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 0.0688 1209816 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0689 1209816 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.2808 4102888 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.2820 4130680 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.2828 4148392 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.2829 4165400 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.2836 4173136 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.2836 4173136 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.3691 4401112 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.3706 4418512 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.3712 4439440 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.3805 4505352 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.3805 4505608 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182 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 0.0688 1209816 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0689 1209816 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.2808 4102888 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.2820 4130680 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.2828 4148392 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.2829 4165400 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.2836 4173136 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.2836 4173136 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.3691 4401112 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.3706 4418512 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.3712 4439440 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.8324 12298576 13. JEventDispatcher->trigger() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:199 0.8326 12299024 14. plgContentJComments->update() /var/www/astarmathsandphysics/libraries/joomla/event/dispatcher.php:160 0.8327 12299024 15. plgContentJComments->onContentAfterDisplay() /var/www/astarmathsandphysics/libraries/joomla/event/event.php:70 0.8329 12307272 16. plgContentJComments->onAfterDisplayContent() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:339 0.8331 12309008 17. JComments::show() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:282 0.8337 12342056 18. JCommentsFactory::getACL() /var/www/astarmathsandphysics/components/com_jcomments/jcomments.php:188 0.8337 12342456 19. spl_autoload_call() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274 0.8338 12342536 20. JLoader::load() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274

Interchanging Functions and Coordinates in Green's Theorem

Green's Theorem states that for differentiable functions  
\[P(x,y), \: Q(x,y)\]
  defined on a region  
\[R\]
  with boundary  
\[C\]
;
\[\oint_C P \; dx + Q \: dy = int \int_R (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} ) \: dx \: dy\]

Interchanging  
\[P,Q\]
  and  
\[x,y\]
  respectively gives
\[\oint_C Q \; dy + P \: dx = int \int_R (\frac{\partial P}{\partial y} - \frac{\partial Q}{\partial x} ) \: dx \: dy\]

The Left hand side is the same as before, but the right hand side is reversed in sign, which amounts to reversing the orientation of the curve.
With  
\[\mathbf{a} = P \mathbf{i} + Q \mathbf{j}\]
  and  
\[\mathbf{r}= x \mathbf{i} +y \mathbf{j} \rightarrow d \mathbf{r}= dx \mathbf{i} +dy \mathbf{j}\]
 
\[\oint_C \mathbf{r} \cdot d \mathbf{r} = - \oint_{-C} \mathbf{r} \cdot d \mathbf{r}\]

Add comment

Security code
Refresh