Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; JCommentsACL has a deprecated constructor in /var/www/astarmathsandphysics/components/com_jcomments/classes/acl.php on line 17 Call Stack: 0.0001 360976 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.1071 1211272 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.1071 1211272 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.2665 4109976 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.2680 4143384 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.2688 4161096 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.2688 4178104 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.2696 4185840 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.2696 4185840 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.3156 4421688 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.3172 4439088 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.3177 4460016 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.4742 9561232 13. JEventDispatcher->trigger() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:199 0.4744 9561680 14. plgContentJComments->update() /var/www/astarmathsandphysics/libraries/joomla/event/dispatcher.php:160 0.4744 9561680 15. plgContentJComments->onContentAfterDisplay() /var/www/astarmathsandphysics/libraries/joomla/event/event.php:70 0.4746 9569928 16. plgContentJComments->onAfterDisplayContent() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:339 0.4749 9571624 17. JComments::show() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:282 0.4755 9604904 18. JCommentsFactory::getACL() /var/www/astarmathsandphysics/components/com_jcomments/jcomments.php:188 0.4756 9605304 19. spl_autoload_call() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274 0.4757 9605384 20. JLoader::load() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274

Relations Between Covariant Components of a Vector in Two Coordinate Systems

Supp pose a vector field  
\[\mathbf{F}\]
  is defined in two coordinate systems  
\[U(u_1,u_2,u_3), \: V(v_1,v_2,v_3)\]
.
Let  
\[\alpha_1, \: \alpha_2, \: \alpha_3\]
  be the coravariant components of  
\[\mathbf{F}\]
  in the  
\[U(u_1,u_2,u_3)\]
  coordinate system, so that
\[\mathbf{F} = \alpha_1 \mathbf{\nabla} u_1 + \alpha_2 \mathbf{\nabla} u_2 + \alpha_3 \mathbf{\nabla} u_3 = \alpha_i \mathbf{\nabla} u_i\]

with the repeated index indicating summation.
In the  
\[V(v_1,v_2,v_3)\]
  system
\[\mathbf{F} = \beta_1 \mathbf{\nabla} v_1 + \beta_2 \mathbf{\nabla} v_2 + \beta_3 \mathbf{\nabla} v_3 = \beta_i \mathbf{\nabla} v_i\]

Hence  
\[\alpha_i \mathbf{\nabla} u_i = \beta_j \mathbf{\nabla} v_j\]

Since  
\[u_i =u_i(v_1,v_2,v_3)\]
  and  
\[v_j =v_j(u_1,u_2,u_3)\]
,
\[\frac{\partial v_j}{\partial x} = \frac{\partial v_j}{\partial u_1} \frac{\partial u_1}{\partial x}+ \frac{\partial v_j}{\partial u_2} \frac{\partial u_2}+ \frac{\partial v_j}{\partial u_3} \frac{\partial u_3}= \frac{\partial v_j}{\partial u_k} \frac{\partial u_k}{\partial x} \]

Let  
\[x=x_1, \: y=x_2 \: z=x_3\]
  then the last result becomes  
\[\frac{\partial v_j}{\partial x_1} = \frac{\partial v_j}{\partial u_k} \frac{\partial u_k}{\partial x_1} \]

and in general  
\[\frac{\partial v_j}{\partial x_k} = \frac{\partial v_j}{\partial u_n} \frac{\partial u_n}{\partial x_k} \]
  (1)
By cycling the  
\[j\]
;s and  
\[k\]
's we obtain nine equations.
We have  
\[\alpha_i \mathbf{\nabla} u_i = \alpha_j \frac{\partial u_j}{\partial x_1} \mathbf{i} + \alpha_j \frac{\partial u_j}{\partial x_2} \mathbf{j} + \alpha_j \frac{\partial u_j}{\partial x_3} \mathbf{k} \]

and  
\[\beta_i \mathbf{\nabla} v_i = \beta_j \frac{\partial u_j}{\partial x_1} \mathbf{i}+ \beta_j \frac{\partial u_j}{\partial x_2} \mathbf{j} + \beta_j \frac{\partial u_j}{\partial x_3} \mathbf{k} \]

Equating components of  
\[ \mathbf{i} , \:\mathbf{j} , \: \mathbf{k} \]
  gives
\[\alpha_i \frac{\partial u_i}{\partial x_k} = \beta_j \frac{\partial v_j}{\partial x_k} \]

Use (1) in this result to get  
\[\alpha_i \frac{\partial u_i}{\partial x_k} = \beta_j \frac{\partial v_j}{\partial u_i} \frac{\partial u_i}{\partial x_k}\]

Equating coefficients of  
\[\frac{\partial u_i}{\partial x_k}\]
  gives  
\[\alpha_i = \beta_j \frac{\partial v_j}{\partial u_i} \]

Now multiply by  
\[\frac{\partial u_j}{\partial v_k}\]
  to get
\[\alpha_i \frac{\partial u_j}{\partial v_k}= \beta_i \frac{\partial v_i}{\partial u_j} \frac{\partial u_j}{\partial v_k} \]

Remeber  
\[\frac{\partial v_i}{\partial u_j} \frac{\partial u_j}{\partial v_k}= \delta_{ik} \]

Then  
\[\alpha_j \frac{\partial u_j}{\partial v_k} =\delta{ik}\]

Then  
\[\alpha_j = \beta_i \frac{\partial v_i}{\partial u_j} \]
  and  
\[\beta_j = \alpha_i \frac{\partial u_i}{\partial u_j} \]

Add comment

Security code
Refresh