6.1 Smooth curves in the plane  (Page 4/4)

1. Restate the definition of tangential direction and unit tangent using the $R^2$ version of the plane instead of the $C$ version. That is, restate the definition in terms of pairs $(x,y)$ of real numbers instead of a complex number $z.$
2. Suppose $\phi :[a,b]\to C$ is a parameterization of a piecewise smooth curve $C,$ and that $t\in (a,b)$ is a point where $\phi$ is differentiable with ${\phi }^{\text{'}}\left(t\right)\ne 0.$ Show that the unit tangent (relative to the parameterization $\phi$ ) to $C$ at $z=\phi \left(t\right)$ exists and equals ${\phi }^{\text{'}}\left(t\right)/\left|{\phi }^{\text{'}}\left(t\right)\right|.$ Conclude that, except possibly for a finite number of points, the unit tangent to $C$ at $z$ is independent of the parameterization.
3. Let $C$ be the graph of the function $f\left(t\right)=\left|t\right|$ for $t\in [-1,1].$ Is $C$ a smooth curve? Is it a piecewise smooth curve?Does $C$ have a unit tangent at every point?
4. Let $C$ be the graph of the function $f\left(t\right)=t^{2/3}={\left(t^{1/3}\right)}^2$ for $t\in [-1,1].$ Is $C$ a smooth curve? Is it a piecewise smooth curve?Does $C$ have a unit tangent at every point?
5. Consider the set $C$ that is the right half of the unit circle in the plane. Let ${\phi }_1:[-1,1]\to C$ be defined by
   $${\phi }_1\left(t\right)=(cos\left(t\frac{\pi }{2}\right),sin\left(t\frac{\pi }{2}\right)),$$
   and let ${\phi }_2:[-1,1]\to C$ be defined by
   $${\phi }_2\left(t\right)=(cos\left(t^3\frac{\pi }{2}\right),sin\left(t^3\frac{\pi }{2}\right)).$$
   Prove that ${\phi }_1$ and ${\phi }_2$ are both parameterizations of $C.$ Discuss the existence of a unit tangent at the point $(1,0)={\phi }_1\left(0\right)={\phi }_2\left(0\right)$ relative to these two parameterizations.
6. Suppose $\phi :[a,b]\to C$ is a parameterization of a curve $C$ from $z_1$ to $z_2.$ Define $\psi$ on $[a,b]$ by $\psi \left(t\right)=\phi (a+b-t).$ Show that $\psi$ is a parameterization of a curve from $z_2$ to $z_1.$

1. Suppose $f$ is a smooth, real-valued function defined on the closed interval $[a,b],$ and let $C\subseteq R^2$ be the graph of $f.$ Show that $C$ is a smooth curve, and find a “natural” parameterization $\phi :[a,b]\to C$ of $C.$ What is the unit tangent to $C$ at the point $(t,f(t\left)\right)?$
2. Let $z_1$ and $z_2$ be two distinct points in $C,$ and define $\phi :[0,1]\to c$ by $\phi \left(t\right)=(1-t)z_1+t{z}_2.$ Show that $\phi$ is a parameterization of the straight line from the point $z_1$ to the point $z_2.$ Consequently, a straight line is a smooth curve. (Indeed, what is the definition of a straight line?)
3. Define a function $\phi :[-r,r]\to R^2$ by $\phi \left(t\right)=(t,\sqrt{r^2-t^2}).$ Show that the range $C$ of $\phi$ is a smooth curve, and that $\phi$ is a parameterization of $C.$
4. Define $\phi$ on $[0,\pi /2)$ by $\phi \left(t\right)=e^{it}.$ For what curve is $\phi$ a parametrization?
5. Let $z_1,z_2,...,z_n$ be $n$ distinct points in the plane, and suppose that the polygonal line joing these points in order never crosses itself.Construct a parameterization of that polygonal line.
6. Let $S$ be a piecewise smooth geometric set determined by the interval $[a,b]$ and the two piecewise smooth bounding functions $u$ and $l.$ Suppose $z_1$ and $z_2$ are two points in the interior $S^0$ of $S.$ Show that there exists a piecewise smooth curve $C$ joining $z_1$ to $z_2,$ i.e., a piecewise smooth function $\phi :[\widehat{a},\widehat{b}]\to C$ with $\phi \left(\widehat{a}\right)=z_1$ and $\phi \left(\widehat{b}\right)=z_2,$ that lies entirely in $S^0.$
7. Let $C$ be a piecewise smooth curve, and suppose $\phi :[a,b]\to C$ is a parameterization of $C.$ Let $[c,d]$ be a subinterval of $[a,b].$ Show that the range of the restriction of $\phi$ to $[c,d]$ is a smooth curve.

Suppose $C$ is a smooth curve, parameterized by $\phi =u+iv:[a,b]\to C.$

1. Suppose that $u^{\text{'}}\left(t\right)\ne 0$ for all $t\in (a,b).$ Prove that there exists a smooth, real-valued function $f$ on some closed interval $[a^{\text{'}},b^{\text{'}}]$ such that $C$ coincides with the graph of $f.$ HINT: $f$ should be something like $v\circ u^{-1}.$
2. What if $v^{\text{'}}\left(t\right)\ne 0$ for all $t\in (a,b)?$

Let $C$ be the curve that is the range of the function $\phi :[-1,1]\to C,$ where $\phi \left(t\right)=t^3+t^{6}i).$

1. Is $C$ a piecewise smooth curve? Is it a smooth curve?What points $z_1$ and $z_2$ does it join?
2. Is $\phi$ a parameterization of $C?$
3. Find a parameterization for $C$ by a function $\psi :[3,4]\to C.$
4. Find the unit tangent to $C$ and the point $0+0i.$

Let $C$ be the curve parameterized by $\phi :[-\pi ,\pi -ϵ]\to C$ defined by $\phi \left(t\right)=e^{it}=cos\left(t\right)+isin\left(t\right).$

1. What curve does $\phi$ parameterize?
2. Find another parameterization of this curve, but base on the interval $[0,1-ϵ].$