Vector-Valued Functions – Change of Parameter & Arc Length

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

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.

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.

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.

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).

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).

Formula to switch from general t to arc length s with reference t0: s=\int{t_0}^t\Bigl|\tfrac{d\mathbf r}{du}\Bigr|\,du.
- Positive change (orientation preserved when t increases).
- Component forms:
  s=\int{t0}^t\sqrt{\left(\tfrac{dx}{du}\right)^2+\left(\tfrac{dy}{du}\right)^2}\,du\quad(\text{2-space})
  s=\int{t0}^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})

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).

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).

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 r0+\dfrac{s}{|\mathbf v|}\,\mathbf v=\mathbf r0+s\,\dfrac{\mathbf v}{|\mathbf v|}.

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.

(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.

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.

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).