Sections 1 to 7: vector functions and space curves, derivatives and integrals, arc length and reparametrization, curvature, the TNB frame ($\mathbf{T}$, $\mathbf{N}$, $\mathbf{B}$), velocity and acceleration, tangential and normal componentsCALC III第1至7节：向量函数与空间曲线、导数与积分、弧长与重参数化、曲率、TNB标架（$\mathbf{T}$、$\mathbf{N}$、$\mathbf{B}$）、速度与加速度、切向与法向分量CALC III
Name:姓名：Date:日期：
PART I · CORE TECHNIQUES第I部分 · 核心技巧Computational fluency · 28 marks计算熟练度 · 28分
Derivatives, Integrals, and the TNB Frame导数、积分与TNB标架
Show all componentwise working. Normalize correctly when computing unit vectors: verify $|\mathbf{T}|=1$ and $|\mathbf{N}|=1$ before moving on. Where a result is a vector, state each component explicitly.写出所有分量计算过程。计算单位向量时须正确规范化：继续解题前验证 $|\mathbf{T}|=1$ 且 $|\mathbf{N}|=1$。若结果为向量，请逐一写出每个分量。
Q1MEDIUMCOREderivatives, integrals, and limits of vector functions向量函数的导数、积分与极限[8 marks]
Let $\mathbf{r}(t)=\langle t^{2},\, e^{2t},\, \ln(t+1)\rangle$.
(a)Find $\mathbf{r}'(t)$ and $\mathbf{r}''(t)$.求 $\mathbf{r}'(t)$ 与 $\mathbf{r}''(t)$。[3]
(b)Evaluate $\displaystyle\int_{0}^{1}\mathbf{r}(t)\,dt$ as a single constant vector.将 $\displaystyle\int_{0}^{1}\mathbf{r}(t)\,dt$ 化为一个常数向量。[3]
(c)State the natural domain of $\mathbf{r}(t)$ and evaluate $\displaystyle\lim_{t\to 0}\mathbf{r}(t)$.写出 $\mathbf{r}(t)$ 的自然定义域，并求 $\displaystyle\lim_{t\to 0}\mathbf{r}(t)$。[2]
Q2MEDIUMCOREunit tangent vector and tangent line to a space curve空间曲线的单位切向量与切线[8 marks]
Let $\mathbf{r}(t)=\langle 3\cos t,\, 3\sin t,\, 4t\rangle$.
(a)Compute $\mathbf{r}'(t)$ and $|\mathbf{r}'(t)|$.计算 $\mathbf{r}'(t)$ 与 $|\mathbf{r}'(t)|$。[2]
(b)Find the unit tangent vector $\mathbf{T}(t)$.求单位切向量 $\mathbf{T}(t)$。[2]
(c)Write a parametric equation for the tangent line to the curve at $t=0$.写出曲线在 $t=0$ 处切线的参数方程。[2]
(d)Find the arc length of one full turn of this helix (from $t=0$ to $t=2\pi$).求该螺旋线转一整圈（从 $t=0$ 到 $t=2\pi$）的弧长。[2]
Q3HARDCOREunit normal vector $\mathbf{N}$ and binormal vector $\mathbf{B}$主法向量 $\mathbf{N}$ 与副法向量 $\mathbf{B}$[12 marks]
For the circular helix $\mathbf{r}(t)=\langle \cos t,\, \sin t,\, t\rangle$, use the unit tangent found from $\mathbf{T}=\mathbf{r}'/|\mathbf{r}'|$.对于圆柱螺旋线 $\mathbf{r}(t)=\langle \cos t,\, \sin t,\, t\rangle$，利用 $\mathbf{T}=\mathbf{r}'/|\mathbf{r}'|$ 求单位切向量。
(b)Compute $\mathbf{T}'(t)$ and $|\mathbf{T}'(t)|$, then write the principal unit normal $\mathbf{N}(t)=\mathbf{T}'(t)/|\mathbf{T}'(t)|$. Verify that $|\mathbf{N}|=1$ and $\mathbf{N}\cdot\mathbf{T}=0$.计算 $\mathbf{T}'(t)$ 与 $|\mathbf{T}'(t)|$，再写出主单位法向量 $\mathbf{N}(t)=\mathbf{T}'(t)/|\mathbf{T}'(t)|$，并验证 $|\mathbf{N}|=1$ 且 $\mathbf{N}\cdot\mathbf{T}=0$。[5]
