13.3 Videos Guide

13.3a

• Arc length

  • $L=\int^{b}_{a}{\left|\mathbf{r}'(t)\right|\ dt}$

Exercises:

• Find the length of the curve $\mathbf{r}(t)=\left\langle 2t,\ t^{2},\ \frac{1}{3}t\right\rangle$, $0\leq t\leq1$.

• Find, correct to four decimal places, the length of the curve of intersection of the cylinder $4x^{2}+y^{2}=4$ and the plane $x+y+z=2$.

13.3b

• Arc length function

  • $s(t)=\int^{t}_{a}{\left|\mathbf{r}'(u)\right|\ du}$

Exercise:

• (a) Find the arc length function for the curve measured from the point $P$ in the direction of increasing $t$ and then reparameterize the curve with respect to arc length starting from $P$, and (b) find the point 4 units along the curve (in the direction of increasing $t$) from $P$.
  $\mathbf{r}(t)=e^{t}\sin t\mathbf{i}+e^{t}\cos t\mathbf{j}+\sqrt{t}e^{t}\mathbf{k}$, $P(0,\ 1,\ \sqrt{2}$)

13.3c

• Curvature

  • For a vector function: $\kappa(t)=\left|\frac{d\mathbf{T}}{ds}\right|=\frac{\left|\mathbf{T}'(t)\right|}{\left|\mathbf{r}'(t)\right|}=\frac{\left|\mathbf{r}'(t)\times\mathbf{r}^{''}(t)\right|}{\left|\mathbf{r}'(t)\right|^{3}}$

  • For a plane curve in $\mathbb{R}^{2}$: $\kappa(x)=\frac{\left|f^{''}(x)\right|}{\left\lbrack 1+\left\lbrack f'(x)\right\rbrack ^{2}\right\rbrack ^{\frac{3}{2}}}$

  • The radius of the osculating circle to a curve at a point is $\rho=\frac{1}{\kappa}$ at that point

• Development of the formula $\kappa(t)=\frac{\left|\mathbf{T}'(t)\right|}{\left|\mathbf{r}'(t)\right|}$

13.3d

• Development of the formula $\kappa(t)=\frac{\left|\mathbf{r}'(t)\times\mathbf{r}^{''}(t)\right|}{\left|\mathbf{r}'(t)\right|^{3}}$

13.3e

• Unit vectors

  • The unit tangent vector: $\mathbf{T}(t)=\frac{\mathbf{r}'(t)}{\left|\mathbf{r}'(t)\right|}$ (from Lesson 13.2)

  • The unit normal vector: $\mathbf{N}(t)=\frac{\mathbf{T}'(t)}{\left|\mathbf{T}'(t)\right|}$

  • The binormal vector: $\mathbf{B}(t)=\mathbf{T}(t)\times\mathbf{N}(t)$

• Planes

  • The direction vector for the normal plane is the unit tangent vector $\mathbf{T}(t)$

  • The direction vector for the binormal plane is the binormal vector $\mathbf{B}(t)$

Exercises:

13.3f

• (a) Find the unit tangent and unit normal vectors $\mathbf{T}(t)$ and $\mathbf{N}(t)$.

(b) Use $\kappa(t)=\frac{\left|\mathbf{T}'(t)\right|}{\left|\mathbf{r}'(t)\right|}$ to find the curvature.

$\mathbf{t}(t)=\left\langle t^{2},\sin t-t,\cos t+t\sin t\right\rangle$, $t>0$

13.3g

• Use $\kappa(t)=\frac{\left|\mathbf{r}'(t)\times\mathbf{r}^{''}(t)\right|}{\left|\mathbf{r}'(t)\right|^{3}}$ to find the curvature.
  $$\mathbf{r}(t)=t\mathbf{i}+t^{2}\mathbf{j}+e^{t}\mathbf{k}$$

13.3h

• Find equations of the normal plane and osculating plane of the curve at the given point.
  $x=\ln t$, $y=2t$, $z=t^{2}$, $(0,\ 2,\ 1)$