📚 Formula & Theorem Quick Reference Handbook | 公式定理速查手册
This quick reference guide collates essential formulas, theorems and quantitative relationships from the Cambridge Pre-U Geography syllabus. It covers population dynamics, urban systems, hydrological processes, coastal geomorphology, soil erosion, climate indices, development metrics and map skills. Use it for revision and to strengthen your confidence in handling numerical data and geographical theorems.
本速查手册汇集剑桥Pre-U地理课程中的关键公式、定理与定量关系，涵盖人口动态、城市体系、水文过程、海岸地貌、土壤侵蚀、气候指数、发展指标与地图技能，助力复习并增强处理地理数据与理解地理定理的信心。
1. Population Change Formulas | 人口变动公式
The fundamental equation for natural increase links births and deaths. Natural increase rate (NIR) is the difference between crude birth rate and crude death rate, measured per 1000 population.
人口自然增长的基本公式将出生与死亡联系起来。自然增长率（NIR）等于粗出生率减去粗死亡率，以每千人为单位。
NIR = CBR – CDR
When net migration is included, the population growth rate becomes (CBR – CDR) + net migration rate, giving the total annual change per 1000.
当包括净迁移时，人口增长率变为（CBR – CDR）+ 净迁移率，得出每千人的年度总变化。
Growth rate = (CBR – CDR) + Net Migration Rate
Doubling time estimates the number of years needed for a population to double at a constant growth rate. The ‘rule of 70’ is widely used.
倍增时间估算人口按恒定增长率翻倍所需的年数，常用’70法则’。
Doubling Time (years) = 70 / Annual Growth Rate (%)
For projecting future population, two simple models are arithmetic growth (constant absolute change) and geometric growth (constant percentage change).
预测未来人口时，两个简单模型为算术增长（绝对增长恒定）和几何增长（增长率恒定）。
Arithmetic: Pₜ = P₀ + b × t
Geometric: Pₜ = P₀ × (1 + r)ᵗ
Where Pₜ is the future population, P₀ the base population, b the annual absolute increase, r the annual growth rate as a decimal, and t the time in years.
式中Pₜ为未来人口，P₀为基期人口，b为年绝对增加量，r为以小数表示的年增长率，t为年数。
2. Demographic Measures & Rates | 人口统计测量与比率
Crude birth rate (CBR) standardises the number of live births relative to the total mid-year population.
粗出生率（CBR）将活产婴儿数与年中总人口对比进行标准化。
CBR = (Live Births / Mid-year Population) × 1000
Similarly, crude death rate (CDR) expresses total deaths per 1000 people.
类似地，粗死亡率（CDR）表示每千人的死亡人数。
CDR = (Deaths / Mid-year Population) × 1000
Infant mortality rate (IMR) focuses on deaths of infants under one year per 1000 live births, a key indicator of health.
婴儿死亡率（IMR）关注每千例活产中一岁以下婴儿的死亡数，是一项关键健康指标。
IMR = (Deaths under 1 year / Live Births) × 1000
Total fertility rate (TFR) is the average number of children a woman would have if she experienced current age-specific fertility rates throughout her reproductive life.
TFR = ∑ (ASFRₐ × 5)
Each ASFRₐ is the age-specific fertility rate per 1000 women in a 5-year age group, and the sum is multiplied by 5 and divided by 1000 to express as births per woman.
| Measure | Formula | Unit |
|---|---|---|
| CBR | (Live births ÷ Population) × 1000 | per 1000 |
| CDR | (Deaths ÷ Population) × 1000 | per 1000 |
| IMR | (Deaths under 1 ÷ Live births) × 1000 | per 1000 live births |
| TFR | ∑ (ASFR × 5) ÷ 1000 | children per woman |
These rates form the backbone of demographic analysis and appear frequently in Pre-U data-response questions.
这些比率构成人口分析的基础，常出现于Pre-U数据分析题中。
3. Urbanisation and City Systems | 城市化与城市体系
Urbanisation level is simply the proportion of a country’s population living in urban areas.
