em dash; sentence case

quarto/differentiable_vector_calculus/vector_valued_functions.qmd

@@ -981,7 +981,7 @@ $$
 \vec{v} \times \vec{c} = GM \hat{x} + \vec{d}.
 $$
 
-As $\vec{x}$ and $\vec{v}\times\vec{c}$ lie in the same plane - orthogonal to $\vec{c}$ - so does $\vec{d}$. With a suitable re-orientation, so that $\vec{d}$ is along the $x$ axis, $\vec{c}$ is along the $z$-axis, then we have $\vec{c} = \langle 0,0,c\rangle$ and $\vec{d} = \langle d ,0,0 \rangle$, and $\vec{x} = \langle x, y, 0 \rangle$. Set $\theta$ to be the angle, then $\hat{x} = \langle \cos(\theta), \sin(\theta), 0\rangle$.
+As $\vec{x}$ and $\vec{v}\times\vec{c}$ lie in the same plane---orthogonal to $\vec{c}$---so does $\vec{d}$. With a suitable re-orientation, so that $\vec{d}$ is along the $x$ axis, $\vec{c}$ is along the $z$-axis, then we have $\vec{c} = \langle 0,0,c\rangle$ and $\vec{d} = \langle d ,0,0 \rangle$, and $\vec{x} = \langle x, y, 0 \rangle$. Set $\theta$ to be the angle, then $\hat{x} = \langle \cos(\theta), \sin(\theta), 0\rangle$.
 
 
 Now
@@ -1662,7 +1662,7 @@ $$
 The first equation relates the steering angle with the curvature. If the steering angle is not changed ($d\alpha/du=0$) then the curvature is constant and the motion is circular. It will be greater for larger angles (up to $\pi/2$). As the curvature is the reciprocal of the radius, this means the radius of the circular trajectory will be smaller. For the same constant steering angle, the curvature will be smaller for longer wheelbases, meaning the circular trajectory will have a larger radius. For cars, which have similar dynamics, this means longer wheelbase cars will take more room to make a U-turn.
 
 
-The second equation may be interpreted in ratio of arc lengths. The infinitesimal arc length of the rear wheel is proportional to that of the front wheel only scaled down by $\cos(\alpha)$. When $\alpha=0$ - the bike is moving in a straight line - and the two are the same. At the other extreme - when $\alpha=\pi/2$ - the bike must be pivoting on its rear wheel and the rear wheel has no arc length. This cosine, is related to the speed of the back wheel relative to the speed of the front wheel, which was used in the initial differential equation.
+The second equation may be interpreted in ratio of arc lengths. The infinitesimal arc length of the rear wheel is proportional to that of the front wheel only scaled down by $\cos(\alpha)$. When $\alpha=0$---the bike is moving in a straight line---and the two are the same. At the other extreme---when $\alpha=\pi/2$---the bike must be pivoting on its rear wheel and the rear wheel has no arc length. This cosine, is related to the speed of the back wheel relative to the speed of the front wheel, which was used in the initial differential equation.
 
 
 The last equation, relates the curvature of the back wheel track to the steering angle of the front wheel. When $\alpha=\pm\pi/2$, the rear-wheel curvature, $k$, is infinite, resulting in a cusp (no circle with non-zero radius will approximate the trajectory). This occurs when the front wheel is steered orthogonal to the direction of motion. As was seen in previous graphs of the trajectories, a cusp can happen for quite regular front wheel trajectories.
@@ -1875,7 +1875,7 @@ $$
 $$
 
 
-We see $\vec\beta'$ is zero (the curve is non-regular) when $\kappa'(s) = 0$. The curvature changes from increasing to decreasing, or vice versa at each of the $4$ crossings of the major and minor axes - there are $4$ non-regular points, and we see $4$ cusps in the evolute.
+We see $\vec\beta'$ is zero (the curve is non-regular) when $\kappa'(s) = 0$. The curvature changes from increasing to decreasing, or vice versa at each of the $4$ crossings of the major and minor axes--there are $4$ non-regular points, and we see $4$ cusps in the evolute.
 
 
 The curve parameterized by $\vec{r}(t) = 2(1 - \cos(t)) \langle \cos(t), \sin(t)\rangle$ over $[0,2\pi]$ is cardiod. It is formed by rolling a circle of radius $r$ around another similar sized circle. The following graphically shows the evolute is a smaller cardiod (one-third the size). For fun, the evolute of the evolute is drawn: