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 360856 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.1025 1211056 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.1025 1211056 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1877 4197176 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1891 4230488 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1899 4248200 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1901 4278720 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1911 4301440 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1911 4301440 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2862 4971176 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2897 5163328 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2902 5184256 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.3020 5378448 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.3020 5378704 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Finding a Volume Using the Divergence Theorem

Theorem
Let  
\[V\]
  be any region of space with a surface  
\[S\]

The volume enclosed by  
\[S\]
  is  
\[V= \frac{1}{3} \int \int_S \mathbf{r} \cdot \mathbf{n} \]
  where  
\[\mathbf{n}\]
  is the outward normal to  
\[S\]

Proof
Use the Divergence Theorem  
\[\int \int \int_V \mathbf{\nabla} \cdot \mathbf{F} dV = \int \int_S \mathbf{F} \cdot \mathbf{n} dV\]

Let  
\[\mathbf{F} = \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}\]

Then
\[\begin{equation} \begin{aligned}\mathbf{\nabla} \cdot \mathbf{r} &= (\frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} \mathbf{j} + \frac{\partial}{\partial z} \mathbf{k}) \cdot (x \mathbf{i} + y \mathbf{j} + z \mathbf{k}) \\ &= 1 + 1+1 =3 \end{aligned} \end{equation}\]

Then  
\[\int \int \int_V 3 dV = \int \int_S \mathbf{r} \cdot \mathbf{n} dS \rightarrow V = \frac{1}{3} \int \int_S \mathbf{r} \cdot \mathbf{n} dS\]

Add comment

Security code
Refresh