SQA Higher Mathematics: Key Terminology Quick Reference Guide | SQA 进阶数学：核心术语速记指南

📚 SQA Higher Mathematics: Key Terminology Quick Reference Guide | SQA 进阶数学：核心术语速记指南

Mastering the precise vocabulary of Higher Mathematics is not just about passing the exam; it is the key to interpreting questions correctly and conveying your reasoning with clarity. This quick-reference guide organises essential terms by topic and pairs each with a memorable mnemonic to speed up your revision. Use it to build confidence and ensure you never confuse a scalar with a vector or a tangent with a normal.

掌握进阶数学的精确术语不只是为了通过考试，更是准确理解题意、清晰表达推理过程的关键。这份速记指南按主题梳理核心词汇，并为每个术语搭配了易记的助记方法，帮你提高复习效率。用它建立信心，确保你再也不会混淆标量与向量，或切线与法线。

1. Differentiation Terminology | 微分学术语速记

Differentiation focuses on rates of change and slopes of curves. The terms below form the foundation of calculus at Higher level.

微分学研究变化率与曲线的斜率。下面这些术语构成了高阶微积分的基础。

Derivative (f'(x) or dy/dx): f the tangent at any point. If s(t) is displacement, the derivative is velocity. Mnemonic: "Derivative = slope of tangent, instant by instant."

导数 (f'(x) 或 dy/dx)：函数的瞬时变化率，给出任意点切线的斜率。若 s(t) 为位移，则导数就是速度。速记：“导数即切线斜率，时刻在变。”

Differentiation from First Principles: Finding the derivative directly from the limit definition: f'(x) = lim_h→0 -f(x))/h. It reminds us that a derivative is the limit of a difference quotient. Mnemonic: "First Principles = limit of tiny changes."

第一原理求导：直接从极限定义求导数：f'(x) = lim_h→0 (f(x+h)-f(x))/h。它提醒我们导数是一个差商的极限。速记：“第一原理=微小变化的极限。”

Stationary Point: A point where f' rizontal, and the function could have a maximum, minimum or point of inflection. Mnemonic: "Stationary means slope zero – think of a car pausing at traffic lights."

驻点：使 f'(x)=0 的点。此处切线水平，函数可能有极大值、极小值或拐点。速记：“驻点意味着斜率为零——想象汽车在红灯前暂停。”

Chain Rule: ou peel layers like an onion: differentiate the outer function, then multiply by the derivative of the inner function. Mnemonic: "Outside derivative times inside derivative."

链式法则：用于复合函数求导：dy/dx = dy/du × du/dx。让你像剥洋葱一样分层：先对外层求导，再乘以内层函数的导数。速记：“外导乘内导。”

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2. Integration Terminology | 积分学术语速记

Integration reverses differentiation and deals with accumulation and area. These terms keep your working accurate.

积分是微分的逆运算，处理累积与面积。熟记这些术语能让你的解答准确无误。

Indefinite Integral: The set of all antiderivatives of a function, written ∫ f(x) dx = F(x) + C. The 'dx' tells you the variable of integration. Mnemonic: "Indefinite integral = general antiderivative; don't forget +C."

不定积分：函数的所有原函数的集合，记作 ∫ f(x) dx = F(x) + C。'dx' 指明积分变量。速记：“不定积分=一般原函数，千万别忘了 +C。”

Constant of Integration (+C): Because the derivative of a constant is zero, every indefinite integral must include an arbitrary constant. Losing +C loses marks. Mnemonic: "+C: the constant that never forgets differentiation's loss."

积分常数 (+C)：由于常数的导数为零，每个不定积分必须包含一个任意常数。忘写 +C 就会失分。速记：“+C：永远不会遗忘微分丢失的常数。”

Definite Integral: Evaluates the net area between a curve and the x-axis from a to b: ∫_a^b f(x) dx = F(b)-F(a). The constant cancels out. Mnemonic: "Definite integral = signed area, boundaries included."

