Year 13 SQA Advanced Higher Mathematics: Key Terminology Quick Reference | Year 13 SQA 进阶数学：词汇术语速记指南

📚 Year 13 SQA Advanced Higher Mathematics: Key Terminology Quick Reference | Year 13 SQA 进阶数学：词汇术语速记指南

Mastering the precise language of Advanced Higher Mathematics is essential for success in the SQA examination. This guide provides a concise yet comprehensive glossary of core terms across all major topics, enabling you to recall definitions quickly and apply them accurately in problem solving.

掌握进阶高等数学的精确语言是 SQA 考试成功的关键。本指南提供了一份简洁而全面的核心术语词汇表，涵盖所有主要主题，帮助您快速回忆定义并在解题中准确应用。

1. Differential Calculus | 微分学

Derivative: The instantaneous rate of change of a function, denoted f'(x) or dy/dx. For y = xⁿ, the derivative is n xⁿ⁻¹.

导数：函数瞬时变化率，记作 f'(x) 或 dy/dx。对 y = xⁿ，导数为 n xⁿ⁻¹。

Stationary point: A point where f'(x) = 0; can be a local maximum, local minimum or a point of inflection.

驻点：满足 f'(x) = 0 的点；可能是局部极大、局部极小或拐点。

Point of inflection: A point where the curvature changes sign; f”(x) = 0 is a necessary condition but a sign change of f”(x) must be verified.

拐点：曲率符号改变的点；f”(x) = 0 是必要条件，但必须验证 f”(x) 变号。

Implicit differentiation: Technique used when y is not given explicitly as a function of x; differentiate both sides with respect to x, applying the chain rule to terms with y.

隐函数微分：当 y 未显式表达为 x 的函数时使用的技巧；关于 x 对等式两边求导，对含 y 的项应用链式法则。

Parametric differentiation: When x and y are given in terms of a parameter t, dy/dx = (dy/dt) / (dx/dt) provided dx/dt ≠ 0.

参数微分：当 x 与 y 用参数 t 表示时，dy/dx = (dy/dt) / (dx/dt)，只要 dx/dt ≠ 0。

Second derivative: The derivative of the derivative, denoted f”(x), d²y/dx²; used to test concavity and classify stationary points.

二阶导数：导数的导数，记作 f”(x)、d²y/dx²；用于检验凹凸性及对驻点分类。

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2. Integral Calculus | 积分学

Indefinite integral: The general antiderivative of a function, represented as ∫ f(x) dx = F(x) + C, where F'(x) = f(x) and C is an arbitrary constant.

不定积分：函数的一般原函数，表示为 ∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)，C 为任意常数。

Definite integral: The signed area under the curve y = f(x) from x = a to x = b, given by ∫_a^b f(x) dx = F(b) – F(a).

定积分：曲线 y = f(x) 从 x = a 到 x = b 的带符号面积，∫_a^b f(x) dx = F(b) – F(a)。

Fundamental Theorem of Calculus: Connects differentiation and integration; if F is an antiderivative of f on [a,b], then ∫_a^b f(x) dx = F(b) – F(a).

微积分基本定理：联系微分与积分；若 F 是 f 在 [a,b] 上的一个原函数，则 ∫_a^b f(x) dx = F(b) – F(a)。

Integration by substitution: Reverse of the chain rule; set u = g(x), then ∫ f(g(x)) g'(x) dx = ∫ f(u) du.

换元积分法：链式法则的逆运算；设 u = g(x)，则 ∫ f(g(x)) g'(x) dx = ∫ f(u) du。

Integration by parts: Derived from the product rule; ∫ u dv = uv – ∫ v du, useful for products of functions and logarithmic terms.

分部积分法：由乘法法则导出；∫ u dv = uv – ∫ v du，常用于函数之积及含对数的项。

Area between curves: The area enclosed between y = f(x) and y = g(x) from a to b is ∫_a^b |f(x) – g(x)| dx.

