Vector-Valued Functions – Change of Parameter & Arc Length

Quick-Check Answers 12.2

• 1. Vector derivatives

  • (a) $9\mathbf i + 6\mathbf j$

  • (b) $\left( \tfrac{\sqrt2}{2},\;\tfrac{\sqrt2}{2} \right)$

• 2. Parametric velocity functions

  • (a) $\mathbf r'(t)=5\mathbf i + (1-2t)\mathbf j$

  • (b) $\mathbf r'(t)=\left( -\tfrac1{t^2},\;\sec^2 t,\;2e^{2t} \right)$

• 3. Dot products / projections

  • (a) $(6,4,2)$

  • (b) $(-4,0,4)$

  • (c) $0$

  • (d) $-28$

• 4. Miscellaneous

  • (a) $\left(1,\;\tfrac13,\;\tfrac2\pi\right)$

  • (b) Position function: $\tfrac{t^2}{2}\,\mathbf i - t^3\mathbf j + e^t\mathbf k + C$

Smooth Parametrizations

• Definition of smoothness

  • A vector-valued function $\mathbf r(t)$ is smooth if $\mathbf r'(t)$ is continuous and $\mathbf r'(t)\neq0$ for every allowable $t$.

  • Smoothness refers to the parametrization, not the geometric curve.

• Geometric meaning

  • Guarantees the tangent vector never vanishes ⇒ no abrupt direction changes.

• Example 1

  • (a) \mathbf r(t)=a\cos t\,\mathbf i + a\sin t\,\mathbf j + ct\,\mathbf k \quad(a>0,\,c>0)

  • $\mathbf r'(t)=(-a\sin t,\;a\cos t,\;c)$ is continuous; all three components cannot vanish simultaneously ⇒ smooth.

  • Graph: circular helix (Fig 12.1.2).

  • (b) $\mathbf r(t)=(t^2,\;t^3)$

  • $\mathbf r'(t)=(2t,\;3t^2)$. Both components zero at $t=0$ ⇒ not smooth.

  • Graph: semicubical parabola (Fig 12.3.1). Sudden reversal of tangent direction across $t=0$.

Arc Length – Vector Viewpoint

• Standard formulas (Theorem 10.1.1)

  • Plane curve (Eq 1): $L=\int_a^b\sqrt{\left(\tfrac{dx}{dt}\right)^2+\left(\tfrac{dy}{dt}\right)^2}\,dt$

  • Space curve (Eq 3): $L=\int_a^b\sqrt{\left(\tfrac{dx}{dt}\right)^2+\left(\tfrac{dy}{dt}\right)^2+\left(\tfrac{dz}{dt}\right)^2}\,dt$

• Vector reinterpretation

  • Write $\mathbf r(t)=(x(t),y(t))$ or $\mathbf r(t)=(x(t),y(t),z(t))$.

  • Then $\bigl|\tfrac{d\mathbf r}{dt}\bigr|$ equals the square-root expressions above.

• Theorem 12.3.1 (Vector arc length)

  • For a smooth curve $C$ given by $\mathbf r(t)$,
    $L=\int_a^b \Bigl|\tfrac{d\mathbf r}{dt}\Bigr|\,dt$

• Example 2 – Helix length

  • Curve $\mathbf r(t)=(\cos t,\sin t,t)$, $t\in[0,\pi]$.

  • $\mathbf r'(t)=(-\sin t,\cos t,1)$ with norm $\sqrt{\sin^2 t+\cos^2 t+1}=\sqrt2$.

  • $L=\int_0^\pi\sqrt2\,dt=\sqrt2\,\pi$.

Arc Length as a Parameter

• Construction steps

  1. Choose a reference point on $C$.

  2. Fix a positive (and opposite negative) traversal direction.

  3. For any point $P$, define $s=\pm$ (signed arc length from reference to $P$).

• Arc-length parametrization ⇒ coordinates become $x(s),y(s),z(s)$.

• Reference & orientation choices give infinitely many valid $s$.

• Example 3 – Circle $x^2+y^2=a^2$ (counter-clockwise, reference at $(a,0)$)

  • Original param: $x=a\cos t,\;y=a\sin t$.

  • Since circumference measured positively CCW, arc length satisfies $s=at\;\Rightarrow\;t=s/a$.

  • Arc-length form: $x=a\cos(s/a),\;y=a\sin(s/a),\quad 0\le s\le2\pi a$.

Changing the Parameter

• General idea: Substitute $t=g(\tau)$ to form $\mathbf r(g(\tau))$.

• Orientation control

  • If $g$ increasing ⇒ orientation preserved.

  • If $g$ decreasing ⇒ orientation reversed.

• Example 4 – Circle $\mathbf r(t)=(\cos t,\sin t)$

  • (a) Want CCW traversal while $\tau\in[0,1]$ ⇒ choose linear $g(\tau)=2\pi\tau$.

  • New form: $\mathbf r(\tau)=(\cos2\pi\tau,\sin2\pi\tau)$.

  • (b) Want clockwise while $\tau\in[0,1]$ ⇒ $g(\tau)=2\pi(1-\tau)$.

  • Equivalent simplified form: $\bigl(\cos2\pi\tau,-\sin2\pi\tau\bigr)$.

