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}.

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)

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

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

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