曲线间面积：y = f(x) 与 y = g(x) 从 a 到 b 围成的面积为 ∫_a^b |f(x) – g(x)| dx。

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3. Vectors in 3D | 三维向量

Position vector: A vector from the origin to a point; denoted r = (x, y, z) or xi + yj + zk.

位置向量：从原点到某点的向量；记作 r = (x, y, z) 或 xi + yj + zk。

Scalar product (dot product): a · b = |a||b| cos θ, where θ is the angle between vectors; also a₁b₁ + a₂b₂ + a₃b₃.

数量积（点积）：a · b = |a||b| cos θ，θ 为向量夹角；也等于 a₁b₁ + a₂b₂ + a₃b₃。

Vector product (cross product): a × b gives a vector perpendicular to both a and b, magnitude |a||b| sin θ; calculated using a determinant.

向量积（叉积）：a × b 得出垂直于 a 和 b 的向量，大小为 |a||b| sin θ；使用行列式计算。

Equation of a line: r = a + t d, where a is a point on the line and d is the direction vector; t is a scalar parameter.

直线方程：r = a + t d，a 为线上一点，d 为方向向量；t 为标量参数。

Equation of a plane: Can be given as r · n = d (normal form) or written in Cartesian form ax + by + cz = k.

平面方程：可表示为 r · n = d（法线式）或笛卡儿形式 ax + by + cz = k。

Angle between vectors: cos θ = (a · b) / (|a||b|), used to find acute or obtuse angle between lines or between a line and a plane.

向量夹角：cos θ = (a · b) / (|a||b|)，用于求两线或直线与平面的锐角或钝角。

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4. Matrices & Systems of Equations | 矩阵与方程组

Matrix multiplication: The product AB is defined if the number of columns of A equals the number of rows of B; not commutative in general.

矩阵乘法：若 A 的列数等于 B 的行数，积 AB 有定义；一般不可交换。

Determinant of a 3×3 matrix: det(A) = a₁₁(a₂₂a₃₃ – a₂₃a₃₂) – a₁₂(a₂₁a₃₃ – a₂₃a₃₁) + a₁₃(a₂₁a₃₂ – a₂₂a₃₁); zero determinant indicates a singular matrix.

3×3 矩阵的行列式：det(A) = a₁₁(a₂₂a₃₃ – a₂₃a₃₂) – a₁₂(a₂₁a₃₃ – a₂₃a₃₁) + a₁₃(a₂₁a₃₂ – a₂₂a₃₁)；行列式为零时矩阵奇异。

Inverse of a matrix: A⁻¹ satisfies A A⁻¹ = I; exists only if det(A) ≠ 0. For 2×2 [a b; c d], A⁻¹ = (1/det) [d -b; -c a].

逆矩阵：A⁻¹ 满足 A A⁻¹ = I；仅当 det(A) ≠ 0 时存在。对 2×2 矩阵 [a b; c d]，A⁻¹ = (1/det) [d -b; -c a]。

Augmented matrix: Representation of a system of equations [A|b] used to perform row operations and solve linear systems via Gaussian elimination.

增广矩阵：方程组的表示 [A|b]，用于通过高斯消元法进行行运算并求解线性方程组。

Row echelon form: A staircase pattern obtained through elementary row operations, where leading coefficients are 1 and rows of zeros are at the bottom.

行阶梯形：通过初等行运算得到的阶梯状形式，首项系数为 1，零行位于底部。

Eigenvalues and eigenvectors: For a square matrix A, λ is an eigenvalue if det(A – λI) = 0; the non-zero vector v satisfying (A – λI)v = 0 is the corresponding eigenvector.

特征值与特征向量：对方阵 A，若 det(A – λI) = 0 则 λ 为特征值；满足 (A – λI)v = 0 的非零向量 v 为相应特征向量。

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5. Complex Numbers | 复数

Imaginary unit i: Defined as i² = -1; a complex number is written z = a + bi, where a, b are real numbers.

