Calculus of Vector-Valued Functions: Curves

Exercises

Plane Curves The following vector-valued functions \({\bm{r}\colon \mathbf{R} \to \mathbf{R}^2}\) parametrically-define a curve in two-dimensional space. Sketch each curve. The notation “\(t \in (2,3]\)” is short-hand for \(2 \lt t \leq 3.\) If no restriction is stated for \(t\) the implication is that the curve is traced out for all real numbers \(t.\)

\(\displaystyle \bm{r}(t) \!=\! \bigl\langle t^2-2t, t+1 \bigr\rangle\)

\(\displaystyle \bm{r}(t) \!=\! \bigl\langle t^3+1, t \bigr\rangle,\) \(t \in (-1,1)\)

\(\displaystyle \bm{r}(t) \!=\! \bigl\langle \cos(t), \sin(t) \bigr\rangle,\) \(t \!\in\! (0,2\pi)\)

\(\displaystyle \bm{r}(t) \!=\! \bigl\langle \cos(2t), \sin(2t) \bigr\rangle,\) \(t \!\in\! (0, 2\pi)\)

\(\displaystyle \bm{r}(t) \!=\! \bigl\langle \cos(t), \sin(2t) \bigr\rangle,\) \(t \!\in\! (0, 2\pi)\)

\(\displaystyle \bm{r}(t) \!=\! \bigl\langle \sin(t), \sin^2(t) \bigr\rangle\)
Space Curves The following vector-valued functions \(\bm{r}\colon \mathbf{R} \to \mathbf{R}^3\) parametrically-define a curve in three-dimensional space. Sketch each curve. If you’re finding it challenging to visualize a curve “all at once”, consider the components of the parameterization two at a time and the three corresponding projections onto the three coordinate planes first.

\(\displaystyle \bm{r}(t) = \bigl\langle 1+t, 2+5t, -1+6t\bigr\rangle\)

\(\displaystyle \bm{r}(t) = \bigl\langle \cos(t), \sin(t), t \bigr\rangle\)

\(\displaystyle \bm{r}(t) = \bigl\langle t, t^2, t^3 \bigr\rangle\)

\(\displaystyle \bm{r}(t) = \bigl\langle 6t, 3t^2, t^3 \bigr\rangle\)

\(\displaystyle \bm{r}(t) = \Bigl\langle (8-t)^{\frac{3}{2}}+3, 0, 8t^{\frac{3}{2}} \Bigr\rangle\)

\(\displaystyle \bm{r}(t) = \bigl\langle t^2-2t, 0, t+1 \bigr\rangle\)

\(\displaystyle \bm{r}(t) = \bigl\langle 2t^2+1, 7-t, t^3 \bigr\rangle\)

\(\displaystyle \bm{r}(t) = \Bigl\langle t^3, t^{-1}, \sqrt{6}t \Bigr\rangle\)

\(\displaystyle \bm{r}(t) = \bigl\langle t, \cos(2t), \sin(2t) \bigr\rangle\)

\(\displaystyle \bm{r}(t) = \Bigl\langle \frac{1}{2}t^2, 2t, \ln\bigl(t^2\bigr) \Bigr\rangle\)

\(\displaystyle \bm{r}(t) = \bigl\langle \cos(t), \sin(t), \ln\bigl(\sec(t)\bigr) \bigr\rangle,\) \(t \in (-1,1)\)

\(\displaystyle \bm{r}(t) = \bigl\langle t^7, t^4, t \bigr\rangle \)

\(\displaystyle \bm{r}(t) = \bigl\langle 10t, \mathrm{e}^t, 50\mathrm{e}^{-t} \bigr\rangle \)

\(\displaystyle \bm{r}(t) = \left\langle \frac{t^2}{1+t}, \frac{t}{1+t}, \frac{1}{1+t} \right\rangle \)

