FP3向量偏微分全解析 | OCR MEI FP3 Vectors & Partial Diff

📘 OCR MEI FP3 是 A-Level 进阶纯数的核心模块，向量代数与偏微分（Partial Differentiation）更是高频考点。本文结合 2010 年 6 月真题评分标准，带你逐一拆解解题思路，避免常见扣分陷阱。

📘 OCR MEI FP3 is a core module of A-Level Further Pure Mathematics, and Vectors + Partial Differentiation are among the most frequently tested topics. This article uses the June 2010 mark scheme to break down key solution techniques and help you avoid common pitfalls that cost marks.

🔢 向量代数 / Vector Algebra

考点一：点到直线的垂直距离

给定点 C 和直线 AB，垂直距离公式为：
d = |AC × AB| / |AB|

解题三步走：① 计算向量 AC 和 AB；② 求叉积 AC × AB（每个分量独立计分，B1 一个分量正确即得分）；③ 分别计算叉积和 AB 的模长，相除即得距离。注意：叉积分量计算错误时，只要模长计算逻辑正确，后续仍有 follow-through 分。

Key Point 1: Perpendicular distance from point to line

Given point C and line AB, the perpendicular distance is d = |AC × AB| / |AB|.

Three-step solution: ① Compute vectors AC and AB; ② Find cross product AC × AB (each component earns B1 if correct); ③ Divide the magnitude of the cross product by |AB|. Tip: Even if a cross product component is wrong, you can still earn follow-through marks if the magnitude calculation is correct.

考点二：标量三重积与体积

四面体体积公式：V = (1/6) |(AC × AB) · AD|

标量三重积的计算是 FP3 必考题型。先算叉积，再点积，最后取绝对值的 1/6。评分标准明确：叉积正确得 M1A1，点积展开得 M1，化简得 A1——每步都有独立分数，即使最终答案出错，中间步骤照样拿分。

Key Point 2: Scalar triple product and volume

Tetrahedron volume: V = (1/6) |(AC × AB) · AD|. The scalar triple product is a guaranteed exam question. Compute the cross product first (M1A1), then the dot product (M1), then simplify (A1). Each step earns independent marks — even if the final answer is wrong, you still get credit for correct intermediate work.

📐 偏微分 / Partial Differentiation

考点三：多变量函数的偏导

对于函数 g(x, y, z) = (y + xyz²)e^(x+2y)：

∂g/∂x = (yz²)(e^(x+2y)) + (y + xyz²)(e^(x+2y)) — 乘积法则 + 链式法则
∂g/∂y = (1 + xz²)(e^(x+2y)) + (y + xyz²)(2e^(x+2y))
∂g/∂z = 2xyz·e^(x+2y)

评分标准中，每个偏导 独立计分 M1A1——三个偏导就是 3×2=6 分。即使一个偏导出错，其他两个对仍能拿满分。考生常犯错误：忘记链式法则中 ∂(x+2y)/∂y = 2 而非 1。

Key Point 3: Partial derivatives of multivariable functions

For g(x, y, z) = (y + xyz²)e^(x+2y), use product rule + chain rule. Each partial derivative earns independent M1A1 marks — 3 derivatives × 2 marks = 6 marks total. Common mistake: forgetting that ∂(x+2y)/∂y = 2 (not 1) in the chain rule.

考点四：法向量与切线

梯度向量 ∇g = (∂g/∂x, ∂g/∂y, ∂g/∂z) 在给定点处的值即为曲面的法向量方向。过该点沿此法向的直线就是法线。真题中常见问法：”证明某点在法线上”——只需验证该点坐标满足法线参数方程。

Key Point 4: Normal vectors and normal lines

The gradient ∇g at a point gives the direction of the normal vector to the surface. The line through the point with this direction is the normal line. A common exam question: “Show that point P lies on the normal line” — simply verify that P’s coordinates satisfy the parametric equation of the normal line.

💡 学习建议 / Study Tips

• 📝 勤练叉积：FP3 向量题中 80% 涉及叉积运算，建议每天手算 3-5 个，培养肌肉记忆。
• 📝 Practice cross products daily: 80% of FP3 vector problems involve them. Hand-calculate 3-5 per day to build muscle memory.
• 🔍 读懂评分标准：OCR MEI 的 Mark Scheme 明确标注了每步的 M1（方法分）和 A1（答案分）——即使算错，只要方法对就有分。
• 🔍 Study the mark scheme: OCR MEI clearly labels M1 (method) and A1 (accuracy) marks — you earn credit for correct methods even with arithmetic errors.
• 🧮 偏微分检查清单：① 确认变量个数 ② 对目标变量求导时其余视为常数 ③ 乘积法则/链式法则逐项检查。
• 🧮 Partial differentiation checklist: ① Identify all variables ② Treat others as constants ③ Apply product/chain rule term by term.

📞 联系方式 / Contact
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