虚数单位 i：定义为 i² = -1；复数写作 z = a + bi，其中 a, b 为实数。

Complex conjugate: For z = a + bi, the conjugate is z̄ = a – bi; product z z̄ = a² + b² is real.

共轭复数：对 z = a + bi，其共轭为 z̄ = a – bi；乘积 z z̄ = a² + b² 为实数。

Modulus and argument: |z| = √(a² + b²) is the distance from origin; arg(z) = θ where tan θ = b/a (taking quadrant into account).

模与辐角：|z| = √(a² + b²) 是到原点的距离；arg(z) = θ，tan θ = b/a（须考虑象限）。

Polar form: z = r (cos θ + i sin θ) = r cis θ, where r = |z| and θ = arg(z). Useful for multiplication and powers.

极坐标形式：z = r (cos θ + i sin θ) = r cis θ，其中 r = |z|，θ = arg(z)。对乘法和乘方运算很有用。

De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for any rational n; used to find powers and roots of complex numbers.

棣莫弗定理：(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)，对任意有理数 n 成立；用于求复数的幂与根。

Fundamental Theorem of Algebra: A polynomial equation of degree n has exactly n roots in the complex number system, counting multiplicities.

代数基本定理：n 次多项式方程在复数范围内恰好有 n 个根（计重数）。

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6. Differential Equations | 微分方程

Order of a differential equation: Determined by the highest derivative present; e.g., d²y/dx² + 3dy/dx = 0 is second order.

微分方程的阶：由出现的最高阶导数决定；如 d²y/dx² + 3dy/dx = 0 是二阶。

General solution: The family of all solutions containing arbitrary constants; for a first-order equation, one constant; for second-order, two constants.

通解：含有任意常数所有解构成的族；一阶方程含一个常数，二阶方程含两个常数。

Particular solution: Obtained by applying initial or boundary conditions to the general solution to determine the constants.

特解：利用初始或边界条件代入通解确定常数后得到的解。

Separable equation: A first-order ODE that can be written dy/dx = g(x)h(y); solved by rewriting as 1/h(y) dy = g(x) dx and integrating both sides.

可分离方程：可写成 dy/dx = g(x)h(y) 的一阶常微分方程；变形为 1/h(y) dy = g(x) dx 后两边积分求解。

Integrating factor: For linear first-order ODE dy/dx + P(x)y = Q(x), the integrating factor is e^{∫ P(x) dx}; multiplies through to create an exact derivative.

积分因子：对一阶线性常微分方程 dy/dx + P(x)y = Q(x)，积分因子为 e^{∫ P(x) dx}；乘到整个方程后可化为全导数。

Second-order homogeneous ODE: a d²y/dx² + b dy/dx + c y = 0; solve using auxiliary equation a m² + b m + c = 0; real distinct, repeated or complex conjugate roots give different forms of solution.

二阶齐次常微分方程：a d²y/dx² + b dy/dx + c y = 0；利用辅助方程 a m² + b m + c = 0 求解；实根不相等、重根或共轭复根对应不同形式的解。

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7. Methods of Proof | 证明方法

Direct proof: Starting from known facts or axioms, use logical steps to arrive at the statement to be proved.

直接证明：从已知事实或公理出发，通过逻辑步骤推出所要证明的命题。

Proof by contradiction: Assume the negation of the statement is true and derive a logical inconsistency, thereby confirming the original statement must be true.

反证法：假设命题的否定为真，推导出逻辑矛盾，从而确认原命题必然为真。

Proof by induction: Used for statements about natural numbers. Base case n = 1 is verified, then assuming true for n = k, prove for n = k+1. Hence true for all n by induction.

数学归纳法：用于关于自然数的命题。验证 n = 1 的基本情况，再假设 n = k 时成立，证明 n = k+1 时也成立。由此归纳得命题对所有 n 成立。

Proof by exhaustion: Check the statement for every possible case within a finite set; often used when a small number of cases covers all possibilities.