Smooth Change of Parameter & Chain Rule

• Theorem 12.3.2 (Chain Rule)
  $\frac{d\mathbf r}{d\tau}=\frac{d\mathbf r}{dt}\,\frac{dt}{d\tau}$.

• Smooth change defined by

  • $dt/d\tau$ continuous and non-zero.

  • Positive change: dt/d\tau>0 (orientation preserved).

  • Negative change: dt/d\tau<0 (orientation reversed).

• Example 5

  • From Example 4: $g(\tau)=2\pi\tau$ ⇒ positive (derivative 2\pi>0).

  • $g(\tau)=2\pi(1-\tau)$ ⇒ negative (derivative -2\pi<0).

Finding Arc-Length Parametrizations (Theorem 12.3.3)

• Formula to switch from general $t$ to arc length $s$ with reference $t<em>0$: $s=\int</em>{t_0}^t\Bigl|\tfrac{d\mathbf r}{du}\Bigr|\,du$.

  • Positive change (orientation preserved when $t$ increases).

  • Component forms:
    $s=\int<em>{t</em>0}^t\sqrt{\left(\tfrac{dx}{du}\right)^2+\left(\tfrac{dy}{du}\right)^2}\,du\quad(\text{2-space})$
    $s=\int<em>{t</em>0}^t\sqrt{\left(\tfrac{dx}{du}\right)^2+\left(\tfrac{dy}{du}\right)^2+\left(\tfrac{dz}{du}\right)^2}\,du\quad(\text{3-space})$

Example 6 – Helix revisited

• Original: $\mathbf r(t)=(\cos t,\sin t,t)$, reference $t_0=0$.

• Speed $|\mathbf r'(u)|=\sqrt2$ ⇒ $s=\sqrt2 t$.

• Solve $t=s/\sqrt2$ and substitute:
  $\mathbf r(s)=\bigl(\cos(s/\sqrt2),\;\sin(s/\sqrt2),\;s/\sqrt2\bigr)$.

• Orientation unchanged (positive change).

Example 7 – Bug problem

• Using arc-length form above. After walking $s=10$ units:

  • $x=\cos(10/\sqrt2)\approx0.705$

  • $y=\sin(10/\sqrt2)\approx0.709$

  • $z=10/\sqrt2\approx7.07$
    ⇒ Final position $\approx(0.705,0.709,7.07)$.

Example 8 – Straight line

• Generic line $\mathbf r=\mathbf r_0+t\mathbf v$.

• Speed $|\mathbf v|$ (constant). Then $s=|\mathbf v|t$ ⇒ $t=s/|\mathbf v|$.

• Arc-length form: $\mathbf r=\mathbf r<em>0+\dfrac{s}{|\mathbf v|}\,\mathbf v=\mathbf r</em>0+s\,\dfrac{\mathbf v}{|\mathbf v|}$.

Example 9 – Concrete line $x=2t+1,\,y=3t-2$

• Direction $\mathbf v=(2,3)$, $|\mathbf v|=\sqrt{13}$.

• Normalize & replace $t\to s$:
  $x=\tfrac{2}{\sqrt{13}}s+1,\quad y=\tfrac{3}{\sqrt{13}}s-2$.

Properties Unique to Arc-Length Parametrizations (Theorem 12.3.4)

• (a) Relationship between general parameter $t$ and $s$:
  $\Bigl|\tfrac{d\mathbf r}{dt}\Bigr|=\tfrac{ds}{dt}.$

• (b) If already using arc length $s$, then unit speed:
  $\Bigl|\tfrac{d\mathbf r}{ds}\Bigr|=1.$

• (c) Conversely, if a parametrization has unit-length tangent vector everywhere, then $s=t-t_0$ is an arc-length parameter.

• Component forms (plane vs space) – helpful for computations:

  • Plane: $\tfrac{ds}{dt}=\sqrt{\left(\tfrac{dx}{dt}\right)^2+\left(\tfrac{dy}{dt}\right)^2}$,
    $\Bigl|\tfrac{d\mathbf r}{ds}\Bigr|^2=\left(\tfrac{dx}{ds}\right)^2+\left(\tfrac{dy}{ds}\right)^2=1.$

  • Space: add the $\left(\tfrac{dz}{dt}\right)^2$ or $\left(\tfrac{dz}{ds}\right)^2$ terms accordingly.

Practical & Conceptual Implications

• Choosing an appropriate parameter (time, angle, arc length) can simplify integration, differentiation, and physical interpretation.

• Arc length parameter offers:

  • Unit-speed motion ⇒ simplifies curvature, torsion, and Frenet frame calculations.

  • Easy measurement of distances traveled along the curve.

• Change-of-parameter technique enables seamless switching among these representations while preserving or reversing orientation as desired.

Connections & Further Context

• Builds on Section 10.1 (parametric equations, circle orientations) and Section 11.5 (vector line equations).

• Foundational for curvature & torsion (Chapter 13) where unit-speed is essential.

• Real-world relevance: trajectory re-parametrization in physics, computer graphics, CNC machining (feed-rate control), robotics path planning.

• Ethical / practical note: Correct parametrization avoids “jerky” motions (non-smooth) that could damage equipment or mislead data interpretation.

• Numerical examples (counter-clockwise vs clockwise) highlight importance of orientation in applied problems (e.g., navigation, animation).