城镇化水平即一国人口中居住在城市地区的比例。
Urbanisation Level (%) = (Urban Population / Total Population) × 100
The rank-size rule describes the expected population of a city given its rank. The simplest form states that the population of the nth largest city is 1/n of the largest city’s population.
位序-规模法则描述给定位序城市的预期人口。最简形式表明第n大城市的人口是最大城市人口的1/n。
Pₙ = P₁ / n
A more general form incorporates an exponent q to reflect deviation from the ideal pattern.
Pₙ = P₁ / n^q
Reilly’s law of retail gravitation calculates the breakpoint between two settlements where customers are drawn equally to either centre.
赖利零售引力定律计算两点间的断裂点，即顾客被两个中心同等吸引的分界。
dₐ = D_AB / (1 + √(P_B / P_A))
Where dₐ is the distance from city A to the breakpoint, D_AB is the total distance between cities A and B, and P_A, P_B are their populations.
式中dₐ为从城市A到断裂点的距离，D_AB为A与B两城间的距离，P_A、P_B为各自的人口。
The gravity model assesses potential interaction between two places, inspired by Newtonian physics.
Interaction I = k × (P_A × P_B) / d^2
A high population product and short distance yield stronger flows of people, goods and information.
人口乘积大、距离短，则人员、货物与信息流更强。
4. Central Place Theory & Thresholds | 中心地理论与门槛值
Christaller’s central place theory organises settlements into a hierarchical network based on market area, transportation and administration principles. The K-value indicates how many lower-order centres are served by one higher-order centre.
克里斯泰勒的中心地理论基于市场、交通和行政原则将聚落组织为等级网络。K值表示一个高等级中心服务于多少个低等级中心。
- K=3 (marketing principle) – 2 lower-order centres surrounding each higher-order centre
- K=4 (transport principle) – 3 lower-order centres
- K=7 (administrative principle) – 6 lower-order centres
每个高等级中心周围分别有2个（营销）、3个（交通）、6个（行政）低等级中心。
The threshold population is the minimum number of customers needed to support a service. It can be estimated by dividing total costs by average spending per customer.
门槛人口是支撑某项服务所需的最低顾客数，可用总成本除以顾客平均消费来估算。
Threshold = (Fixed Costs + Expected Profit) / Average Spending per Customer
Range of a good is the maximum distance consumers will travel, influenced by transport costs and the good’s characteristics.
商品的范围是消费者愿意出行的最大距离，受交通成本和商品特性影响。
5. Hydrological Formulae | 水文公式
Discharge is a fundamental river metric, calculated as the product of cross-sectional area and mean velocity.
流量是河流基本度量，等于过水断面面积与平均流速的乘积。
Q = A × V
Hydraulic radius relates the cross-sectional area of flow to its wetted perimeter, indicating channel efficiency.
水力半径将水流横截面积与湿周关联，反映河道效率。
R = A / P
Where A is the cross-sectional area (m2) and P is the wetted perimeter (m). A larger R generally means less energy lost to friction.
式中A为横截面积（平方米），P为湿周（米）。R越大通常意味着摩擦能量损失越小。
Manning’s equation estimates mean velocity in an open channel using roughness, hydraulic radius and slope.
曼宁公式利用糙率、水力半径和坡降估算明渠平均流速。
v = (1/n) × R^(²/₃) × S^(¹/₂)
Where n is Manning’s roughness coefficient, R the hydraulic radius, and S the channel slope. The exponent ²/₃ on R and ¹/₂ on S reflect empirical adjustments.
式中n为曼宁糙率系数，R为水力半径，S为河道比降。R的²/₃次方和S的¹/₂次方为经验调整。
Runoff coefficient expresses the proportion of precipitation that appears as direct runoff.
C = Runoff / Precipitation
Values range from near 0 on permeable forested slopes to over 0.9 on impermeable urban surfaces.