(c)Compute the binormal vector $\mathbf{B}(t)=\mathbf{T}(t)\times\mathbf{N}(t)$ and verify $|\mathbf{B}|=1$.计算副法向量 $\mathbf{B}(t)=\mathbf{T}(t)\times\mathbf{N}(t)$，并验证 $|\mathbf{B}|=1$。[5]
PART II · DEFINITIONS AND PROOF第II部分 · 定义与证明Rigorous arguments · 26 marks严格论证 · 26分
Proofs and Derivations证明与推导
Supply complete proofs. Every claim about perpendicularity or magnitude must be established by computation, not just asserted. Mark calls (M1/A1/R1/B1) indicate the intended structure.给出完整证明。关于垂直性或模长的每项论断必须通过计算来建立，不能仅凭断言。评分标记（M1/A1/R1/B1）说明预期结构。
Q4HARDPROOFconstant-magnitude vector perpendicular to its derivative模长恒定的向量与其导数垂直[8 marks]
Throughout this question, $\mathbf{r}(t)$ is a differentiable vector function.本题中，$\mathbf{r}(t)$ 为可微向量函数。
(a)Prove that if $|\mathbf{r}(t)|=c$ is constant, then $\mathbf{r}(t)\cdot\mathbf{r}'(t)=0$ for all $t$. Your proof must use the product rule for the dot product; do not assume the result.证明：若 $|\mathbf{r}(t)|=c$ 为常数，则对所有 $t$ 有 $\mathbf{r}(t)\cdot\mathbf{r}'(t)=0$。证明必须使用点积的乘积法则，不得直接援引结论。[4]
(b)State the converse and prove it.写出逆命题并加以证明。[2]
(c)Deduce that the unit tangent vector $\mathbf{T}(t)$ is always perpendicular to $\mathbf{T}'(t)$, and explain in one sentence why this is the geometric basis for defining $\mathbf{N}=\mathbf{T}'/|\mathbf{T}'|$.由此推断单位切向量 $\mathbf{T}(t)$ 始终与 $\mathbf{T}'(t)$ 垂直，并用一句话解释这为何是定义 $\mathbf{N}=\mathbf{T}'/|\mathbf{T}'|$ 的几何依据。[2]
Q5HARDPROOFproduct rules for dot and cross products of vector functions向量函数点积与叉积的乘积法则[8 marks]
Let $\mathbf{u}(t)$ and $\mathbf{v}(t)$ be differentiable vector functions in $\mathbb{R}^{3}$, and write their components $\mathbf{u}=\langle u_{1},u_{2},u_{3}\rangle$, $\mathbf{v}=\langle v_{1},v_{2},v_{3}\rangle$.设 $\mathbf{u}(t)$ 与 $\mathbf{v}(t)$ 为 $\mathbb{R}^{3}$ 中的可微向量函数，分量写作 $\mathbf{u}=\langle u_{1},u_{2},u_{3}\rangle$，$\mathbf{v}=\langle v_{1},v_{2},v_{3}\rangle$。
(a)Using only the componentwise definition of the dot product, prove the product rule $\dfrac{d}{dt}[\mathbf{u}\cdot\mathbf{v}]=\mathbf{u}'\cdot\mathbf{v}+\mathbf{u}\cdot\mathbf{v}'$.仅使用点积的分量定义，证明乘积法则 $\dfrac{d}{dt}[\mathbf{u}\cdot\mathbf{v}]=\mathbf{u}'\cdot\mathbf{v}+\mathbf{u}\cdot\mathbf{v}'$。[4]
(b)State (but do not prove) the product rule for $\dfrac{d}{dt}[\mathbf{u}\times\mathbf{v}]$ and give one example showing that the order of the factors in the first term cannot be reversed.写出（但无需证明） $\dfrac{d}{dt}[\mathbf{u}\times\mathbf{v}]$ 的乘积法则，并举一例说明第一项中因子的顺序不可对调。[2]
