📚 IGCSE Edexcel Geography: Formula & Theorem Quick Reference Handbook | IGCSE Edexcel 地理：公式定理速查手册
Mastering key formulas and theorems is essential for success in IGCSE Edexcel Geography. This quick reference handbook compiles must-know equations and principles across physical and human geography topics, from population change to river processes. Keep it handy for revision and exam practice.
掌握关键公式和定理对于 IGCSE Edexcel 地理考试至关重要。本速查手册汇总了自然与人文地理必知的方程和原理，涵盖人口变化、河流过程等主题。方便复习与备考使用。
1. Population Density & Demographic Change | 人口密度与人口变化公式
Population density describes how crowded an area is, calculated by dividing the total population by the land area. The result is usually expressed as people per square kilometre.
Population Density = Total Population / Area (km²)
人口密度描述了一个地区的拥挤程度，通过总人口除以土地面积计算。结果通常以每平方公里的人数表示。
Natural increase is the difference between the number of live births and deaths in a year, ignoring migration. It shows how the population size changes through reproduction alone.
Natural Increase = Birth Rate – Death Rate
自然增长是一年内活产婴儿数与死亡人数之间的差值，不考虑迁移。它反映了仅由出生和死亡导致的人口规模变化。
Net migration adds the movement of people into and out of a region. When net migration is positive, more people are arriving than leaving.
Net Migration = Immigration – Emigration
净迁移量计算的是迁入和迁出一个地区的人数差额。净迁移量为正时，迁入者多于迁出者。
The overall population growth rate combines natural change and net migration, then relates them to the total population. It is expressed as a percentage.
Population Growth Rate (%) = [(Births – Deaths) + Net Migration] / Total Population × 100
整体人口增长率将自然变动与净迁移结合起来，再与总人口相比。通常以百分数表示。
2. Settlement & Service Thresholds | 聚落与服务阈值
A settlement’s function depends on the minimum number of customers needed to keep a service profitable. This minimum is called the threshold population. Below this level, the service is likely to close.
Threshold = Minimum population required to support a service
聚落的功能依赖于维持某项服务盈利所需的最低顾客数量，这个最低人数即为服务阈值。低于这一水平，服务就可能关闭。
The range of a service is the maximum distance people are prepared to travel to use it. High-order goods (e.g., furniture) have a larger range than low-order goods (e.g., bread).
Range = Maximum distance consumers will travel to access a service
服务范围是指人们愿意前往使用的最大距离。高级商品（如家具）的范围大于低级商品（如面包）。
3. River Discharge & Hydraulic Geometry | 河流流量与水力学几何
Discharge is the volume of water flowing past a point in a channel each second. It requires measuring both the cross-sectional area of the stream and the average flow velocity.
Discharge, Q = A × V
流量是每秒流过河道某一点的水体积。需要测量河道的横截面积和平均流速。
In the formula, A represents the cross-sectional area in square metres (m²), and V is the mean velocity in metres per second (m/s). Discharge is recorded in cubic metres per second (m³/s or m³ s⁻¹).
Q (m³ s⁻¹) = A (m²) × V (m/s)
公式中，A 代表横截面积，单位为平方米；V 是平均流速，单位为米每秒。流量的单位是立方米每秒。
The hydraulic radius helps describe channel efficiency. A larger hydraulic radius means less friction and faster flow for a given cross-sectional area.
Hydraulic Radius, R = A / Wetted Perimeter
水力半径有助于描述河道输水效率。水力半径越大，意味着在给定横截面积下摩擦越小，流速越快。
4. River Gradient & Velocity Measurement | 河流梯度与流速测量
Stream gradient is the steepness of a river channel. It is calculated as the vertical fall over a measured horizontal distance. Gradient is often expressed as a percentage or as a ratio.
Gradient = Vertical Interval / Horizontal Distance
河流梯度是指河道的陡峭程度，通过垂直落差与水平距离之比计算。梯度通常用百分数或比值表示。
For example, a 5-metre vertical drop over 200 metres of channel gives a gradient of 0.025, or 2.5 %. On a map, gradient can be written as 1 in 40.
