📝 Change of parameter for curves (25 MCQs)

📖 Explanation: When substituting $t = s^{1/3}$, the new vector function becomes $\mathbf{R}(s) = \langle s^{1/3}, s^{2/3} \rangle$. Since $s = t^3$ is strictly increasing for positive values, orientation is preserved. However, the derivative magnitude changes because the chain rule introduces a factor of $dt/ds$, causing non-linear speed variation relative to the new parameter.

📖 Explanation: The cross product being zero indicates that the velocity vectors are collinear or parallel. This is a necessary condition for two parametrizations to trace the same geometric curve with consistent orientation. It does not imply identical parameters or equal accelerations, as reparametrization affects higher-order derivatives through the chain rule and product rule.

📖 Explanation: Reparametrization preserves the image set (geometry) but alters the velocity vector magnitude unless specifically designed for arc length. Applying the chain rule gives $d\mathbf{r}/du = (d\mathbf{r}/dt)(dt/du)$. Since $dt/du = 1/(2\sqrt{u})$, the speed varies with position. Geometric invariance does not imply kinematic invariance; confusing these is a fundamental misconception in vector calculus.

📖 Explanation: Orientation preservation in reparametrization depends solely on the sign of the derivative of the parameter transformation function. If \phi'(s) > 0, the mapping is strictly increasing, preserving direction. A negative derivative would reverse orientation. Continuity alone allows decreasing functions, and equal speed is unrelated to directional consistency along the curve.

📖 Explanation: For a cubic spline component, \|\mathbf{r}'(t)\| typically involves $\sqrt{(at^2+bt+c)^2 + ...}$, resulting in the square root of a degree-four polynomial. Such integrals generally yield elliptic integrals, which lack closed-form expressions in elementary functions. This necessitates numerical inversion or approximation techniques for practical arc-length parametrization in engineering simulations involving complex trajectories.

📖 Explanation: As $t \to \infty$, $u = e^{-t} \to 0^+$. The original helix extends indefinitely along the z-axis. The reparametrized version maps this infinite extension to a finite interval near $u=0$. While the geometric set traced is identical for corresponding domains, the limiting behavior in parameter space differs fundamentally, compressing infinite traversal into a boundary point.

📖 Explanation: By definition, arc length parametrization means the parameter itself measures distance traveled. Therefore, the rate of change of position with respect to arc length is identically unity: $\| d\mathbf{r}/ds \| = 1$. Regardless of the original speed profile $v(t)$, plotting speed against arc length always yields a constant value of one, distinguishing it from time-based representations.

📖 Explanation: The unit tangent vector \mathbf{T} = \mathbf{r}'(t)/\|\mathbf{r}'(t)\| is a geometric property independent of parametrization. Under any smooth bijection with positive derivative, the chain rule introduces a scalar factor that cancels during normalization. Both linear transformations preserve orientation and smoothness, so $\mathbf{T}$ remains identical at corresponding geometric points regardless of scaling or translation.

📖 Explanation: Curvature is geometrically intrinsic, but its computational formula depends on parametrization. The simple form \kappa = \|\mathbf{r}''(s)\| applies only to arc length. For arbitrary $t$, one must use \kappa = \|\mathbf{r}' \times \mathbf{r}''\| / \|\mathbf{r}'\|^3. Using the unadjusted formula with a general parameter produces erroneous values that vary with speed, misleading physical interpretations of flow dynamics.

📖 Explanation: A cusp represents a genuine geometric singularity where the curve fails to have a well-defined tangent line. Reparametrization by a smooth bijection cannot create a non-zero velocity where none existed geometrically. While some algebraic singularities are artifacts of parametrization, true cusps persist because no smooth invertible transformation can map a zero-velocity point to a regular point while preserving the curve's local structure.

📖 Explanation: Kepler’s second law states that radius vectors sweep equal areas in equal times for gravitational orbits. However, for arbitrary parametrization, the area sweep rate is \frac{1}{2}\|\mathbf{r} \times \mathbf{r}'\|. To achieve uniform area parametrization, one must integrate this quantity. This differs from arc length and requires computing the magnitude of the cross product, linking geometry directly to the desired physical constraint.

📖 Explanation: While symmetric speed profiles suggest potential geometric symmetry, the actual position depends on integrating velocity from a specific starting point. Even if \|\mathbf{r}'(a+h)\| = \|\mathbf{r}'(a-h)\|, the directions of \mathbf{r}' may differ asymmetrically. Additionally, the base point $\mathbf{r}(a)$ might not lie on an axis of symmetry. Thus, speed symmetry is necessary but insufficient for concluding spatial symmetry without verifying directional and positional constraints.

📖 Explanation: Line integrals depend on geometric measure $ds$, but numerical efficiency benefits from adaptive sampling. When the integrand $f$ changes rapidly in a localized region, concentrating parameter samples there reduces discretization error. Uniform arc length ignores integrand behavior, potentially missing critical features. Adaptive reparametrization aligns computational resources with functional complexity, optimizing accuracy without increasing total sample count unnecessarily.

📖 Explanation: For $u(t) = t + \sin(t)$ to serve as a valid reparametrization, it must be strictly monotonic (bijective) on the domain. The derivative u'(t) = 1 + \cos(t) must be positive everywhere. Since $\cos(t) \geq -1$, u'(t) \geq 0, but equals zero at odd multiples of $\pi$. Strict positivity requires avoiding these points or restricting domain. Without monotonicity, the mapping folds, violating the definition of a proper parametrization.

