alevel高数应不应该选？一文详解alevel高数考纲/alevel高数教材/alevel高数真题领取！ |
| 时间：2024-10-29 11:07:08 作者：网络 来源：网络 |
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首先看一下高数的学习内容，以CIE考试局为例，高数分为四个部分：Further Pure Mathematics 1, Further Pure Mathematics 2, Further Probability and Statistics, Further Mechanics. 每个部分都是对Alevel数学中所学内容对加深和拓展，具体大纲如下：
Further Pure Mathematics:
1. Complex Numbers:
(1) Argand diagrams,
(2) Roots of complex numbers,
(3) De Moivre's theorem.
2. Matrices and Transformations:
(1) Matrix algebra,
(2) Determinants,
(3) Inverse matrices,
(4) Linear transformations.
3. Proof:
(1) Direct and indirect proofs,
(2) Proof by contradiction,
(3) Mathematical induction.
4. Differential Equations:
(1) First and second-order differential equations,
(2) Homogeneous and non-homogeneous equations.
5. Polar Coordinates and Hyperbolic Functions:
(1) Conversion between Cartesian and polar coordinates,
(2) Hyperbolic functions, their derivatives, and integrals.
6. Further Differentiation and Integration:
(1) Differentiation rules,
(2) Integration techniques.
7. Differential Equations:
Second-order linear differential equations, variation of parameters.
8. Further Complex Numbers:
(1) Complex functions,
(2) Complex integration,
(3) Residues.
9. Further Algebra:
(1) Matrices, determinants, eigenvalues, eigenvectors,
(2) Vector spaces.
Further Probability and Statistics
1. Probability:
(1) Counting principles,
(2) Conditional probability,
(3) Binomial distribution
(4) Poisson distribution.
2. Statistical Distributions:
(1) Normal distribution,
(2) Hypothesis testing,
(3) Confidence intervals.
3. Regression and Correlation:
(1) Linear regression,
(2) Correlation coefficient,
(3) Least squares method.
4. Probability Generating Functions:
Generating functions for discrete random variables.
5. Multivariate Distributions:
Joint and conditional probability distributions, covariance, independence.
Further Mechanics
1. Vectors in Mechanics:
(1) Scalars and vectors, vector algebra,
(2) Resolution of forces,
(3) Equilibrium,
(4) Moments.
2. Kinematics:
(1) Displacement, velocity, acceleration,
(2) Projectile motion.
3. Newton's Laws of Motion:
(1) Force, mass, inertia, friction,
(2) Momentum and impulse.
4. Work, Energy, and Power:
(1) Work done by a force,
(2) Kinetic and potential energy,
(3) Power.
5. Circular Motion and Simple Harmonic Motion:
(1) Centripetal force,
(2) Angular displacement,
(3) Simple harmonic motion,
(4) Damping.
6. Elasticity and Fluid Mechanics:
(1) Hooke's Law,
(2) Young modulus,
(3) Fluid pressure,
(4) Archimedes' principle.
可以看出高数知识点的深度和难度还是比较大的，因此在选择这门课时要进行多方考虑。首先，高数对于那些对数学有强烈兴趣和天赋的学生来说是一个很好的选择。学习高数扩展了高中数学的范围，涵盖了更高级的数学概念和技巧。选择高数可以让学生更深入地探索数学的美妙世界，并为将来在数学相关领域进一步学习和研究打下坚实的基础。
👉重点：
复杂的微积分：包括多变量微积分和微分方程，学生需要更深入地理解和运用微积分。
高级的纯数学：如群论、向量空间等，需要学生具备更高的抽象思维和数学建模能力。
高阶力学：学生将面对更复杂的运动问题，需要应用微积分和代数知识解决。
👉难点：
数学证明的深度：进阶数学更注重学生对于定理和推论的深刻理解，能够独立完成数学证明。
抽象思维：学生需要适应更为抽象和复杂的数学概念，具备将其运用到实际问题的能力。
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