数值从透水森林坡地的接近0到不透水城市地表的超过0.9不等。
6. Coastal and Fluvial Processes | 海岸与河流过程公式
Wave energy is proportional to the square of wave height, explaining the much greater erosive power of storm waves.
波浪能量与波高的平方成正比，这解释了风暴浪的侵蚀力为何大得多。
E ∝ H^2
Longshore drift volume can be roughly estimated if wave height, angle and sediment concentration are known, though field measurement is preferred.
Q_l = k × H_b^2 × sin(2α_b)
Where Q_l is the longshore sediment transport rate, H_b is breaking wave height, α_b is the breaker angle relative to the shoreline, and k is an empirical coefficient.
式中Q_l为沿岸输沙率，H_b为破波波高，α_b为破波与岸线夹角，k为经验系数。
In rivers, the competent velocity is the minimum velocity needed to transport a particle of a given diameter; the Hjulstrom curve shows critical erosion and deposition velocities for different grain sizes.
在河流中，起动流速是搬运特定粒径颗粒所需的最小流速；尤斯特罗姆曲线显示了不同粒径的临界侵蚀与沉积流速。
Stream gradient is the vertical fall per unit horizontal distance, typically expressed as a ratio or percentage.
Gradient = Vertical Interval (VI) / Horizontal Equivalent (HE)
A steep gradient implies higher energy and greater potential for vertical erosion.
陡峭的河流梯度意味着更高的能量和更大的下切侵蚀潜力。
7. Slope Stability and Mass Movement | 边坡稳定性与块体运动
Slope stability is often expressed as a factor of safety (F), the ratio of resisting forces to driving forces. A value above 1.0 indicates stability.
边坡稳定性常用安全系数（F）表示，即抗滑力与滑动力之比。数值大于1.0表明稳定。
F = Resisting Forces / Driving Forces
A more detailed limit equilibrium formula incorporates cohesion, effective stress and internal friction angle.
更详细的极限平衡公式包含粘聚力、有效应力和内摩擦角。
F = (c × L + (W cos θ – uL) tan φ) / (W sin θ)
Where c is cohesion, L is the length of the slip surface, W is the weight of the slice, θ is the slope angle, u is pore water pressure, and φ is the angle of internal friction. Increased pore pressure from heavy rainfall reduces F, triggering landslides.
式中c为粘聚力，L为滑动面长度，W为滑体重量，θ为坡角，u为孔隙水压力，φ为内摩擦角。强降雨导致的孔隙压力增加会降低F，触发滑坡。
Shear strength (τ) obeys the Coulomb-Terzaghi equation, essential for understanding failure.
τ = c + (σ – u) tan φ
Where σ is the total normal stress. Vegetation can enhance c and reduce u, improving stability.
式中σ为总法向应力。植被可增强c、减小u，从而提高稳定性。
8. Soil Erosion Equations | 土壤侵蚀方程
The Universal Soil Loss Equation (USLE) predicts average annual soil loss from rill and interrill erosion on a hillslope.
通用土壤流失方程（USLE）预测坡面细沟和细沟间侵蚀的年均土壤流失量。
A = R × K × LS × C × P
- R – rainfall erosivity factor (MJ mm ha^(-1) h^(-1) yr^(-1))
- K – soil erodibility factor (t ha h ha^(-1) MJ^(-1) mm^(-1))
- LS – slope length and steepness factor (dimensionless)
- C – cover and management factor (0 to 1)
- P – support practice factor (0 to 1)
- A – computed soil loss (t ha^(-1) yr^(-1))
各因子：R（降雨侵蚀力因子）、K（土壤可蚀性因子）、LS（坡度坡长因子）、C（覆盖与管理因子）、P（水土保持措施因子），A为计算出的土壤流失量（吨/公顷·年）。
The revised RUSLE has improved subfactors, but the multiplicative structure remains unchanged.
修订版RUSLE细化了子因子，但乘积结构不变。
Published by TutorHao | Pre-U Geography Revision Series | aleveler.com更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导