📖 Explanation: Substituting $t = \pm\sqrt{s}$ leads to ambiguity, but even considering $t \geq 0$, $x = s^{3/2}$ and $y = s$. Then $dx/ds = (3/2)s^{1/2} \to 0$ and $dy/ds = 1$. So \mathbf{R}'(0) = \langle 0,1 \rangle seems defined. However, the original curve has a cusp: left and right derivatives differ in direction. The reparametrization $s=t^2$ collapses both branches onto $s \geq 0$, losing directional information. Thus, while $\mathbf{R}(s)$ appears differentiable, it misrepresents the original geometry’s singularity.

📖 Explanation: Ease-in-ease-out requires smooth acceleration and deceleration. Mathematically, this corresponds to \phi'(0) = \phi'(1) = 0 (zero initial/final velocity) and \phi''(t) changing sign appropriately. A sigmoidal or polynomial function satisfying boundary derivative conditions achieves this. Constant derivative implies uniform motion, linear implies constant acceleration, and discontinuities cause jerky motion. Matching human perceptual expectations demands careful design of the parameter transformation’s differential properties.

📖 Explanation: Acceleration decomposes into tangential and normal components: $\mathbf{a} = a_T \mathbf{T} + a_N \mathbf{N}$. The normal component $a_N = \kappa v^2$ relates to geometry and speed squared, while tangential component $a_T = dv/dt$ depends explicitly on how speed changes with the chosen parameter. Different parametrizations yield different $dv/dt$ even at the same geometric point, making tangential acceleration non-invariant, whereas curvature and normal direction remain geometric.

📖 Explanation: Reparametrization changes only the label assigned to points on a curve, leaving geometric quantities invariant. Reference frame transformations involve changing coordinates in spacetime, affecting measurements like time intervals and lengths relativistically. Confusing internal parameter freedom with external observer dependence is a category error. Arc length parametrization simplifies equations but does not confer physical frame invariance; that requires proper tensorial representation under coordinate transformations.

📖 Explanation: Here, \|\mathbf{r}'(t)\| = \sqrt{2}e^t, so $s = \sqrt{2}(e^t - 1)$. Inversion gives $t = \ln(1 + s/\sqrt{2})$, which is straightforward. However, for more complex curves like elliptical helices, $s(t)$ involves non-elementary integrals, making analytical inversion impossible. Even when integrable, transcendental equations often resist closed-form solutions. Thus, inversion is typically the bottleneck in explicit arc-length parametrization, requiring numerical methods in practice.

📖 Explanation: The correct formula is \kappa = \|\mathbf{r}' \times \mathbf{r}''\| / \|\mathbf{r}'\|^3. In 2D, \|\mathbf{r}' \times \mathbf{r}''\| = |x'y'' - y'x''|. The student’s denominator lacks one power of speed. At $t=1$, \|\mathbf{r}'\| = \sqrt{5}, so their result exceeds true curvature by exactly $\sqrt{5}$. This systematic overestimation arises from dimensional inconsistency: curvature has units of inverse length, but their expression has inverse length divided by speed, revealing fundamental misunderstanding of derivative scaling.

📖 Explanation: Tool chatter arises from abrupt force changes, which relate to acceleration discontinuities. While $C^1$ continuity prevents instantaneous velocity jumps, $C^2$ ensures smooth acceleration profiles, minimizing inertial shocks. In precision machining, jerk continuity ($C^3$) further improves surface finish, but $C^2$ is the minimum threshold to avoid destructive vibrations. Reparametrization must therefore preserve or enhance smoothness class beyond mere geometric continuity to satisfy dynamic mechanical constraints.

📖 Explanation: Angular position is $\theta = t^3$, so angular speed is $d\theta/dt = 3t^2$. As $t \to 0$, this tends to zero quadratically. Uniform circular motion has constant $d\theta/dt$. Graphically, plotting angular speed versus time shows a parabola touching zero at origin, contrasting sharply with a horizontal line for uniform motion. This illustrates how nonlinear reparametrization distorts temporal perception of rotational dynamics despite identical geometric trajectory.

📖 Explanation: Under reparametrization $t = \phi(s)$, the first derivative transforms via chain rule, but the second derivative involves both \phi''(s) and (\phi'(s))^2 terms due to product rule. Showing that extraneous terms cancel in the curvature formula requires meticulous expansion. Faà di Bruno generalizes this, but even basic multivariable chain rule suffices. Ignoring these higher-order transformation terms is a common pitfall; recognizing their structured cancellation confirms geometric invariance rigorously.

📖 Explanation: Arc length parametrization requires \|\mathbf{r}'(t)\| > 0 everywhere to define $s(t)$ as a strictly increasing bijection. If the original path contains stationary points (e.g., stops or cusps), $s(t)$ plateaus, making inversion impossible. Assuming regularity without verification leads to division by zero or non-invertible mappings. Real-world paths often include dwell periods, necessitating preprocessing to remove or handle singularities before attempting arc-length conversion for constant-speed execution.

📖 Explanation: Although \mathbf{r}'(2\pi n) = \mathbf{0}, the unit tangent \mathbf{T}(t) = \mathbf{r}'(t)/\|\mathbf{r}'(t)\| approaches $\langle 1, 0 \rangle$ as $t \to 2\pi n^\pm$. The geometric tangent exists as a limiting direction despite vanishing velocity. Reparametrization by arc length yields $\mathbf{R}(s)$ with \mathbf{R}'(s) continuous and non-zero at corresponding points, effectively regularizing the parametrization while preserving the geometric tangent. This distinguishes parametric singularity from geometric singularity.