Multivariable Calculus 13.1 Textbook Notes

Status: Adult

13.1 Curves in Space and their Tangents

A curve in space is given by a vector-valued function $r(t)=〈f(t),g(t),h(t)〉$of a parameter $t$ defined on some interval $(a,b)$ or $[a,b]$ of $\mathbb{R}$. When one plots the graph of a function $y=g(t)$ of one variable $t$, one plots both the $t$ and the $y$-axis. When one plots the “graph” of a vector-valued function $r(t)=〈f(t),g(t),h(t)〉$, however, one often omits the $t$-axis, and only plots the collection of points $(f(t),g(t),h(t))$ for $t\in (a,b)$. What one sees is a curve, but for a specific point on this curve, the value $t$ that gives rise to this point is not plotted (it is possible that more than one values of $t$ may give rise to the same point).

If one treats a parametric curve $r(t)=〈f(t),g(t)〉$ as an ${\mathbb{R}}^2$-valued function, it is possible
to plot the “graph” of this function including the $t$-axis. This GeoGebra/3d plot plots the ${\mathbb{R}}^2$-
valued functions $r_2(t)=〈sin(2t),cos(2t)〉$ and $r(t)=〈sin(t),cos(t)〉$ over $0\le t\le 2\pi$ (plotted
as parametric curves $〈sin(2t),cos(2t),t〉$ and $〈sin(t),cos(t),t〉$ in 3D and treating the $z$-axis for
the $t$-axis). They are different functions, but both $(sin(2t),cos(2t))$ and $(sin(t),cos(t))$ lie on
the circle $x^2+y^2=1$ as one can see from the top view over the z-axis and one would get both
parametric curves as the curve $x^2+y^2=1$ if one does not plot the $t$-axis.

Likewise, two different vector-valued functions may produce the same curve in space, as is
the case of $r(t)=〈\cos t,\sin t,t〉$ and $r_3(t)=〈\cos t^3,\sin t^3,t^3〉$. Note that $r_3(t)=r(t^3)$, which
means that the point $r_3(t)$ is simply the point on the first curve at time $t^3$.

So we use the terminology curve for a vector-valued function of one variable, but there is
this subtle difference between its geometric interpretation and its property as a function.

• Read the definition of the derivative of a vector-valued function $r(t)=〈f(t),g(t),h(t)〉$and its interpretation as the velocity vector if a particle moves with time t according to this function.
• Why does the definition of a smooth curve require that $\frac{dr}{dt}$ be continuous and never 0? Plot the curve $r(t)=(t^2,t^3,0)$ and investigate whether it is smooth at the point corresponding to t = 0.
• Study the definition of a piecewise smooth curve. Is the curve $r(t)=(t^2,t^3,0)$ piecewise smooth according to this definition?
• Interpret $||\frac{dr}{dt}\Vert$ as the speed and $\frac{d^{2}r}{d{t}^2}=r^{\prime \prime }(t)$ as the acceleration if a particle moves with time $t$ according to $r(t)$.
• Read about the differentiation rules, paying particular attention to the Dot Product Rule, Cross Product Rule, and Chain Rule.
• Study the property of a differentiable vector-valued function $r(t)$ satisfying ‖r(t)‖ equals a constant (Hint: $r(t)\cdot r(t)=a$ constant, so its derivative in $t$ will be = 0).

Reading Questions/Quizzes

1. Given $r(t)=〈\cos t,\sin t,t〉$ and $r_3(t)=〈\cos t^3,\sin t^3,t^3〉$. Plot both curves for $-1\le t\le 1$ (you may use GeoGebra or Desmos) and compute $\frac{dr(t)}{dt},||\frac{dr(t)}{dt}||,\frac{d^{2}r(t)}{d{t}^2},\frac{d{r}_3(t)}{dt},||\frac{d{r}_3(t)}{dt}||,\frac{d^2{r}_3(t)}{d{t}^2}$. Does $r(t)$ travel with constant velocity, or constant speed, or constant acceleration? What about $r_3(t)$?
   1. $\frac{dr(t)}{dt}=<-\sin t,\cos t,1>$
   2. $||\frac{dr(t)}{dt}||=\sqrt{{\sin }^{2}t+{\cos }^{2}t+1}=\sqrt{2}$
   3. $\frac{d^{2}r(t)}{d{t}^2}=<-\cos t,-\sin t,1>$
   4. $\frac{d{r}_3(t)}{dt}=<-3t^2\sin t^3,3t^2\cos t^3,3t^2>$
   5. $||\frac{d{r}_3(t)}{dt}||=\sqrt{9t^4{\sin }^2{t}^3+9t^4{\cos }^2{t}^3+3t^2}=\sqrt{9t^4({\sin }^2{t}^3+{\cos }^2{t}^3)+3t^2}=t\sqrt{9t^2+3}$
   6. $||\frac{d^2{r}_3(t)}{d{t}^2})||=<-6t\sin t^3-9t^4\cos t^3,6t\cos t^3-9t^4\sin t^3,6t>$
   7. $r(t)$ does not travel with constant velocity, travels with constant speed, does not travel with constant acceleration
   8. $r_3(t)$ does not travel with constant velocity, speed, or acceleration
2. Set $v(t)=\Vert r\prime (t)\Vert =\sqrt{r\prime (t)\cdot r\prime (t)}$. Derive a formula for $v\prime (t)$ in terms of $r\prime (t)$ and $r\prime \prime (t)$
   1. $v^{\prime }(t)=\frac{r^{\prime \prime }(t)\cdot r^{\prime }(t)}{\sqrt{r^{\prime }(t)\cdot r^{\prime }(t)}}$