Lecture 007

Vector Valued Function

Vector Valued Function: A function $r: \mathbb{N} \rightarrow \overrightarrow{v}$ such that $\overrightarrow{r}(t) = f(t)\overrightarrow{i} + g(t)\overrightarrow{j} + h(t)\overrightarrow{k}$ in $\mathbb{R^3}$.

• Space curve: graph of a vector-valued function in $\mathbb{R^3}$.

• Plane curve: graph of a vector-valued function in $\mathbb{R^2}$.

• Example: $\langle{\cos(t), \sin(t), t}\rangle$ ($\langle{\cos(2t), \sin(2t), 2t}\rangle$ is the same space curve, but different vector function), but is not the same as $\langle{2\cos(t), 2\sin(t), 2t}\rangle$

Limits of Vector Valued Function

Limits of Vector Valued Function: $\lim_{t \rightarrow a} \overrightarrow{r}(t) = \lim_{t \rightarrow a}f(t)\overrightarrow{i} + \lim_{t \rightarrow a}g(t)\overrightarrow{j} + \lim_{t \rightarrow a}h(t)\overrightarrow{k}$ given that the limits all exist.

Limit Exist: if limit exists from both side, and equal to each other, and not approaching infinity, then the limit exist.

Continuity: $\overrightarrow{r}(t)$ is continuous at $t=a$ if $\lim_{t \rightarrow a} \overrightarrow{v}(t) = \overrightarrow{r}(a)$

• ie. if one of limit does not exists, or infinity, then the limit of the whole vector function does not exist

Derivatives of Vector Valued Function

Definition 1: $\overrightarrow{r}(t) = \lim_{\Delta t \rightarrow 0} \frac{\overrightarrow{r}(t + \Delta t) - \overrightarrow{r} (t)}{\Delta t}$

Definition 2: given $\overrightarrow{r}(t) = \langle{f(t), g(t), h(t)}\rangle \text{ with f, g, h differentiable}$, then $\overrightarrow{r}'(t) = \langle{f'(t), g'(t), h'(t)}\rangle$

Principal Unit Tangent Vector: Principal Unit Tangent Vector $\overrightarrow{T}(t)$ at $t = t_0$ is $\frac{\overrightarrow{r}'(t_0)}{\|\overrightarrow{r}'(t_0)\|}$

Dot Product of Vector Valued Function: $\overrightarrow{r_1} \cdot \overrightarrow{r_2} = f_1f_2 + g_1g_2 + h_1h_2$

• Dot Product Derivative $(\overrightarrow{r_1} \cdot \overrightarrow{r_2})' = \overrightarrow{r_1} \cdot \overrightarrow{r_2}' + \overrightarrow{r_1}' \cdot \overrightarrow{r_2} = f_1f_2' + f_1'f_2 + g_1g_2' + g_1'g_2 + h_1h_2' + h_1'h_2$

• This is similar to multiplication rule for derivative.

Cross Product Derivative: $(\overrightarrow{r_1}(t) \times \overrightarrow{r_2}(t))' = \overrightarrow{r_1}(t) \times \overrightarrow{r_2}'(t) + \overrightarrow{r_1}'(t) \times \overrightarrow{r_2}(t)$

Integrals of Space Curves $\int \overrightarrow{r}(t) dt = \langle{\int f(t) dt, \int g(t)dt, \int h(t) dt}\rangle$