Definitiones de functiones vectorial

Definition

• Un function vectorial $\mathbf{r}$ ha un dominio que es un subconjunto de numeros real e un rango que es un collection de vectores.
• Pro cata numero real $t$ in le dominio de $\mathbf{r}$, le valor de $\mathbf{r}(t)$ es un vector.
• Forma: $$ \begin{aligned} \mathbf{r}: \mathbb{R} & \longrightarrow \mathbb{R}^{n} \ t & \longmapsto \mathbf{r}(t)=\left(f_{1}(t), f_{2}(t), \ldots, f_{n}(t)\right) \end{aligned} $$
• Le functiones componente $f_{i}: \mathbb{R} \longrightarrow \mathbb{R}$ son functiones real del variabile real $t$.

Exemplo

• Un exemplo de un function vectorial es: $$ \mathbf{r}(t)=\langle\cos t, \operatorname{sen} t, t\rangle $$
• Con functiones componente $f_{1}(t)=\cos t, f_{2}(t)=\operatorname{sen} t$ e $f_{3}(t)=t$.

Limite de un Function Vectorial

• Si $\mathbf{r}(t)=\left\langle f_{1}(t), f_{2}(t), \ldots, f_{n}(t)\right\rangle$, alora: $$ \lim _{t \rightarrow a} \mathbf{r}(t)=\left\langle\lim {t \rightarrow a} f{1}(t), \lim {t \rightarrow a} f{2}(t), \ldots, \lim {t \rightarrow a} f{n}(t)\right\rangle $$
• Isto es solmente valide si le limites del functiones componente existe.

Exemplo

• Un calculation de limite es: $$ \lim _{t \rightarrow 0}\left\langle 1+t^{3}, t e^{-t}, \frac{\operatorname{sen} t}{t}\right\rangle=\left\langle\lim _{t \rightarrow 0}\left(1+t^{3}\right), \lim _{t \rightarrow 0} t e^{-t}, \lim _{t \rightarrow 0} \frac{\operatorname{sen} t}{t}\right\rangle=\langle 1,0,1\rangle $$

Continuitate de Functiones Vectorial

• Un function vectorial $\mathbf{r}$ es continue in $a$ si: $$ \lim _{t \rightarrow a} \mathbf{r}(t)=\mathbf{r}(a) $$
• $\mathbf{r}(t)=\left\langle f_{1}(t), f_{2}(t), \ldots, f_{n}(t)\right\rangle$ es continue in $a$ si le functiones componente $f_{1}, f_{2}, \ldots, f_{n}$ es continue in $a$.

Derivation de un Function Vectorial

• Si $\mathbf{r}(t)=\left\langle f_{1}(t), f_{2}(t), \ldots, f_{n}(t)\right\rangle$, ubi $f_{1}, f_{2}, \ldots, f_{n}$ son functiones derivabile, alora: $$ \mathbf{r}^{\prime}(t)=\left\langle f_{1}^{\prime}(t), f_{2}^{\prime}(t), \ldots, f_{n}^{\prime}(t)\right\rangle $$

Exemplo

• Si $\mathbf{r}(t)=\left\langle t, e^{-3 t}, t^{2}\right\rangle$, alora $\mathbf{r}^{\prime}(t)=\left\langle 1,-3 e^{-3 t}, 2 t\right\rangle$.

Integral de Functiones Vectorial

• Le integral indefinite de $\mathbf{r}$ es un altere function vectorial $\mathbf{R}$ tal que $\mathbf{R}^{\prime}(t)=\mathbf{r}(t)$.
• Si $\mathbf{r}(t)=\left\langle f_{1}(t), f_{2}(t), \ldots, f_{n}(t)\right\rangle$, alora: $$ \int \mathbf{r}(t) d t=\left\langle\int f_{1}(t) d t, \int f_{2}(t) d t, \ldots, \int f_{n}(t) d t\right\rangle $$
• Le integral definite de $\mathbf{r}$ inter $a$ e $b$ es: $$ \int_{a}^{b} \mathbf{r}(t) d t=\left\langle\int_{a}^{b} f_{1}(t) d t, \int_{a}^{b} f_{2}(t) d t, \ldots, \int_{a}^{b} f_{n}(t) d t\right\rangle $$

Exemplo

• Un calculation de integral es: $$ \begin{aligned} \int_{0}^{1}\left(t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}\right) d t & =\left(\int_{0}^{1} t d t\right) \mathbf{i}+\left(\int_{0}^{1} t^{2} d t\right) \mathbf{j}+\left(\int_{0}^{1} t^{3} d t\right) \mathbf{k} \ & =\left.\frac{t^{2}}{2}\right|{0} ^{1} \mathbf{i}+\left.\frac{t^{3}}{3}\right|{0} ^{1} \mathbf{j}+\left.\frac{t^{4}}{4}\right|_{0} ^{1} \mathbf{k} \ & =\frac{1}{2} \mathbf{i}+\frac{1}{3} \mathbf{j}+\frac{1}{4} \mathbf{k} \end{aligned} $$