Chapter Review

• The vector function \(\mathbf{r}=\mathbf{r}(t)=\left\langle x(t), y( t) \right\rangle\;=\;x(t)\mathbf{i}+y(t)\mathbf{j, }\) \(a\leq t\leq b,\) traces out a plane curve. (p. 758)
• The vector function \(\mathbf{r}=\mathbf{r}(t)=\left\langle x( t), y( t), z( t) \right\rangle\;=\;x(t) \mathbf{i}+y(t)\mathbf{j}+z(t)\mathbf{k}\), \(a\leq t\leq b,\) traces out a space curve. (p. 759)
• Limit of a vector function (p. 761)
• A vector function \(\mathbf{r}=\mathbf{r}(t)\) is continuous at a real number \(t_{0}\) if:

• \(\mathbf{r}(t_{0})\)is defined
• \(\lim\limits_{t\rightarrow t_{0}}\mathbf{r}(t) \hbox{exists }\)
• \(\lim\limits_{t\rightarrow t_{0}}\mathbf{r}(t)=\mathbf{r}(t_{0})\)(p. 761)

• The derivative of the vector function \(\mathbf{r=r}( t) \) is \(\mathbf{r}^{\prime} (t)=\lim\limits_{h\rightarrow 0}\dfrac{ \mathbf{r}(t+h)-\mathbf{r}(t)}{h},\) provided the limit exists. (p. 762)

Derivative formulas: (p. 763)

• Sum formula: \(\dfrac{d}{dt}\left[ \mathbf{u}(t)+\mathbf{v}(t)\right]\;=\; \dfrac{d}{dt}\mathbf{u}(t)+\dfrac{d}{dt}\mathbf{v}(t)=\mathbf{u^{\prime} }(t)+ \mathbf{v^{\prime} }(t)\)
• Scalar multiplication formula: \(\dfrac{d}{dt}\left[ f(t)\mathbf{u}(t) \right]\;=\;\left[ \dfrac{d}{dt}f( t) \right] \mathbf{u}( t) +f( t) \!\left[ \dfrac{d}{dt}\mathbf{u}( t) \right]\;=\;f^{\prime} (t)\mathbf{u}(t)+f(t)\mathbf{u}^{\prime} (t)\)
• Dot product formula: \(\dfrac{d}{dt}\left[ \mathbf{u}( t) \,{\cdot}\, \mathbf{v}( t) \right]\;=\;\left[ \mathbf{u}(t)\,{\cdot}\, \mathbf{v} (t)\right] ^{\prime}\;=\;\mathbf{u^{\prime} }(t)\,{\cdot}\, \mathbf{v}(t)+\mathbf{u} (t)\,{\cdot}\, \mathbf{v}^{\prime} (t)\)
• Cross product formula: \(\dfrac{d}{dt}\left[ \mathbf{u}( t) \times \mathbf{v}( t) \right]\;=\;\left[ \mathbf{u}(t)\times \mathbf{ v}(t)\right] ^{\prime}\;=\;\mathbf{u^{\prime} }(t)\times \mathbf{v}(t)+\mathbf{u} (t)\times \mathbf{v}^{\prime} (t)\)
• Chain Rule: \(\dfrac{d}{dt} \mathbf{v} (f (t))\;=\;\mathbf{v}\prime (f(t)) \,{\cdot}\, f^{\prime}(t)\)

• Tangent vector to a curve (p. 768)
• Smooth curve (p. 768)
• Unit tangent vector: \(\mathbf{T}(t)=\dfrac{\mathbf{r}^{\prime} (t)}{ \left\Vert \mathbf{r}^{\prime} (t)\right\Vert }\) (p. 768)
• Principal unit normal vector: \(\mathbf{N}(t)=\dfrac{\mathbf{T} ^{\prime} (t)}{\left\Vert \mathbf{T}^{\prime} (t)\right\Vert }\) (p. 769)
• Arc length \(s\) of a vector function: \(s=\int_{a}^{b}\left\Vert \mathbf{r}^{\prime} (t)\right\Vert dt\) (p. 770)