(c)Let $\mathbf{u}(t)=\langle t,\, t^{2},\, 0\rangle$ and $\mathbf{v}(t)=\langle \sin t,\, 0,\, 1\rangle$. Verify the product rule for $\dfrac{d}{dt}[\mathbf{u}\cdot\mathbf{v}]$ by computing both sides independently at $t=0$.设 $\mathbf{u}(t)=\langle t,\, t^{2},\, 0\rangle$，$\mathbf{v}(t)=\langle \sin t,\, 0,\, 1\rangle$。在 $t=0$ 处独立计算两侧，验证 $\dfrac{d}{dt}[\mathbf{u}\cdot\mathbf{v}]$ 的乘积法则。[2]
Q6HARDPROOFdecomposition of acceleration into tangential and normal components加速度的切向与法向分量分解[10 marks]
A particle moves along a smooth curve $\mathbf{r}(t)$. Let $v=|\mathbf{r}'(t)|$ denote the speed, $\mathbf{T}$ the unit tangent, and $\mathbf{N}$ the principal unit normal.一质点沿光滑曲线 $\mathbf{r}(t)$ 运动。设 $v=|\mathbf{r}'(t)|$ 为速率，$\mathbf{T}$ 为单位切向量，$\mathbf{N}$ 为主单位法向量。
(a)Starting from $\mathbf{v}=v\mathbf{T}$, differentiate to show that the acceleration decomposes as
$$\mathbf{a}=a_{T}\mathbf{T}+a_{N}\mathbf{N},\qquad a_{T}=\frac{dv}{dt},\quad a_{N}=\kappa v^{2},$$
where $\kappa=|\mathbf{T}'|/v$ is the curvature. Clearly justify each step.从 $\mathbf{v}=v\mathbf{T}$ 出发，对其微分，证明加速度可分解为
$$\mathbf{a}=a_{T}\mathbf{T}+a_{N}\mathbf{N},\qquad a_{T}=\frac{dv}{dt},\quad a_{N}=\kappa v^{2},$$
其中曲率 $\kappa=|\mathbf{T}'|/v$。每一步须清晰说明理由。[6]
(b)Using the result of part (a), express $a_{T}$ and $a_{N}$ in terms of $\mathbf{r}'$ and $\mathbf{r}''$ only (no $\mathbf{T}$ or $\mathbf{N}$):
$$a_{T}=\frac{\mathbf{r}'\cdot\mathbf{r}''}{|\mathbf{r}'|},\qquad a_{N}=\frac{|\mathbf{r}'\times\mathbf{r}''|}{|\mathbf{r}'|}.$$
You may use $\mathbf{a}=a_{T}\mathbf{T}+a_{N}\mathbf{N}$ and properties of the dot and cross products.利用(a)的结论，仅用 $\mathbf{r}'$ 与 $\mathbf{r}''$ 表达 $a_{T}$ 与 $a_{N}$（不得含 $\mathbf{T}$ 或 $\mathbf{N}$）：
$$a_{T}=\frac{\mathbf{r}'\cdot\mathbf{r}''}{|\mathbf{r}'|},\qquad a_{N}=\frac{|\mathbf{r}'\times\mathbf{r}''|}{|\mathbf{r}'|}.$$
可使用 $\mathbf{a}=a_{T}\mathbf{T}+a_{N}\mathbf{N}$ 及点积与叉积的性质。[4]
PART III · APPLICATIONS AND SYNTHESIS第III部分 · 应用与综合Extended problems · 28 marks综合题 · 28分
Curvature, Motion in Space, and Arc-Length Reparametrization曲率、空间运动与弧长重参数化
Carry exact forms throughout; simplify only at the final step. For curvature problems, state which formula you use and show the cross product or magnitude computation in full.全程保持精确形式，仅在最后一步化简。对于曲率题，须注明所用公式，并完整写出叉积或模长的计算过程。
Q7MEDIUMAPPLIEDcurvature of a plane curve and a space curve平面曲线与空间曲线的曲率[8 marks]
Recall the curvature formulas: for a space curve, $\kappa=|\mathbf{r}'\times\mathbf{r}''|/|\mathbf{r}'|^{3}$; for a plane curve $y=f(x)$, $\kappa=|y''|/(1+y'^{2})^{3/2}$.回顾曲率公式：对于空间曲线，$\kappa=|\mathbf{r}'\times\mathbf{r}''|/|\mathbf{r}'|^{3}$；对于平面曲线 $y=f(x)$，$\kappa=|y''|/(1+y'^{2})^{3/2}$。