Gradient (%) = (Vertical Interval / Horizontal Distance) × 100
例如，河道在 200 米水平距离内下降了 5 米，梯度为 0.025，即 2.5%。在地图上，可表示为 1：40。
Average surface velocity is measured during fieldwork by timing a float over a set distance. The true mean velocity is usually less than the surface speed, so a correction factor (typically 0.8 or 0.85) is applied.
Velocity = Distance / Time
平均表流流速在野外调查中通过测量浮标移动一定距离所需时间获得。真实平均流速通常小于表流，因此需要乘以修正系数（一般为 0.8 或 0.85）。
5. Bradshaw Model & Channel Changes | 布拉德肖模型与河道变化
The Bradshaw model is a conceptual theorem summarising how river characteristics change from source to mouth. Downstream, discharge, occupied channel width, and depth all increase, while bedload size and slope gradient decrease.
Downstream: Q ↑, Velocity ↑, Width ↑, Depth ↑, Load particle size ↓, Gradient ↓
布拉德肖模型是一个概念性定律，总结了从河源到河口河流特征的变化。下游方向，流量、水流宽度和深度增大，而河床粒径和坡度减小。
These predictable relationships are used to interpret fieldwork data. For instance, if the average bedload diameter does not decrease downstream, local inputs from tributaries or human activity may be responsible.
The model supports the idea of an idealised, unmanaged river profile.
这些可预测的关系常用来分析实地调查数据。例如，如果下游平均河床粒径并未减小，可能是支流汇入或人为活动所致。该模型描述了理想化、未受干预的河道纵剖面。
6. Hjulström Curve & Critical Erosion Velocity | 尤斯特罗姆曲线与临界侵蚀速度
The Hjulström curve illustrates the relationship between water velocity and particle entrainment, transport, and deposition. It shows that sand-sized particles are easiest to erode, while clays and larger pebbles require higher velocities.
Erosion threshold ≈ 20–30 cm/s for fine sand; >100 cm/s for clay and boulders
尤斯特罗姆曲线展示了水流速度与颗粒起动、搬运和沉积之间的关系。图中显示，砂粒最容易被侵蚀，而黏土和较大卵石则需要更高的流速才能起动。
Once in motion, finer particles can remain suspended at very low velocities. The curve provides a critical framework for understanding sediment dynamics in river channels, but no single formula captures the entire relationship.
Deposition occurs when velocity falls below the settling velocity of a given particle size.
一旦起动，细颗粒物质即使在很低流速下也能保持悬浮。该曲线是理解河道沉积动力学的关键框架，但没有单一公式可以概括所有关系。沉积作用发生在流速低于某粒径颗粒沉降速度时。
7. Coastal Wave Energy & Longshore Drift | 海岸波浪能量与沿岸漂移
Wave energy depends primarily on wave height, wave period, and water density. A simplified relationship shows that energy is proportional to the square of the wave height. This explains why storm waves are so destructive.
Wave Energy, E ∝ H²
波浪能量主要取决于波高、波浪周期和水密度。一个简化关系显示，能量与波高的平方成正比，这解释了为何风暴浪破坏力极强。
Longshore drift is the movement of sediment along a coast by wave action. The volume of sediment transported is affected by wave energy and the angle at which waves approach the shore. While no simple formula is used at IGCSE, the process is modelled by the product of longshore current velocity and available sediment.
Sediment transport rate ≈ Longshore current velocity × Sediment availability
沿岸漂移是波浪作用导致泥沙沿岸线移动的过程。输沙量受波浪能量和波浪入射角影响。尽管 IGCSE 不用复杂公式，但这一过程可模型化为沿岸流速度与可搬运泥沙量的乘积。
8. Map Skills: Scale, Area & Gradient | 地图技能：比例尺、面积与梯度
Map scale expresses the ratio between distances on a map and the corresponding real-world distances. A scale of 1:50,000 means 1 cm on the map represents 50,000 cm (or 500 m) on the ground.
Real Distance = Map Distance × Scale Factor
地图比例尺表示图上距离与对应实际距离的比值。1:50,000 意味着图上的 1 厘米代表实地 500 米。
Calculating area from a map often uses grid squares. In a 1:25,000 map, each 4-cm grid square (2 cm × 2 cm) covers 1 km². You count full and partial squares to estimate irregular areas.
