Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0001 360976 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0898 1209816 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0898 1209816 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1586 4102888 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1600 4136232 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1608 4153944 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1608 4170952 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1616 4178688 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1616 4178688 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2082 4408296 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2098 4425696 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2103 4446624 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2190 4512872 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2190 4513128 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Relationships Between Derivatives of Coordinate Systems

Given two coordinate systems  
\[U(u_1 ,u_2 ,u_3) , \: V(v_1 ,v_2 ,v_3)\]
, we can express the derivatives of one in terms of the derivatives of the other.
Let  
\[(x,y,z)=(x(u_1,u_2,u_3),y(u_1,u_2,u_3),z(u_1,u_2,u_3))\]

and  
\[(x,y,z)=(x(v_1,v_2,v_3),y(v_1,v_2,v_3),z(v_1,v_2,v_3))\]

The transformations are onto and one to one, so there exists a transformation from  
\[U\]
  to  
\[V\]
.
We can write  
\[(u_1,u_2,u_3)=(u_1(v_1,v_2,v_3),u_2(v_1,v_2,v_3),u_3(v_1,v_2,v_3))\]

The transformation from  
\[V\]
  to  
\[U\]
  also exists and is one to one.
Differentiating,  
\[d \mathbf{r} =\frac{\partial \mathbf{r}}{\partial u_1} d u_1 + \frac{\partial \mathbf{r}}{\partial u_2} d u_2 + \frac{\partial \mathbf{r}}{\partial u_3} d u_3 = \frac{\partial \mathbf{r}}{\partial u_i} d u_i\]

where the repeated use of the index  
\[i\]
  indicates summation. Similarly  
\[d \mathbf{r} = \frac{\partial \mathbf{r}}{\partial v_i}dv_i\]

We can equate these two expressions, so  
\[\frac{\partial \mathbf{r}}{\partial u_i}du_i = \frac{\partial \mathbf{r}}{\partial v_i}dv_i\]

\[u_i =u_i (v_1,v_2,v_3) \rightarrow du_i =\frac{\partial u_i}{\partial v_j} dv_j\]

Combining the last two equations gives  
\[\frac{\mathbf{r}}{\partial v_j} dv_j = \frac{\mathbf{r}}{\partial u_j}\frac{\partial u_i}{\partial v_j} dv_j \]

Cancelling then gives  
\[\frac{\mathbf{r}}{\partial v_j} = \frac{\mathbf{r}}{\partial u_j}\frac{\partial u_i}{\partial v_j} \]

Add comment

Security code
Refresh