(a)Find the curvature $\kappa$ of the parabola $y=x^{2}$ at the vertex $x=0$ and at $x=1$.求抛物线 $y=x^{2}$ 在顶点 $x=0$ 及 $x=1$ 处的曲率 $\kappa$。[4]
(b)Find the curvature of the circular helix $\mathbf{r}(t)=\langle a\cos t,\, a\sin t,\, bt\rangle$ (with $a,b>0$) using the cross-product formula. Show the cross product $\mathbf{r}'\times\mathbf{r}''$ in full and simplify to a closed form in $a$ and $b$.用叉积公式求圆柱螺旋线 $\mathbf{r}(t)=\langle a\cos t,\, a\sin t,\, bt\rangle$（$a,b>0$）的曲率。完整写出叉积 $\mathbf{r}'\times\mathbf{r}''$，并化简为关于 $a$ 与 $b$ 的封闭表达式。[4]
Q8HARDAPPLIEDcurvature via the cross-product formula for a polynomial space curve用叉积公式求多项式空间曲线的曲率[8 marks]
Consider the space curve $\mathbf{r}(t)=\langle t,\, t^{2},\, \tfrac{2}{3}t^{3}\rangle$.考虑空间曲线 $\mathbf{r}(t)=\langle t,\, t^{2},\, \tfrac{2}{3}t^{3}\rangle$。
(a)Compute $\mathbf{r}'(t)$ and $\mathbf{r}''(t)$.计算 $\mathbf{r}'(t)$ 与 $\mathbf{r}''(t)$。[1]
(b)Compute the cross product $\mathbf{r}'(t)\times\mathbf{r}''(t)$ explicitly.显式计算叉积 $\mathbf{r}'(t)\times\mathbf{r}''(t)$。[3]
(c)Compute $|\mathbf{r}'(t)\times\mathbf{r}''(t)|$ and $|\mathbf{r}'(t)|$. Hence find the curvature $\kappa(t)$.计算 $|\mathbf{r}'(t)\times\mathbf{r}''(t)|$ 与 $|\mathbf{r}'(t)|$，进而求曲率 $\kappa(t)$。[3]
(d)Evaluate $\kappa(0)$ and interpret geometrically.求 $\kappa(0)$ 并给出几何解释。[1]
Q9HARDAPPLIEDarc-length reparametrization of a helix螺旋线的弧长重参数化[6 marks]
Consider the helix $\mathbf{r}(t)=\langle 3\cos t,\, 3\sin t,\, 4t\rangle$ from Question 2.考虑第2题中的螺旋线 $\mathbf{r}(t)=\langle 3\cos t,\, 3\sin t,\, 4t\rangle$。
(a)Using $|\mathbf{r}'(t)|$ from Question 2, write the arc length function $s(t)=\int_{0}^{t}|\mathbf{r}'(u)|\,du$ and solve for $t$ in terms of $s$.利用第2题中的 $|\mathbf{r}'(t)|$，写出弧长函数 $s(t)=\int_{0}^{t}|\mathbf{r}'(u)|\,du$，并将 $t$ 用 $s$ 表示。[2]
(b)Write out $\mathbf{r}(s)$, the arc-length parametrization of the helix.写出螺旋线的弧长参数化 $\mathbf{r}(s)$。[2]
(c)Verify that $\left|\dfrac{d\mathbf{r}}{ds}\right|=1$.验证 $\left|\dfrac{d\mathbf{r}}{ds}\right|=1$。[2]
Q10HARDAPPLIEDmotion in space: velocity, acceleration, and $a_T$/$a_N$ components空间运动：速度、加速度与 $a_T$/$a_N$ 分量[6 marks]
A particle has position $\mathbf{r}(t)=\langle t^{2}-1,\; 2t,\; t^{2}\rangle$ at time $t\ge 0$.一质点在时刻 $t\ge 0$ 的位置为 $\mathbf{r}(t)=\langle t^{2}-1,\; 2t,\; t^{2}\rangle$。
(a)Find the velocity $\mathbf{v}(t)$, acceleration $\mathbf{a}(t)$, and speed $v(t)=|\mathbf{v}(t)|$.求速度 $\mathbf{v}(t)$、加速度 $\mathbf{a}(t)$ 及速率 $v(t)=|\mathbf{v}(t)|$。[2]