Calculus with Vector Functions

Let’s say our generic vector function form is $\vec{r}(t)=⟨f(t),g(t),h(t)⟩$

Limits of Vector Functions

$$\begin{array}{rl}\underset{t\to a}{\lim }\vec{r}(t)& =\underset{t\to a}{\lim }⟨f(t),g(t),h(t)⟩\\ & =⟨\underset{t\to a}{\lim }f(t),\underset{t\to a}{\lim }g(t),\underset{t\to a}{\lim }h(t)⟩\\ & =\underset{t\to a}{\lim }f(t)\vec{i}+\underset{t\to a}{\lim }g(t)\vec{j}+\underset{t\to a}{\lim }h(t)\vec{k}\end{array}$$

Derivatives of Vector Functions

$$\begin{array}{rl}{\vec{r}}^{\prime }(t)& =⟨f^{\prime }(t),g^{\prime }(t),h^{\prime }(t)⟩\\ & =f^{\prime }(t)\vec{i}+g^{\prime }(t)\vec{j}+h^{\prime }(t)\vec{k}\end{array}$$

Properties of Derivatives

• Addition: $\frac{d}{dt}(\vec{u}+\vec{v})={\vec{u}}^{\prime }+{\vec{v}}^{\prime }$
• Constant multiple:$(c\vec{u}{)}^{\prime }=c{\vec{u}}^{\prime }$
• Chain rule: $\frac{d}{dt}(\vec{u}\cdot f(t))=f^{\prime }(t)\cdot {\vec{u}}^{\prime }f(t)$
• Product rule:

$$\frac{d}{dt}(f(t)\vec{u}(t))=f^{\prime }(t)\vec{u}(t)+f(t){\vec{u}}^{\prime }(t)$$ $$\frac{d}{dt}(\vec{u}\cdot \vec{v})={\vec{u}}^{\prime }\cdot \vec{v}+\vec{u}\cdot {\vec{v}}^{\prime }$$ $$\frac{d}{dt}(\vec{u}\times \vec{v})={\vec{u}}^{\prime }\times \vec{v}+\vec{u}\times {\vec{v}}^{\prime }$$

Integrals of Vector Functions

Indefinite:

$$\begin{array}{rl}\int \vec{r}(t)dt& =⟨\int f(t)dt,\int g(t)dt,\int h(t)dt⟩\\ & =\int f(t)dt\vec{i}+\int g(t)dt\vec{j}+\int h(t)dt\vec{k}+\vec{c}\end{array}$$

Definite:

$$\begin{array}{rl}{\int }_a^b\vec{r}(t)dt& =⟨{\int }_a^{b}f(t)dt,{\int }_a^{b}g(t)dt,{\int }_a^{b}h(t)dt⟩\\ & \\ & ={\int }_a^{b}f(t)dt\vec{i}+{\int }_a^{b}g(t)dt\vec{j}+{\int }_a^{b}h(t)dt\vec{k}\\ & =\left(⟨\int f(t)dt,\int g(t)dt,\int h(t)dt⟩\right){|}_a^b\\ & =\left(\int f(t)dt\vec{i},\int g(t)dt\vec{j},\int h(t)dt\vec{j}\right){|}_a^b\end{array}$$

Smooth Curve

• Any curve for which $\vec{r}(t)$ is continuous and $\vec{r}(t)\ne 0$ for any $t$ (except maybe endpoints)
• A helix is an example of smooth curve