一元向量值函数
时间:2022-10-10 19:30:00 sitemap
一元向量值函数
- 定义
- 极限
- 导数(导数)
-
- 定义
- 计算
- 性质
- 几何意义
定义
设数集 D ? R D\subset \mathbb{R} D?R,则称映射 f : D → R n \boldsymbol{f}:D \to \mathbb{R}^n f:D→Rn 为一元向量值函数,记为
r = f ( t ) , t ∈ D \boldsymbol{r} = \boldsymbol{f}(t) , t \in D r=f(t),t∈D
其中, D D D 为函数定义域, t t t 为自变量, r \boldsymbol{r} r 为因变量。
极限
以三维向量为例,设 f ( t ) = ( f 1 ( t ) , f 2 ( t ) , f 3 ( t ) ) \boldsymbol{f}(t) = \left(f_1(t), f_2(t),f_3(t)\right) f(t)=(f1(t),f2(t),f3(t)),则
lim t → t 0 f ( t ) = ( lim t → t 0 f 1 ( t ) , lim t → t 0 f 2 ( t ) , lim t → t 0 f 3 ( t ) ) \lim_{t \to t_0} \boldsymbol{f}(t) = \left( \lim_{t \to t_0} f_1(t),\lim_{t \to t_0} f_2(t),\lim_{t \to t_0} f_3(t) \right) t→t0limf(t)=(t→t0limf1(t),t→t0limf2(t),t→t0limf3(t))
导数(导向量)
定义
设 f ( t ) \boldsymbol{f}(t) f(t) 在点 t 0 t_0 t0 的某一邻域内有定义,若 lim Δ t → 0 Δ r Δ t = f ( t 0 + Δ t ) + f ( t 0 ) Δ t \lim\limits_{\Delta t \to 0} \dfrac{\Delta\boldsymbol{r}}{\Delta t} = \dfrac{\boldsymbol{f} (t_0+\Delta t) + \boldsymbol{f} (t_0)}{\Delta t} Δt→0limΔtΔr=Δtf(t0+Δt)+f(t0) 存在,则称此极限向量为一元向量函数 r = f ( t ) \boldsymbol{r} = \boldsymbol{f}(t) r=f(t) 在 t 0 t_0 t0 处的导数(导向量),记作 f ′ ( t 0 ) \boldsymbol{f}^{'}(t_0) f′(t0) 或 d r d t ∣ t = t 0 \left.\dfrac{d \boldsymbol{r}}{dt}\right\vert_{t=t_0} dtdr∣∣∣∣t=t0。
计算
f ′ ( t 0 ) = f 1 ′ ( t 0 ) i + f 2 ′ ( t 0 ) j + f 3 ′ ( t 0 ) k \boldsymbol{f}^{'}(t_0) = f_1^{'}(t_0) \boldsymbol{i} + f_2^{'}(t_0) \boldsymbol{j} + f_3^{'}(t_0) \boldsymbol{k} f′(t0)=f1′(t0)i+f2′(t0)j+f3′(t0)k
性质
d d t C = 0 d d t [ c u ( t ) ] = c u ′ ( t ) d d t [ u ( t ) ± v ( t ) ] = u ′ ( t ) ± v ′ ( t ) d d t [ φ ( t ) u ( t ) ] = φ ′ ( t ) u ( t ) + φ ( t ) u ′ ( t ) d d t [ u ( t ) v ( t ) ] = u ′ ( t ) v ( t ) + u ( t ) v ′ ( t ) d d t [ u ( t ) × v ( t ) ] = u ′ ( t ) × v ( t ) + u ( t ) × v ′ ( t ) d d t u [ φ ( t ) ] = φ ′ ( t ) u ′ [ φ ( t ) ] \begin{aligned} & \frac{d}{dt} \boldsymbol{C} = 0 \\ & \frac{d}{dt} \left[ c\boldsymbol{u}(t) \right] = c\boldsymbol{u}^{'}(t) \\ & \frac{d}{dt} \left[ \boldsymbol{u}(t) \pm \boldsymbol{v}(t) \right] = \boldsymbol{u}^{'}(t) \pm \boldsymbol{v}^{'}(t) \\ & \frac{d}{dt} \left[ \varphi(t)\boldsymbol{u}(t) \right] = \varphi^{'}(t)\boldsymbol{u}(t) + \varphi(t)\boldsymbol{u}^{'}(t) \\ & \frac{d}{dt} \left[ \boldsymbol{u}(t)\boldsymbol{v}(t) \right] = \boldsymbol{u}^{'}(t)\boldsymbol{v}(t) + \boldsymbol{u}(t)\boldsymbol{v}^{'}(t) \\ & \frac{d}{dt} \left[ \boldsymbol{u}(t) \times \boldsymbol{v}(t) \right] = \boldsymbol{u}^{'}(t) \times \boldsymbol{v}(t) + \boldsymbol{u}(t) \times \boldsymbol{v}^{'}(t) \\ & \frac{d}{dt} \boldsymbol{u}\left[\varphi(t)\right] = \varphi^{'}(t)\boldsymbol{u}^{'}\left[\varphi(t)\right] \end{aligned} dtdC=0dtd[cu(t)]=cu′(t)dtd[u(t)±v(t)]=u′(t)±v′(t)dtd[φ(t)u(t)]=φ′(t)u(t)+φ(t)u′(t)dtd[u(t)v(t)]=u′(t)v(t)+u(t)v′(t)dtd[u(t)×v(t)]=u′(t)×v(t)+u(t)×v′(t)dtdu[φ(t)]=φ′(t)u′[φ(t)]
几何意义
导向量是向量值函数终端曲线 Γ \mathit{\Gamma} Γ 在点 M M M 处的一个切向量,其指向与 t t t 的增长方向( t t t 增大时点 M M M 的移动方向)一致。