穷举证明：检查有限集合内的每一种可能情况；常用于少量情况可涵盖所有可能时。

Counterexample: To disprove a universal statement, it is sufficient to exhibit a single example that satisfies the hypothesis but not the conclusion.

反例：要证伪一个全称命题，只需给出一个满足假设但不满足结论的实例即可。

Proof of irrationality: Classic contradiction proof that √2 is irrational; assumes √2 = p/q in lowest terms and shows this leads to both p and q being even, contradicting the assumption.

无理数证明：经典的反证法证明 √2 为无理数；假设 √2 = p/q 为最简分数，推导出 p 和 q 均为偶数，与假设矛盾。

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8. Binomial Theorem & Series Expansions | 二项式定理与级数展开

Binomial theorem for positive integer n: (a + b)ⁿ = Σ_r=0ⁿ ⁿCᵣ aⁿ⁻ʳ bʳ, where ⁿCᵣ = n!/(r!(n-r)!).

正整数指数的二项式定理：(a + b)ⁿ = Σ_r=0ⁿ ⁿCᵣ aⁿ⁻ʳ bʳ，其中 ⁿCᵣ = n!/(r!(n-r)!)。

Binomial expansion for rational exponent: (1 + x)ⁿ = 1 + n x + [n(n-1)/2!] x² + … for |x| < 1, where n can be a rational number; the series is infinite.

有理数指数的二项展开：对 |x| < 1，(1 + x)ⁿ = 1 + n x + [n(n-1)/2!] x² + ...，n 可为有理数；级数无穷。

Maclaurin series: A Taylor series expansion about x = 0: f(x) = f(0) + f'(0) x + f”(0)/2! x² + … + f⁽ⁿ⁾(0)/n! xⁿ + …

麦克劳林级数：在 x = 0 处的泰勒级数展开：f(x) = f(0) + f'(0) x + f”(0)/2! x² + … + f⁽ⁿ⁾(0)/n! xⁿ + …

Standard Maclaurin series: eˣ = 1 + x + x²/2! + x³/3! + … ; sin x = x – x³/3! + x⁵/5! – … ; cos x = 1 – x²/2! + x⁴/4! – … (all valid for all real x).

标准麦克劳林级数：eˣ = 1 + x + x²/2! + x³/3! + …；sin x = x – x³/3! + x⁵/5! – …；cos x = 1 – x²/2! + x⁴/4! – …（均对所有实数 x 成立）。

Radius of convergence: The interval |x| < R for which a power series converges; determined by the ratio test or root test.

收敛半径：幂级数收敛的区间 |x| < R 中的 R；可用比值法或根值法确定。

Partial fractions in expansion: Rewriting a rational function as a sum of simpler fractions to allow term-by-term binomial expansion for series representation.

部分分式在展开中的应用：将有理函数改写为较简单分式之和，以便逐项二项展开得到级数表示。

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9. Partial Fractions & Algebraic Manipulation | 部分分式与代数运算

Proper rational function: Degree of numerator is less than degree of denominator; if not, perform polynomial division first.

有理真分式：分子次数小于分母次数；否则先进行多项式除法。

Linear factors: For denominator (ax + b)(cx + d), assume A/(ax + b) + B/(cx + d); solve for constants A and B.

线性因子：分母为 (ax + b)(cx + d) 时，假设形式为 A/(ax + b) + B/(cx + d)；求解常数 A 和 B。

Repeated linear factors: For (ax + b)² in denominator, use A/(ax + b) + B/(ax + b)².

重复线性因子：分母含 (ax + b)² 时，使用 A/(ax + b) + B/(ax + b)² 的形式。

Irreducible quadratic factor: For (ax² + bx + c) that cannot be factorised over reals, use (Ax + B)/(ax² + bx + c).

不可约二次因子：对不能在实数范围分解的 (ax² + bx + c)，使用 (Ax + B)/(ax² + bx + c) 的形式。

Cancelling and equating coefficients: After setting up identity, multiply through by denominator and compare coefficients of like powers or substitute convenient x-values to find unknown constants.