\(\displaystyle \bm{r}(t) = \bigl\langle t, \mathrm{e}^t, t^2 \bigr\rangle \)

\(\displaystyle \bm{r}(t) = \Bigl\langle 2t, 2t^3, \sqrt{6}t^2 \Bigr\rangle \)

\(\displaystyle \bm{r}(t) = \bigl\langle t^2, \ln(t), t\ln(t) \bigr\rangle \)
Determine a vector-valued function that describes the following curves.
1. The line that passes through the points \((1,3,-2)\) and \((2,-1,3).\)
2. The intersection of the cylinder \(x^2+y^2=1\) and the plane \(y+z=2.\)
3. The intersection of the paraboloid \(4y=x^2+z^2\) and the plane \(y=x.\)
4. The intersection of the quartic surfaces \(x^2+y^2=1\) and \(x^2-y^2=z^2.\)
5. Consider again the line that passes through the points \((1,3,-2)\) and \((2,-1,3)\) and the parameterization you wrote down previously. Write down two other parameterizations of this same line: one that starts at a different point and travels along at a different “speed” than your previous parameterization, and one that travels in the reverse direction as your previous parameterization.
Determine the coordinates of the point(s) at which each of the following space curves intersects the surface.
1. The curve with parameterization \(\bm{r}(t) = \bigl\langle \ln(t), \mathrm{e}^t, \ln\bigl(t^2\bigr) \bigr\rangle\) and the plane given by \(x+z = 6.\)
2. The unit helix with parameterization \(\bm{r}(t) = \bigl\langle \cos(t), \sin(t), t \bigr\rangle\) and the sphere of radius \(\sqrt{17}.\)
3. The curve with parameterization \(\bm{r}(t) = \bigl\langle t^2+1, t, t-t^2 \bigr\rangle\) and the plane given by \(-3x+12y-5z=12.\)
Closed Curves The following are examples of closed curves, curves that form a closed loop and “retrace” themselves at their parameter increases. For each of these closed curves, figure out the smallest non-negative values of their parameter that trace out a single loop of the curve.

\(\displaystyle \bm{r}(t) = \bigl\langle \cos(t), \sin(t), \cos(2t) \bigr\rangle \)

\(\displaystyle \bm{r}(t) = \bigl\langle \sin(t)+2\sin(2t), \cos(t)-2\cos(2t), -3\sin(3t) \bigr\rangle \)

\(\displaystyle \bm{r}(t) = \bigl\langle \cos(3t)\bigl(2+\cos(2t)\bigr), \sin(3t)\bigl(2+\cos(2t)\bigr), \sin(4t) \bigr\rangle \)
Surfaces The following vector-valued functions \(\bm{r}\colon \mathbf{R}^2 \to \mathbf{R}^3\) parametrically-define a surface in three-dimensional space. Sketch each surface.

\(\displaystyle \bm{r}(s,t) = 2\cos(s)\mathbf{i} + t\mathbf{j} +2\sin(s)\mathbf{k}\)

\(\displaystyle \bm{r}(s,t) = \Bigl\langle s+t, 1+2s+3t, 4-s \Bigr\rangle \)

\(\displaystyle \bm{r}(s,t) = \Bigl\langle s\cos(t), s\sin(t), s \Bigr\rangle \)

