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Find the arc length of the given curve on the specified interval in Exercises 1 to 6. footnote #
\((2\,\cos t,2\,\sin t,t)\), for \(0\leq t\leq 2\pi\)
\((1,3t^2,t^3)\), for \(0\leq t\leq 1\)
\((\sin 3t,\,\cos 3t,2t^{3/2})\), for \(0\leq t\leq 1\)
\(\left(t+1,{\displaystyle\frac{2\sqrt{2}}{3}}t^{3/2}+7,{\displaystyle\frac{1}{2}}t^2 \right)\), for \(1\leq t\leq 2\)
\((t,t,t^2)\), for \(1\leq t\leq 2\)
\((t,t\,\sin t,t\,\cos t)\), for \(0\leq t\leq \pi\)
Find the arc length of \(\textbf{c}(t)=(t, |t|)\) for \(-1 \leq t\leq 1\).
Recall from Section 12.4 that a rolling circle of radius \(R\) traces out a cycloid, which can be parametrized by \(\textbf{c}(t)= (Rt-R\sin t, R- R\cos t)\). One arch of the cycloid is completed from \(t=0\) to \(t= 2\pi\). Show that the length of this arch is always 4 times the diameter of the rolling circle.
Let \(C\) be the line segment connecting the point \(\textbf{p} =(1, 2, 0)\) to the point \(\textbf{q}=(0, 1, -1)\).
Compute the length of the curve \(\textbf{c}(t)=(\log(\sqrt{t}), \sqrt{3}t, \frac{3}{2}t^2)\) for \(1 \leq t \leq 2\).
Find the length of the path \({\bf c}(t)\), defined by \({\bf c}(t)=(2\,\cos t,2\,\sin t,t)\), if \(0\leq t\leq 2\pi\) and \({\bf c}(t)=(2,t-2\pi,t)\), if \(2\pi\leq t \leq 4\pi\).
Let \({\bf c}\) be the path \({\bf c} (t)=(t,t \,\sin t, t \,\cos t)\). Find the arc length of \({\bf c}\) between the two points \((0, 0, 0)\) and \((\pi,0,-\pi)\).
Let \({\bf c}\) be the path \({\bf c} (t) = (2t, t^2 , \log t)\), defined for \(t > 0\). Find the arc length of \({\bf c}\) between the points \((2, 1, 0)\) and \((4, 4, \log 2)\).
The arc-length function \(s (t)\) for a given path \({\bf c} (t)\), defined by \(s (t) = \int_a^t \| {\bf c} ' (\tau) \| \,d \tau\), represents the distance a particle traversing the trajectory of \({\bf c}\) will have traveled by time \(t\) if it starts out at time \(a\); that is, it gives the length of \({\bf c}\) between \({\bf c} (a)\) and \({\bf c} (t)\). Find the arc-length functions for the curves \({\alpha} (t) = (\cosh t, \,\sinh t,t)\) and \({\beta} (t) = (\cos t , \,\sin t,t)\), with \(a=0.\)
Let \({\bf c} (t)\) be a given path, \(a \le t \le b\). Let \(s=\alpha (t)\) be a new variable, where \(\alpha\) is a strictly increasing \(C^1\) function given on \([a,b]\). For each \(s\) in \([\,\alpha (a), \,\alpha (b)]\) there is a unique \(t\) with \(\,\alpha (t) =s\). Define the function \({\bf d}\colon [ \,\alpha (a), \,\alpha (b)] \to {\mathbb R}^3\) by \({\bf d} (s)={\bf c} (t)\).
(c) Let \(s = \,\alpha (t) = \int_a^t \|{\bf c}' (\tau) \|\, d \tau\). Define \({\bf d}\) as above by \({\bf d} (s) = {\bf c} (t)\). Show that \[ \Big\|\frac{d}{ds} {\bf d} (s)\Big\|=1. \]
The path \(s \mapsto {\bf d} (s)\) is said to be an arc-length reparametrization of \({\bf c}\) (see also Exercise 17).
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Exercises 16, 17, and 20–23 develop some of the classic differential geometry of curves.
Let \({\bf c} \colon\, [a,b] \to {\mathbb R}^3\) be an infinitely differentiable path (derivatives of all orders exist). Assume \({\bf c}' (t) \ne {\bf 0}\) for any \(t\). The vector \({\bf c}' (t) / \| {\bf c}' (t) \| = {\bf T} (t)\) is tangent to \({\bf c}\) at \({\bf c} (t)\), and, because \(\|{\bf T} (t) \| =1, {\bf T}\) is called the unit tangent to \({\bf c}\).
Show that any line \({\bf l}(t)=\textbf{x}_0+t\textbf{v}\), where \(\textbf{v}\) is a unit vector, has zero curvature.
Consider the parametrization of the unit circle given by \(\textbf{c}(t)=(\cos t, \sin t)\).
If \({\bf T}' (t) \ne {\bf 0}\), it follows from Exercise 16 that \({\bf N} (t) = {\bf T}' (t) / \|{\bf T}' (t) \|\) is normal (i.e., perpendicular) to \({\bf T} (t); {\bf N}\) is called the principal normal vector. Let a third unit vector that is perpendicular to both \({\bf T}\) and \({\bf N}\) be defined by \({\bf B} = {\bf T} \times {\bf N}; {\bf B}\) is called the binormal vector. Together, \({\bf T}, {\bf N},\) and \({\bf B}\) form a right-handed system of mutually orthogonal vectors that may be thought of as moving along the path (Figure 12.12). Show that
If \({\bf c} (s)\) is parametrized by arc length, we use the result of Exercise 20(c) to define a scalar-valued function \(\tau\), called the torsion, by \[ \frac{d {\bf B}}{ds} = - \tau {\bf N}. \]
Show that if a path lies in a plane, then the torsion is zero. Do this by demonstrating that \({\bf B}\) is constant and is a normal vector to the plane in which \({\bf c}\) lies. (If the torsion is not zero, it gives a measure of how fast the curve is twisting out of the plane of \({\bf T}\) and \({\bf N}\).)
In special relativity, the proper time of a path \({\bf c} \colon [a,b] \to {\mathbb R}^4 \) with components given by \({\bf c} ( \lambda) = ( x (\lambda), y (\lambda),\) \(z (\lambda), t (\lambda))\) is defined to be the quantity \[ \int_a^b \sqrt{-[x' (\lambda)]^2 - [ y' (\lambda)]^2 - [ z' (\lambda)]^2 + c^2 [t' (\lambda)]^2} \, d \lambda, \] where \(c\) is the velocity of light, a constant. In Figure 12.24, show that, using self-explanatory notation, the “twin paradox inequality” holds: \[ \hbox{proper time (AB) + proper time (BC) < proper time (AC)}. \]
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The early Greeks knew that a straight line was the shortest possible path between two points. Euclid, in his book Optics, stated the “principle of the reflection of light”—that is, light traveling in a plane travels in a straight line, and when it is reflected across a mirror, the angle of incidence equals the angle of reflection.
The Greeks could not have had a proof that straight lines provided the shortest path between two points because they, in the first place, had no definition of the length of a path. They saw this property of straight lines as more or less “obvious.”
Using the justification of arc length in this section and the triangle inequality of Section 11.5, argue that if \({\bf c}_{0}\) is the straight-line path \(c_{0}(t)=t {\rm P} + (1 - t)\)Q between P and Q in \({\mathbb R}^{3}\), then \[ L({\bf c}_{0}) \le L({\bf c}) \] for any other path \({\bf c}\) joining P and Q.