) 与迁出人数 (E) 的差额占年中总人口 (P) 的千分比。正值表示净迁入，负值代表净迁出。该指标不反映国内迁移，但对于理解国家或区域尺度的人口变化非常关键。

Population density = P / A

Population density is the total number of people (P) divided by the land area (A), usually given as persons per km². It provides a crude measure of the pressure on land and resources, though it can be misleading without considering the distribution of habitable or productive land. Physiologic density (people per unit of arable land) is a useful refinement.

人口密度是总人口数 (P) 除以土地面积 (A)，通常以人/平方公里表示。它粗略反映了土地与资源所承受的压力，但若不考虑宜居面积或耕地分布可能会产生误导。生理密度 (每单位耕地面积人口) 是一种更有用的细化指标。

9. Rank-Size Rule | 位序−规模法则

Pₙ = P₁ / n

The rank-size rule describes an ideal urban hierarchy where the population of a city (Pₙ) is inversely proportional to its rank (n) in the national urban system. The largest city (rank 1) has population P₁, the second largest has roughly P₁/2, the third P₁/3, and so on. Countries with mature, well-integrated economies, such as the United States, often follow this pattern closely. A significant deviation, such as a primate city that overwhelmingly dominates the urban landscape, indicates a different set of historical, political or economic forces.

位序−规模法则描述了一种理想的城市等级体系：城市的人口规模 (Pₙ) 与其在国家城市体系中位序 (n) 成反比。首位城市人口为 P₁，第二位城市人口约为 P₁/2，第三位约为 P₁/3，依此类推。经济成熟且体系完善的国家，如美国，往往较符合该法则。若出现显著的偏离，例如首位城市一家独大的现象，则反映出不同的历史、政治或经济驱动力。

10. Central Place Theory Principles | 中心地理论原理

k = 3 (marketing), k = 4 (transport), k = 7 (administrative)

Christaller’s central place theory explains the size, number and distribution of settlements based on their function as market centres providing goods and services. The theory identifies three ordering principles. The marketing principle (k=3) maximises consumer convenience; the transport principle (k=4) minimises transport costs by placing lower-order centres along routes linking higher-order places; and the administrative principle (k=7) ensures that lower-order centres fall entirely within the territory controlled by a single higher-order centre. Each k-value determines the number of settlements served by one central place. The theory assumes an isotropic plain and rational consumer behaviour, which limit its real-world applicability but make it a powerful analytical starting point.

克里斯泰勒的中心地理论从市场中心提供商品与服务的角度解释了聚落的规模、数量和分布。理论提出三种组织原则：市场原则 (k=3) 追求消费者便利的最大化；交通原则 (k=4) 通过将低级中心布局在连接高级中心的交通线上以降低运输成本；行政原则 (k=7) 确保次级中心完全从属于一个高级中心的管理区域。每个 k 值决定了一个中心地所服务的聚落数量。该理论假设均质平原和理性消费行为，虽然限制了现实适用性，但仍是强有力的分析起点。

11. Weber’s Least Cost Theory | 韦伯工业区位论

Weber’s model of industrial location explains the optimum location of a factory in terms of minimising three basic costs: transport, labour and agglomeration. The material index (MI) is a key concept used to determine whether an industry is raw-material oriented (MI > 1) or market-oriented (MI < 1). It is the ratio of the weight of localised raw materials to the weight of the finished product. Industries with a high MI, such as sugar refining, locate near raw materials to reduce bulk transport costs. If labour costs are sufficiently low in a particular location, they may pull the industry away from the least-transport-cost point, leading to a locational deflection. Agglomeration economies provide further adjustments by offering shared infrastructure and a pool of skilled workers.

韦伯的工业区位模型通过运输、劳动力和集聚三个基本成本的最小化来解释工厂的最优选址。原料指数 (MI) 是判断工业属于原料指向 (MI > 1) 还是市场指向 (MI < 1) 的关键概念，其值为地方性原料重量与成品重量之比。原料指数较高的行业 (如制糖业) 倾向于靠近原料地以削减运输成本。如果某地劳动力成本足够低廉，可能将厂址拉离运输成本最低点，形成区位偏转。集聚经济通过共享基础设施和熟练劳动力池，进一步调整最优区位。

12. Urban Land-Use Models | 城市土地利用模型

Three classic urban land-use models provide simplified representations of the internal structure of cities. Burgess’s concentric zone model (1925) depicts a city growing outward in five concentric rings, from the CBD and transition zone to commuter suburbs, based on ideas of invasion and succession. Hoyt’s sector model (1939) modifies this by suggesting that high-rent residential, industrial and transport corridors develop in wedge-shaped sectors along major routes. Harris and Ullman’s multiple nuclei model (1945) argues that cities grow around several discrete centres (nuclei) rather than a single CBD, reflecting specialised activities such as airports, universities and industrial parks. These models are theorems in spatial organisation and remain useful for explaining patterns of urban growth, though they were developed from specific historical contexts and should be applied critically.

三种经典的城市土地利用模型以简化的方式呈现了城市内部结构。伯吉斯的同心圆模型 (1925) 基于侵入和演替的概念，将城市描述为由 CBD、过渡带到通勤郊区依次向外扩展的五个同心环带。霍伊特的扇形模型 (1939) 则指出，高租金住宅、工业和交通廊道往往沿主要路线呈楔形扇区发展。哈里斯和厄尔曼的多核心模型 (1945) 认为城市围绕多个离散的中心 (核心) 而非单一 CBD 成长，体现了机场、大学、工业园区等专门化活动。这些模型作为空间组织的定理，至今仍有助于解释城市增长形态，但因源自特定历史背景，需批判性地应用。

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