\(\displaystyle \bm{r}(s,t) = \Bigl\langle s\cos(t), s\sin(t), t \Bigr\rangle \)
Write down formulas for a vector-valued function \(\bm{r}(s,t)\) that represents the plane that passes through a point \(P_0\) that contains two nonparallel vectors \(\bm{u}\) and \(\bm{v}.\)
Find a parametric representation of each of the following surfaces.
1. The sphere \( x^2+y^2+z^2 = 49\)
2. The plane containing the point \((1,2,3)\) that is spanned by the vectors \(\bigl\langle 3,-3,1 \bigr\rangle\) and \(\bigl\langle -3,-2,1 \bigr\rangle\)
3. The portion of the previous plane that lies within the first octant
4. The cylinder \(x^2+y^2=4\) for \(0 \leq z \leq 1\)
5. The elliptic paraboloid \(z=x^2+2y^2\)
6. The surface \(z = 2\sqrt{x^2+y^2},\) i.e. the top half of the cone \(z^2 = 4x^2+4y^2\)
7. The surface generated by rotating the planar curve \(y=\sin(x)\) for \(0\leq \theta \leq 2\pi\) about the \(x\)-axis
Determine \(\lim\limits_{t \to 0} \bm{r}(t)\) for \(\bm{r}(t) = \bigl(1+t^3\bigr)\mathbf{i} +t\mathrm{e}^{-t}\mathbf{j} +\frac{\sin(t)}{t}\mathbf{k}.\)
Find a parameterization of the line tangent to the helix \({\bm{r}(t) = \bigl\langle 2\cos(t), \sin(t), t \bigr\rangle}\) at the point \((0,1,\pi/2).\)
The position vector of an object moving in a plane is given by \(\bm{r}(t) = \bigl\langle t^3, t^2 \bigr\rangle.\) Calculate its velocity, speed, and acceleration when \(t=1\) and illustrate them geometrically.
Calculate the velocity vector, speed, and acceleration vector of a particle with position vector \(\bm{r}(t) = \bigl\langle t^2, \mathrm{e}^t, t\mathrm{e}^t \bigr\rangle.\)
Sketch the curve with parameterization \(\bm{r}(t) = \sqrt{t}\mathbf{i} +(2-t)\mathbf{j}.\) Calculate \(\bm{r}'(t),\) and add the vector \(\bm{r}'(1)\) emanating from the point \(\bm{r}(1)\) to the sketch.
For each of the parametrically-defined curves defined in the problem labelled Space Curves, calculate \(\bm{r}'(t),\) and add the vectors \(\bm{r}'(0)\) and \(\bm{r}'(1)\) each respectively emanating from initial points \(\bm{r}(0)\) and \(\bm{r}(1)\) to your sketch.

Additionally, for each of the previous examples, determine a parameterization of the line tangent to the curve at the point \(\bm{r}(1).\)
The curves with parameterizations \(\bm{r}_1(t) = \bigl\langle t^2+1, t, 2\sin\bigl(\frac{\pi}{3}t\bigr) \bigr\rangle\) and \(\bm{r}_2(t) = \bigl\langle t, \frac{t}{3}, \sqrt{t} \bigr\rangle\) intersect at a single point. Calculate the angle at which they intersect.
A moving particle starts at an initial position \(\bm{r}(0)=\langle 1,0,0\rangle\) with initial velocity \(\bm{v}(0) = \langle 1,-1,1\rangle.\) Its acceleration is \(\bm{a}(t) = \langle 4t, 6t, 1\rangle.\) Determine formulas for its velocity and position at time \(t\).
A projectile is fired with angle of elevation \(\alpha\) and initial velocity \(v_0.\) Assuming that air resistance is negligible and the only external force is due to gravity, find the position function \(\bm{r}(t)\) of the projectile. What value of \(\alpha\) maximizes the range \(d\) of the projectile?
A projectile is fired with angle of elevation π/6 and initial speed 150 m/s from a position 10m above level ground. Where does the projectile hit the ground, and with what speed?
Suppose the vector-valued function \(\bm{r}(t) = \bigl\langle 3t^2, t, t^2-9t\bigr\rangle\) describes the position of a particle at time \(t.\) At what time \(t\) is the speed of the particle minimal?
Determine the derivative \(\bm{r}'(t)\) of \({\bm{r}(t) = \bigl(1+t^3\bigr)\mathbf{i} +t\mathrm{e}^{-t}\mathbf{j} +\sin(2t)\mathbf{k},}\) and calculate the unit tangent vector to the corresponding curve at the point where \(t=0.\)
Calculate formulas for the unit normal and binormal vectors of the helix \({\bm{r}(t) = \bigl\langle \cos(t), \sin(t), t \bigr\rangle.}\) Then determine equations for the normal plane and osculating plane at the point \((0,1,\pi/2).\)
For each of the parametrically-defined curves defined in the problem labelled Space Curves, calculate the unit tangent, unit normal, and unit binormal vectors to the curve at the point where \(t=1\) and add those vectors to your sketch.