消分母并比较系数：建立恒等式后，乘公分母，通过比较同次幂系数或代入方便的 x 值求解未知常数。

Improper fractions: Use algebraic long division to write as a polynomial plus a proper fraction, then decompose the proper part.

假分式：使用代数长除法写成多项式加真分式的形式，然后对真分式进行分解。

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10. Advanced Functions & Graphs | 进阶函数与图像

Odd and even functions: Even: f(-x) = f(x) (symmetric about y-axis); Odd: f(-x) = -f(x) (rotationally symmetric about origin).

奇函数与偶函数：偶函数：f(-x) = f(x)（关于 y 轴对称）；奇函数：f(-x) = -f(x)（关于原点旋转对称）。

Inverse function: f⁻¹(x) reverses the effect of f; graph is a reflection of y = f(x) across the line y = x. One-to-one function is required for inverse to exist.

反函数：f⁻¹(x) 逆转 f 的作用；图像是 y = f(x) 关于直线 y = x 的反射。仅当函数为一一对应时反函数存在。

Modulus function: |x| gives the absolute value; graph is V-shaped; equations with |f(x)| = a lead to f(x) = ±a; inequalities require careful case work.

模函数：|x| 给出绝对值；图像呈 V 形；方程 |f(x)| = a 等价于 f(x) = ±a；不等式需要分情况讨论。

Asymptote: A line that the graph approaches but never touches; vertical asymptote where denominator is zero; horizontal asymptote from limits at infinity.

渐近线：图像无限接近但永不相交的直线；分母为零处产生垂直渐近线；趋于无穷时的极限给出水平渐近线。

Transformations of graphs: y = f(x) + a (vertical shift), y = f(x + a) (horizontal shift), y = a f(x) (vertical stretch), y = f(ax) (horizontal compression).

图像变换：y = f(x) + a（上下平移），y = f(x + a)（左右平移），y = a f(x)（纵向伸缩），y = f(ax)（横向压缩）。

Logarithmic and exponential graphs: y = ln x passes through (1,0), domain x > 0; y = eˣ passes through (0,1); they are inverses, symmetric about y = x.

对数与指数图像：y = ln x 过 (1,0)，定义域 x > 0；y = eˣ 过 (0,1)；两者互为反函数，关于 y = x 对称。

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11. Numerical Methods & Iteration | 数值方法与迭代

Iterative formula: An equation of the form xₙ₊₁ = g(xₙ) used to approximate roots; if the sequence converges, it approaches a fixed point where x = g(x).

迭代公式：形如 xₙ₊₁ = g(xₙ) 的公式，用于逼近根；若序列收敛，将趋近满足 x = g(x) 的不动点。

Newton-Raphson method: xₙ₊₁ = xₙ – f(xₙ)/f'(xₙ); provides rapid convergence to a root provided the initial guess is sufficiently close and f'(xₙ) ≠ 0.

牛顿-拉弗森法：xₙ₊₁ = xₙ – f(xₙ)/f'(xₙ)；只要初始猜测足够接近且 f'(xₙ) ≠ 0，可快速收敛到根。

Fixed point iteration: Rearranging f(x) = 0 into the form x = g(x); convergence depends on |g'(x)| < 1 near the root.

不动点迭代：将 f(x) = 0 改写为 x = g(x) 的形式；收敛性依赖于在根附近 |g'(x)| < 1。

Trapezium rule: Approximates the definite integral by trapping the area under the curve into a series of trapezia: ∫_a^b f(x) dx ≈ (h/2)[ y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ ].

梯形法则：用一系列梯形逼近定积分：∫_a^b f(x) dx ≈ (h/2)[ y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ ]。

Numerical integration error: Can be estimated by comparing results with different step sizes; the error in the trapezium rule is often proportional to h².

数值积分误差：可通过比较不同步长的结果进行估计；梯形法则的误差常与 h² 成正比。

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