Area (km²) = Number of grid squares × Area per square
从地图估算面积时常使用方格法。在 1:25,000 地图上，每个 4 厘米见方的方格代表 1 平方公里。通过数完整和部分方格来估算不规则形状的面积。
Gradient on a map is found by dividing the vertical interval (height difference between contours) by the horizontal equivalent (measurable distance). It is often reported as a ratio, fraction, or percentage.
Map Gradient = Vertical Rise / Horizontal Run
地图上的梯度通过等高线间的高差（垂直间隔）除以水平距离求得，通常以比、分数或百分数给出。
9. Weather & Climate: Humidity & Precipitation | 天气与气候：湿度与降水
Relative humidity indicates how close the air is to saturation. It is the ratio of the actual water vapour content to the maximum possible at that temperature, expressed as a percentage.
Relative Humidity (%) = (Actual Vapour Pressure / Saturation Vapour Pressure) × 100
相对湿度表示空气接近饱和的程度，是实际水汽压与该温度下饱和水汽压之比，以百分数表示。
Precipitation intensity is simply the amount of rainfall collected over a set time. It helps describe storm characteristics and is crucial for flood studies.
Intensity = Total Precipitation (mm) / Duration (hours)
降水强度就是一定时间内收集的雨量，有助于描述暴雨特征，对洪水研究至关重要。
10. Development Indicators & Growth Rates | 发展指标与增长率
Annual economic growth is measured by the change in gross domestic product (GDP). The growth rate is commonly calculated as the percentage increase from one year to the next.
GDP Growth Rate (%) = [(GDPyear2 – GDPyear1) / GDPyear1] × 100
年度经济增长率通过国内生产总值（GDP）的变化衡量。常用公式为今年与去年 GDP 差值除以去年 GDP，再乘以 100%。
The ‘rule of 70’ estimates how many years it takes for a quantity to double, given a constant annual growth rate. This theorem is widely used for population and economic projections.
Doubling Time (years) ≈ 70 / Growth Rate (%)
“70 法则”用于估算在固定年增长率下一个量翻一番所需的年数，在人口和经济增长预测中广泛使用。
The Human Development Index (HDI) is a composite statistic, not a simple formula, but it integrates life expectancy, education (mean and expected years of schooling), and GNI per capita into a single score between 0 and 1.
HDI = (Life Expectancy Index × Education Index × Income Index)^(1/3)
人类发展指数（HDI）是一个复合指标，它将预期寿命、教育指数（平均和预期受教育年限）以及人均国民总收入整合为一个 0 到 1 之间的得分。
11. Energy & Resource Efficiency | 能源与资源使用效率
Efficiency measures how well a system converts input energy into useful output. For electricity generation, a fossil fuel power station may have an efficiency around 35–40%.
Efficiency (%) = (Useful Energy Output / Total Energy Input) × 100
效率衡量系统将输入能量转化为有用输出的表现。燃煤电厂的能量转化效率通常只有 35–40%。
Energy density describes the amount of energy stored per unit mass of a fuel. It helps compare resources such as wood, coal, nuclear fuel, and batteries.
Energy Density (J/kg or MJ/kg) = Total Energy / Mass
能量密度是指单位质量燃料储存的能量，可用于比较木材、煤、核燃料和电池等能源。
12. Fieldwork & Statistical Measures | 实地调查与统计量度
When analysing fieldwork data, central tendency measures summarise a dataset. The mean is the arithmetic average, and it is the most common statistic for river width, pebble size, or questionnaire responses.
Mean = Sum of all values / Number of values
在分析实地调查数据时，集中趋势量度可概括数据集。平均值是最常用的统计量，用于表示河宽、砾石尺寸或问卷回答。
The median is the middle value when data are ordered, and the mode is the most frequent value. Range describes dispersion by subtracting the smallest value from the largest.
Range = Highest value – Lowest value
中位数是将数据排序后位于中间的值，众数是出现频率最高的值。全距计算最大值与最小值的差值，反映数据分散程度。
Spearman’s rank correlation coefficient may be used to test relationships between two variables (e.g., distance downstream and pebble roundness). The calculation is not listed here, but the interpretation follows: a value near +1 indicates a strong positive association.
Rs = 1 – (6 ΣD²) / [n(n² – 1)]
斯皮尔曼等级相关系数可检验两个变量（如下游距离与砾石磨圆度）之间的关系。该公式常用于高阶分析，接近 +1 的值表示强正相关。
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