Note that this task will be arduous for any curve with parameterization \(\bm{r}\) for which \(|\bm{r}'(t)|\) cannot be expressed without a radical. As part of this exercise, identify and prioritize the curves for which this task is not arduous.
Calculate the length of the arc of the helix \(\bm{r}(t) = \bigl\langle \cos(t), \sin(t), t \bigr\rangle\) between the points \((1,0,0)\) and \((1,0,\pi/2).\) Then reparameterize the curve with respect to its arclength starting from the point \((1,0,0).\)
Compute the arclength of the segment of the curve parameterized as \(\bigl\langle 2t, \frac{1}{3}t^3, t^2 \bigr\rangle\) for \(t \in [0, 2].\)
Compute the arclength of the segment of the curve parameterized as \(\bigl\langle \cos(t), \sin(t), \ln(\cos(t))\bigr\rangle\) for \(t \in [0, \pi/3].\)
For each of the closed parametrically-defined curves defined in the problem labelled Closed Curves, compute the arclength of the curve, manually if you can, but using a computer if you must.
For each of the closed parametrically-defined curves defined in the problem labelled Space Curves, write down the integral for the arclength function \(\int_0^t |\bm{r}'(s)|\,\mathrm{d}s.\) If an antiderivative of \(|\bm{r}'(s)|\) appears to be expressible in terms of elementary functions, evaluate the integral. I.e. if you can take the integral, then do so.
Show that the curvature of a circle of radius \(R\) is \(\frac{1}{R}.\)
Calculate a formula for the curvature of the twisted cubic \(\bm{r}(t) = \bigl\langle t, t^2, t^3 \bigl\rangle\) at a generic point on the curve, then use that formula to calculate the curvature specifically at the point \((0,0,0).\)
Calculate the curvature of the parabola \(y=x^2\) at the points \((0,0)\) and \((1,1)\) and \((2,4).\) Sketch the parabola’s osculating circle at \((0,0).\)
Determine the unit tangent, normal, and binormal vectors, and the curvature of the curve with parameterization \(\bm{r}(t) = \bigl\langle t, \sqrt{2}\ln(t), 1/t \bigr\rangle\) at the point \((1,0,1).\)
Calculate a formula for the torsion of the helix \(\bm{r}(t) = \bigl\langle \cos(t), \sin(t), t \bigr\rangle.\)
For each of the parametrically-defined curves defined in the problem labelled Space Curves, determine a formula for the curvature and the torsion of the curve, and use that formula to calculate the curvature and torsion at the point on the curve corresponding to \(t=1.\)

