Exercises

01

Parametric curve: derivative rules for vector products

Use the algebraic properties of dot and cross product to prove the derivative product rules without using the component descriptions of those products. (Imitate the proof of the standard product rule.)

02

$\mathbf{T}$ and $\mathbf{N}$ for a circle

Compute $T$ and $N$ for the circle in the $yz$-plane given parametrically by $(0,R\cos (\theta ),R\sin (\theta ))$.

03

Derivative of reparametrized curve

Suppose $\mathbf{r}(t)$ is a parametric curve, and $t=t(s)$ is a function of another parameter $s$. Under what condition on $t(s)$ does the composite curve $\mathbf{r}(t(s))$ have unit speed everywhere?

04

Curvature of a line

Use $\mathbf{r}(t)=(x_0,y_0,z_0)+t\mathbf{v}$ for a vector-valued parametric line. Compute the curvature of this line using the definition of curvature.

05

Curvature of a circle

Write component formulas for a vector-valued parametric circle of radius $R$ in ${\mathbb{R}}^3$. Compute the curvature of this circle using the definition of curvature.

06

Curvature of a helix

Compute the curvature of the helix given parametrically by $(r\cos (t),r\sin (t),at)$ using the efficient formula. Also compute the unit tangent and principal normal vectors, $\mathbf{T}(t)$ and $\mathbf{N}(t)$. Compute $\mathbf{N}$ once using the definition, and again using the efficient formula.

07

Curvature of a graph

Use the efficient formula for curvature to derive a formula for the curvature of the graph of a function $f:\mathbb{R}\to \mathbb{R}$, namely the points $(x,y)$ satisfying $y=f(x)$. Start by finding a parametric curve whose image is this graph.

08

Normal vector to $(t,e^t,t)$ at $t=1$

Compute the principal normal $\mathbf{N}(t)$ using the efficient formula for the parametric curve given by $(t,e^t,t)$ at the point $t=1$.

09

Acceleration on a circle

Write a parametric curve traversing a circle of radius $R$ in ${\mathbb{R}}^3$ at constant speed. Compute the decomposition of acceleration into components: $\mathbf{a}=a_{\mathbf{T}}\mathbf{T}+a_{\mathbf{N}}\mathbf{N}$.