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 0.0513 1209816 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0513 1209816 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1242 4124800 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1256 4152592 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1263 4170304 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1266 4200824 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1276 4223544 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1276 4223544 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1848 4895224 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1885 5087376 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1891 5108304 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2005 5303736 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2005 5303992 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 0.0513 1209816 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0513 1209816 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1242 4124800 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1256 4152592 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1263 4170304 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1266 4200824 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1276 4223544 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1276 4223544 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1848 4895224 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1885 5087376 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1891 5108304 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.3407 13734384 13. JEventDispatcher->trigger() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:199 0.3410 13734784 14. plgContentJComments->update() /var/www/astarmathsandphysics/libraries/joomla/event/dispatcher.php:160 0.3410 13734784 15. plgContentJComments->onContentAfterDisplay() /var/www/astarmathsandphysics/libraries/joomla/event/event.php:70 0.3412 13743032 16. plgContentJComments->onAfterDisplayContent() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:339 0.3414 13744768 17. JComments::show() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:282 0.3420 13777816 18. JCommentsFactory::getACL() /var/www/astarmathsandphysics/components/com_jcomments/jcomments.php:188 0.3420 13778216 19. spl_autoload_call() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274 0.3421 13778296 20. JLoader::load() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274

Derivation of Volume Formula Using Divergence Theorem

Theorem
Let  
\[S\]
  be the boundary surface of a volume  
\[V\]
  andlket  
\[\mathbf{n}\]
  be the outward normal. Then
\[V= \int \int_S x \: dy \: dz = \int int_S y \: dx \: dz = \int \int_S z \: dx \: dy = \frac{1}{3} \int \int_S x \: dy \: dz + y \: dx \: dz + z \: dx \: dy\]

Proof
From the divergence theorem  
\[\int \i \int_S (\frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} dx \: dy \: dz = \int \int _S F_1 dy \: dz + F_2 \: dx \: dz + F_3 \: dx \: dy\]

If  
\[\mathbf{\nabla} \cdot \mathbf{F}=1\]
  then again from the divergence theorem  
\[\int \int \int_V \mathbf{\nabla} \cdot \mathbf{F} \: dx \: dy \: dz = \int \int \int_V dx \: dy \: dz = \int \int_S F_1 \: dy \: dz + F_2 \; dx \: dz + F_3 \: dy \: dz\]

But  
\[V = \int \int \int_V dx \: dy \: dz\]

Hence  
\[V= \int \int_S F_1 \: dy \: dz + F_2 \; dx \: dz + F_3 \: dy \: dz\]

Take  
\[\mathbf{F} = x \mathbf{i}\]
  then  
\[\mathbf{\nabla} \cdot \mathbf{F} =1\]
  and  
\[V= \int \int_S x \: dy \: dz\]

Take  
\[\mathbf{F} = y \mathbf{j}\]
  then  
\[\mathbf{\nabla} \cdot \mathbf{F} =1\]
  and  
\[V= \int \int_S y \: dx \: dz\]

Take  
\[\mathbf{F} = y \mathbf{i}\]
  then  
\[\mathbf{\nabla} \cdot \mathbf{F} =1\]
  and  
\[V= \int \int_S y \: dx \: dz\]

Hence
\[V= \frac{1}{3} \int \int_S x \: dy \: dz + y \: dx \: dz + z \: dx \: dy\]

Add comment

Security code
Refresh