Problems & Challenges

James Stewart A projectile is fired from the origin with angle of elevation \(\alpha\) and initial speed \(v_0.\) Assuming that air resistance is negligible and that the only force acting on the projectile is gravity \(g,\) the position vector of the projectile is \[\mathbf{r}(t) = \Bigl(v_0\cos(\alpha)t\Bigr)\mathbf{i} + \Bigl(v_0\sin(\alpha)t - \frac{1}{2}gt^2\Bigr) \mathbf{j}.\] The maximum horizontal distance the projectile can travel is \(R = v_0^2/g\) and is achieved when \(\alpha = 45°.\)
1. At what angle should the projectile be fired to achieve maximum height and what is the maximum height?
2. Fix the initial speed \(v_0\) and consider the parabola \(x^2 + 2Ry - R^2 = 0.\) Show that the projectile can hit any target inside or on the boundary of the region bounded by the parabola and the \(x\)-axis, and that it can’t hit any target outside this region.
3. Suppose that the gun is elevated to an angle of inclination \(\alpha\) in order to aim at a target that is suspended at a height \(h\) directly over a point \(D\) units downrange. The target is released at the instant the gun is fired. Show that the projectile always hits the target, regardless of the value \(v_0,\) provided the projectile does not hit the ground “before” \(D.\)
James Stewart A projectile is fired from the origin down an inclined plane that makes an angle \(\theta\) with the horizontal. The angle of elevation of the gun and the initial speed of the projectile are \(\alpha\) and \(v_0\) respectively.
1. Find the position vector of the projectile and the parametric equations of the path of the projectile as functions of the time \(t.\) (Ignore air resistance.)
2. Show that the angle of elevation \(\alpha\) that will maximize the downhill range is the angle halfway between the plane and the vertical.
3. Suppose the projectile is fired up an inclined plane whose angle of inclination is \(\theta\). Show that, in order to maximize the (uphill) range, the projectile should be fired in the direction halfway between the plane and the vertical.
4. In a paper presented in 1686, Edmond Halley summarized the laws of gravity and projectile motion and applied them to gunnery. One problem he posed involved firing a projectile to hit a target a distance \(R\) up an inclined plane. Show that the angle at which the projectile should be fired to hit the target but use the least amount of energy is the same as the angle in the previous part. (Use the fact that the energy needed to fire the projectile is proportional to the square of the initial speed, so minimizing the energy is equivalent to minimizing the initial speed.)
James Stewart A projectile of mass \(m\) is fired from the origin at an angle of elevation \(\alpha\). In addition to gravity, assume that air resistance provides a force that is proportional to the velocity and that opposes the motion. Then, by Newton’s Second Law, the total force acting on the projectile satisfies the equation \[ m\frac{\mathrm{d}^2\mathbf{R}}{\mathrm{d}t^2} = -mg\mathbf{j} - k\frac{\mathrm{d}\mathbf{R}}{\mathrm{d}t} \] where \(\mathbf{R}\) is the position vector and \(k \gt 0\) is the constant of proportionality.
1. Show that this equation can be integrated to obtain the equation \[ \frac{\mathrm{d}\mathbf{R}}{\mathrm{d}t} + \frac{k}{m}\mathbf{R} = \mathbf{v}_0-gt\mathbf{j} \] where \(\mathbf{v}_0 = \mathbf{v}(0) = \frac{\mathrm{d}\mathbf{R}}{\mathrm{d}t}(0).\)
2. Multiply both sides of the equation in the last part by \(\mathrm{e}^{(k/m)t}\) and show that the left-hand side of the resulting equation is the derivative of the product \(\mathrm{e}^{(k/m)t}\mathbf{R}(s).\) Then integrate to find an expression for the position vector \(\mathbf{R}(r).\)
Recall the formulas for the curvature \(\kappa\) and torsion \(\tau\) at a point on a curve with smooth parameterization \(\bm{r}.\)

\(\displaystyle \kappa = \frac{\big|\bm{r}' \times \bm{r}''\big|}{\big|\bm{r}'\big|^3} \)

\(\displaystyle \tau = \frac{\big(\bm{r}' \times \bm{r}''\big) \cdot \bm{r}'''}{\big|\bm{r}' \times\bm{r}'' \big|^2} \)

First prove that these formulas are true from the definitions of curvature and torsion in terms of the tangent, normal, and binormal vector at a point. Then, recalling the geometric interpretations of the dot product, cross product, and scalar triple product, determine some geometric intuition for these formulas.
Suppose that \(\bm{r}\) is a vector-valued function for which \(\bm{r}'\) and \(\bm{r}''\) exist. Prove the following facts:

\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}t} \Bigl(\bigl|\bm{r}(t)\bigr|\Bigr) = \frac{1}{|\bm{r}(t)|}\Bigl(\bm{r}(t) \cdot \bm{r}'(t)\Bigr)\)

\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}t} \Bigl(\bm{r}(t) \times \bm{r}'(t)\Bigr) = \bm{r}(t) \times \bm{r}''(t)\)

\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}t} \biggl(\bm{r}(t) \cdot \Bigl(\bm{r}'(t) \times \bm{r}''(t)\Bigr)\biggr) = \bm{r}(t) \cdot \Bigl(\bm{r}'(t) \times \bm{r}'''(t)\Bigr) \)
Suppose that \(\bm{a}\) and \(\bm{b}\) and \(\bm{c}\) are vector-valued functions. Determine an expression for the following derivative:

\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}t} \biggl(\bm{a}(t) \cdot \Bigl(\bm{b}'(t) \times \bm{c}''(t)\Bigr)\biggr) \)
James Stewart In designing transfer curves to connect sections of straight railroad tracks, it’s important to realize that the acceleration of the train should be continuous so that the reactive force exerted by the train on the track is also continuous. This will be the case if the curvature varies continuously.
1. A logical candidate for a transfer curve to join existing tracks given by lines \(y=1\) for \(x\leq0\) and \(y = \sqrt{2} - x\) for \(x \gt 1/\sqrt{2}\) might be the function \(f(x) = \sqrt{1-x^2}\) for \(0 \lt x \lt 1/\sqrt{2}.\) It looks reasonable at first glance. Show that the function \[ F(x) = \begin{cases} 1 \quad&\text{ if } x \leq 0 \\ f(x) \quad&\text{ if } 0 \lt x \lt 1/\sqrt{2} \\ \sqrt{2}-x \quad&\text{ if } x \geq 1/\sqrt{2} \end{cases} \] is continuous and has continuous slope, but does not have continuous curvature. Therefore, \(f\) is not an appropriate transfer curve.
2. Find a fifth-degree polynomial to serve as a transfer curve between the following straight line segments: \(y=0\) for \(x\leq 0\) and \(y=x\) for \(x \geq 1.\) Could this be done with a fourth-degree polynomial? Use a graphing calculator or computer to sketch the graph of the “connected” function and check to see that it looks like the one in the figure.
For a given smooth curve \(C\) let \(\operatorname{\mathcal{E}}(C)\) denote the locus of the centers of curvature of \(C\) — the curve that is “traced out” by the centers of curvature. The curve \(\operatorname{\mathcal{E}}(C)\) is called the evolute of \(C.\)
1. For a smooth curve \(C\) and a smooth parameterization \(\bm{r}(t)\) of \(C\) show that \(\operatorname{\mathcal{E}}(C)\) has parameterization \({\bm{r}(t) + \frac{1}{\kappa(t)}\mathbf{N}(t).}\)
2. Determine a parameterization for the evolute of \(\bigl\langle \cos(t), \sin(t), t\bigr\rangle.\)
3. For a given curve \(C,\) an involute of \(C\) is any curve \(S\) for which \(\operatorname{\mathcal{E}}(S) = C.\) Determine a parameterization for any involute of \(\bigl\langle \cos(t), \sin(t), t\bigr\rangle.\)
4. Does there exist a curve \(C\) such that \(\operatorname{\mathcal{E}}(C) \neq C\) but \(\operatorname{\mathcal{E}}\bigl(\operatorname{\mathcal{E}}(C)\bigr) = C?\)
5. Let \(\operatorname{\mathcal{E}}^n\) denote the operation of applying the evolute operation \(\operatorname{\mathcal{E}}\) \(n\) times. In general, for each \(n,\) does there exist a curve \(C\) such that \(\operatorname{\mathcal{E}}^n(C) = C\) but \(\operatorname{\mathcal{E}}^k(C) \neq C\) for all \(k \lt n?\)