Vector Functions

$$\begin{gathered} f:\mathbb{R}\to {\mathbb{R}}^n \\ \vec{r}(t)=⟨x(t),y(t),z(t)⟩ \end{gathered}$$

Properties

• Domain of $\vec{r}(t)$ = Domain of $\vec{x}(t)$ $\cap$ Domain of $\vec{y}(t)$ $\cap$ Domain of $\vec{z}(t)$
• Limit:
  • ${\lim }_{t\to t_0}\vec{r}(t)=\vec{a}$
  • $⟺{\lim }_{t\to t_0}\Vert \vec{r}(t)-\vec{a}\Vert =0$
  • $⟺{\lim }_{t\to t_0}x(t)=a_1,{\lim }_{t\to t_0}y(t)=a_2,{\lim }_{t\to t_0}z(t)=a_3$
  • $⟹$ the limit $\vec{r}(t)$ only exists of the limit for $x(t)$, $y(t)$, and $z(t)$ exist

Derivative

• Geometric significance: The slope of a linear line $\vec{L}(t)$ through $\vec{r}(a)$ where $\vec{r}(t)\simeq \vec{L}(t)$ near $t=a$
• ${\vec{r}}^{\prime }(t)=\frac{\text{d}}{\text{d}t}\vec{r}(t)={\lim }_{h\to 0}\frac{\vec{r}(t+h)-\vec{r}(t)}{h}$
  • $\vec{r}(t)$ is differentiable at $t$ if this limit exists
• $\vec{r}(t)$ is differentiable $⟺$ it’s components are differentiable
• ${\vec{r}}^{\prime }(t)=⟨x^{\prime }(t),y^{\prime }(t),z^{\prime }(t)⟩$

Integrals

• Antiderivative: $⟨x(t),y(t),z(t)⟩\to ⟨X(t),Y(t),Z(t)⟩+\vec{C}$
  • One constant per interval of continuity (same as the planar integral)
• The most general antiderivative $\equiv$ Indefinite integral
  • $\int \vec{r}(t)dt=\vec{R}(t)+\vec{C}$
• $\int \vec{r}(t)dt=⟨\int x(t)dt,\int y(t)dt,\int z(t)dx⟩$
• FTC
  • $\vec{r}(t)$ is continuous $⟹{\int }_a^b\vec{r}(t)dt=\vec{R}(b)-\vec{R}(a)$
  • ${\vec{r}}^{\prime }(t)$ is continuous $⟹\vec{r}(t_0)+{\int }_{t_0}^t{\vec{r}}^{\prime }(\tau )d\tau$
• Application: Newton’s Law
  • $\vec{r}(t)=$ position vector
  • ${\vec{r}}^{\prime }(t)=\vec{v}(t)=$ velocity
  • ${\vec{r}}^{\prime \prime }(t)={\vec{v}}^{\prime }(t)=\vec{a}(t)=$ acceleration
  • $\frac{\text{d}}{\text{d}t}\Vert \vec{v}(t)\Vert =\frac{\vec{v}(t)\cdot {\vec{v}}^{\prime }(t)}{\Vert \vec{v}(t)\Vert }$: Use for minimizing speed