定积分：计算曲线与 x 轴之间从 a 到 b 的净面积：∫_a^b f(x) dx = F(b)-F(a)。积分常数在此抵消。速记：“定积分=带符号的面积，边界包含在内。”

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3. Vectors Terminology | 向量术语速记

Vectors have both magnitude and direction, making them essential for describing positions, displacements and forces in space.

向量既有大小又有方向，是描述空间中位置、位移和力的关键。

Vector Notation (Component Form): A 3D vector is written as ai + bj + ck or as a column vector. i, j, k are unit vectors along the axes. Mnemonic: "i j k: the three directions of space."

向量记法（分量形式）：三维向量写作 ai + bj + ck 或列向量。i, j, k 是沿坐标轴的单位向量。速记：“i j k：空间的三个方向。”

Magnitude of a Vector: The length of a vector v = ai+bj+ck is |v| = √a²+b²+c². It is always non‑negative. Mnemonic: "Magnitude = 3D Pythagoras – square, sum, square root."

向量的模：向量 v = ai+bj+ck 的长度为 |v| = √a²+b²+c²，恒为非负数。速记：“模=三维勾股定理——平方、求和、开根号。”

Scalar (Dot) Product: a·b = |a||b| a₁b₁+a₂b₂+a₃b₃. It tests perpendicularity: a·b = 0 means vectors are orthogonal. Mnemonic: "Dot product connects magnitude and angle; zero dot means right angle."

数量积（点积）：a·b = |a||b| cos θ，也可通过分量计算 a₁b₁+a₂b₂+a₃b₃。它可检验垂直：a·b = 0 意味着两向量正交。速记：“点积连接大小与角度；点积为零即垂直。”

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4. Matrices Terminology | 矩阵术语速记

Matrices simplify solving linear systems and performing transformations. Knowing the language of matrices avoids confusion with their operations.

矩阵可以简化线性方程组的求解和变换的执行。熟悉矩阵语言能避免运算时的混淆。

Order of a Matrix: Given as rows × columns. A 2×3 matrix has 2 rows and 3 columns. Mnemonic: "Row first, column second – RC like remote control."

矩阵的阶：表示为行数 × 列数。2×3 矩阵有 2 行 3 列。速记：“先行后列——记作 RC，像遥控器 (Remote Control)。”

Transpose (A^T): Flips rows into columns. If A = [1 2; 3 4], A^T = [1 3; 2 4]. Mnemonic: "Transpose tows the rows into columns."

转置 (A^T)：将行翻转为列。若 A = [1 2; 3 4]，则 A^T = [1 3; 2 4]。速记：“转置把行拖到列的位置。”

Determinant (for 2×2): t = ad - bc. A zero determinant means the matrix is singular and has no inverse. Mnemonic: "Determinant = ad minus bc – if zero, matrix is singular."

行列式（2×2）：对于矩阵 [[a, b], [c, d]]，det = ad - bc。行列式为零意味着矩阵是奇异的，不存在逆矩阵。速记：“行列式 = ad 减 bc ——若为零，矩阵奇异。”

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5. Trigonometric Functions Terminology | 三角函数术语速记

Trigonometry at Higher level goes beyond right triangles into periodic functions and exact values in radian measure.

进阶阶段的三角函数已超越直角三角形，进入了周期函数和弧度制下的精确值。

Radian Measure: an arc equal to the radius. π rad = 180°. Radian mode is essential for calculus. Mnemonic: "Radians: the circle's natural language – 2π around."

弧度制：1 弧度的角所对的弧长等于半径。π 弧度 = 180°。微积分中必须使用弧度模式。速记：“弧度：圆的自然语言——一圈 2π。”

Exact Values: Know sin, cos and tan of 0, π/6, π/4, π/3 and π/2 without a calculator. Order sin: 0, ½, √2/2, √3/2, 1. Mnemonic: "For sine, the sequence goes 0, half, root two over two, root three over two, one – think of a rising slope."

精确值：熟记 0、π/6、π/4、π/3 和 π/2 的正弦、余弦和正切值，无需计算器。正弦值的顺序：0, ½, √2/2, √3/2, 1。速

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