Tài liệu về nhà:
$\left\{\begin{matrix} x = r\: \cos \varphi \: \sin \theta \\ y = r\: \sin \varphi \: \sin \theta \\ z = r\: \cos \theta \end{matrix}\right.$ và $\left\{\begin{matrix}\overrightarrow{e_{r}}=cos\varphi\: sin\theta\overrightarrow{i}+sin\varphi\: sin\theta\overrightarrow{j}+cos\theta\overrightarrow{k}\\\overrightarrow{e_\theta}= cos\varphi\: cos\theta\overrightarrow{i}+ sin\varphi cos\theta\overrightarrow{j}-sin\theta\overrightarrow{k}\\\overrightarrow{e_\varphi}=-sin\varphi\overrightarrow{i}+cos\varphi\overrightarrow{j}\end{matrix}\right.$
+ Vector vận tốc $(\overrightarrow{v})$ trong tọa độ cầu:
Ta có: $v=v_xi+v_yi+v_zk=\frac{dv}{dt}$ với $v_x=\frac{dx}{dt},\: v_y=\frac{dy}{dt},\: v_z=\frac{dz}{dt}$
$\Rightarrow \overrightarrow{v}=\frac{dx}{dt}.\overrightarrow{i}+\frac{dy}{dt}.\overrightarrow{j}+\frac{dz}{dt}.\overrightarrow{k}$
$=r(cos\varphi\: cos\theta \:\overrightarrow{i}+sin\varphi\: cos\theta\: \overrightarrow{j}-sin\theta\: \overrightarrow{k}).\frac{d\theta}{dt}+r(-sin\varphi\: sin \theta\: \overrightarrow{i}+ cos\varphi\: sin\theta\: \overrightarrow{j}).\frac{d\varphi}{dt}+(cos\varphi\: sin\theta\overrightarrow{i}+sin\varphi\: sin\theta\overrightarrow{j}+cos\theta\overrightarrow{k}).\frac{dr}{dt}=r\:\overrightarrow{e_\theta}\:\overset{*}{\theta}+r\:sin\theta\:\overrightarrow{e_\varphi}\:\overset{*}{\varphi}+\overrightarrow{e_r}\overset{*}{r}$
+ Vector gia tốc $(\overrightarrow{a})$ trong tọa độ cầu:
Ta có: $a=a_xi+a_ỵ+a_zk=\frac{dv}{dt}$ với $a_x=\frac{d^2x}{dt^2},\: a_y=\frac{dv_y}{dt}=\frac{d^2y}{dt^2},\: a_z=\frac{dv_z}{dt}=\frac{d^2z}{dt^2},\:$
$\Rightarrow \overrightarrow{a}= \frac{d\overrightarrow{v}}{dt}=d(r\: \overrightarrow{e_\theta})\overset{*}{\theta}+r\: \overrightarrow{e_\theta}\: \overset{**}{\theta}+ d(r\:sin\theta\:\overrightarrow{e_\varphi})\:\overset{*}{\varphi}+r\:sin\theta\:\overrightarrow{e_\varphi}\:\overset{**}{\varphi}+d(\overrightarrow{e_r})\overset{*}{r}+\overrightarrow{e_r}\overset{**}{r}=r\: \overrightarrow{e_\theta}\overset{*}{r}\overset{*}{\theta}+r\: d( \overrightarrow{e_\theta})\overset{*}{\theta}+r\: \overrightarrow{e_\theta}\: \overset{**}{\theta}+r\:sin\theta\:\overrightarrow{e_\varphi}\:\overset{*}{r}\:\overset{*}{\varphi}+r\:cos\theta\:\overrightarrow{e_\varphi}\:\overset{*}{\theta}\:\overset{*}{\varphi}+r\:sin\theta\: d(\overrightarrow{e_\varphi})\:\overset{*}{\varphi}+r\:sin\theta\:\overrightarrow{e_\varphi}\:\overset{**}{\varphi}+d(\overrightarrow{e_r})\overset{*}{r}+\overrightarrow{e_r}\overset{**}{r}$
Với $\left\{\begin{matrix} d(\overrightarrow{e_r})=r\:\overrightarrow{e_\theta}\:\overset{*}{\theta}+r\:sin\theta\:\overrightarrow{e_\varphi}\:\overset{*}{\varphi}\\d(\overrightarrow{e_\theta})= cos\theta\: \overrightarrow{e_\varphi}\overset{*}{\varphi}-\overrightarrow{e_r}\overset{*}{\theta}\\d(\overrightarrow{e_\varphi})=-(cos\varphi\: \overrightarrow{i}+sin\varphi\: \overrightarrow{j})\overset{*}{\varphi}\end{matrix}\right.$
Bài viết đã được chỉnh sửa nội dung bởi Mr nhan: 19-09-2013 - 13:00