• Parameter \(t\) is arc length if and only if \(\left\Vert \mathbf{r}^{\prime} (t)\right\Vert\;=\;1.\) (p. 775)
• Curvature of a curve: \(\kappa\;=\;\left\Vert \dfrac{d\mathbf{T}}{ds} \right\Vert\;=\;\dfrac{\left\Vert \mathbf{T}^{\prime} ( t) \right\Vert }{\left\Vert \mathbf{r}^{\prime} ( t) \right\Vert },\) where the parameter \(s\) is arc length. (p. 777)
• Curvature of a space curve: \(\kappa\;=\;\dfrac{\left\Vert \mathbf{r}^{\prime} (t)\times \mathbf{r}^{\prime \prime} (t)\right\Vert }{\left\Vert \mathbf{r}^{\prime} ( t) \right\Vert ^{3}}\) (p. 778)
• Curvature of a plane curve given by \(y=f(x)\): \(\kappa\;=\;\dfrac{ \left\vert y^{\prime \prime} \right\vert }{[ 1+( y^{\prime} ) ^{2}] ^{3/2}}\) (p. 779)

Properties of an osculating circle (p. 780)

• The osculating circle contains a point \(P\) on a curve \(C\).
• The tangent line to \(C\) at \(P\) and the tangent line to the osculating circle at \(P\) are the same.
• The radius \(\rho \) of the osculating circle is \(\rho\;=\;\dfrac{1}{\kappa }\), where \(\kappa \neq 0\) is the curvature of the curve \(C.\)
• The center of the osculating circle lies on the line in the direction of the principal unit normal vector \(\mathbf{N}\) to \(C\) at \(P\).

• Position: \(\mathbf{r}(t)=x(t)\mathbf{i}+y(t)\mathbf{j}+z(t)\mathbf{k}\) (p. 785)
• Velocity: \(\mathbf{v}(t)=\mathbf{r}^{\prime} (t)=\dfrac{dx}{dt}\mathbf{i }+\dfrac{dy}{dt}\mathbf{j}+\dfrac{dz}{dt}\mathbf{k}\) (p. 785)
• Acceleration: \(\mathbf{a}(t)=\mathbf{r}^{\prime \prime} (t)=\dfrac{d^{2}x }{dt^{2}}\mathbf{i}+\dfrac{d^{2}y}{dt^{2}}\mathbf{j}+\dfrac{d^{2}z}{dt^{2}} \mathbf{k}\) (p. 785)
• Speed: \(v(t)=\;\left\Vert \mathbf{v}(t)\right\Vert\;=\;\left\Vert \mathbf{r} ^{\prime} (t) \right\Vert \) (p. 785)
• Tangential and normal components of an acceleration vector: \( \mathbf{a} (t)=a_{\mathbf{T}}\mathbf{T}(t)+a_{\mathbf{N}}\mathbf{N}(t),\) where \(a_{\mathbf{T}}=v^{\prime }(t)=\dfrac{dv}{dt}\qquad \hbox{and}\qquad a_{\mathbf{N}}=[ v(t)] ^{2}\kappa \) (p. 789)

• If \(\mathbf{r}(t)=x(t)\mathbf{i}+y(t)\mathbf{j}+z(t)\mathbf{k}\), then \( \int \mathbf{r}(t)dt=\left[ \int x(t)dt\right] \mathbf{i}+\left[ \int y(t)dt \right] \mathbf{j}+\left[ \int z(t)dt\right] \mathbf{k}\) (p. 796)
• Projectile motion (p. 797)

• The orbit of each planet is an ellipse with the Sun at a focus. (p. 801)
• A line joining the Sun to a planet sweeps out equal areas in equal times. (p. 801)
• The square of the period of revolution of a planet is proportional to the cube of the length of the semimajor axis of the planet's elliptical orbit. (p. 801)