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: 0.0000 360976 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0525 1209816 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0525 1209816 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1187 4190336 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1200 4223680 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1207 4241392 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1208 4271912 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1218 4294632 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1218 4294632 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1693 4966728 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1728 5158880 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1733 5179808 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.1841 5375384 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.1841 5375640 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: 0.0000 360976 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0525 1209816 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0525 1209816 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1187 4190336 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1200 4223680 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1207 4241392 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1208 4271912 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1218 4294632 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1218 4294632 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1693 4966728 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1728 5158880 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1733 5179808 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.3200 13805728 13. JEventDispatcher->trigger() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:199 0.3202 13806128 14. plgContentJComments->update() /var/www/astarmathsandphysics/libraries/joomla/event/dispatcher.php:160 0.3202 13806128 15. plgContentJComments->onContentAfterDisplay() /var/www/astarmathsandphysics/libraries/joomla/event/event.php:70 0.3204 13814376 16. plgContentJComments->onAfterDisplayContent() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:339 0.3206 13816112 17. JComments::show() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:282 0.3212 13849160 18. JCommentsFactory::getACL() /var/www/astarmathsandphysics/components/com_jcomments/jcomments.php:188 0.3212 13849560 19. spl_autoload_call() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274 0.3214 13849640 20. JLoader::load() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274

Integral of Tangential Component of a Force Along a Curve

Given a curve  
\[\mathbf{r}=\mathbf{r}(x,y)\]
  and a vector field  
\[\mathbf{F}(x,y)=F_1(x,y) \mathbf{i} + F_2 (x,y) \mathbf{j}\]
, to evaluate the integral  
\[\int_C F_T d r\]
  where  
\[F_T\]
  is the component of  
\[\mathbf{F}\]
  along the curve, write  
\[\int_C F_T d r = \int_C \mathbf{F} \cdot d \mathbf{r}=\int_C (F_1 \mathbf{i} + F_2 \mathbf{j}) \cdot (dx \mathbf{i} +dy \mathbf{j})=\int_C F_1 dx + F_2 dy\]

We can parametrize the curve  
\[\mathbf{r}=\mathbf{r}(x(s),y(s))\]
  then
Where  
\[s\]
  is the distance along the curve.
\[\int_C F_T d r=\int_C F_1 dx + F_2 dy =\int_C (F_1 (s)\frac{dx}{ds} + F_2(s) \frac{dy}{ds}) ds\]
.
Example: Evaluate  
\[\int_C F_T d r\]
  along thureve  
\[y=x\]
  from  
\[(0,0)\]
  to  
\[(1,1)\]
  for the vector field  
\[\mathbf{F}=(x^2 -y^2 ) \mathbf{i} + 2xy \mathbf{j}\]

For the line  
\[y=x\]
  use the parametrization  
\[x=\frac{s}{\sqrt{2}} , y=\frac{s}{\sqrt{2}}\]

When  
\[(x,y)=(0,0),(1,1)\]
   
\[s=0, \sqrt{2}\]
  respectively.
The integral becomes
\[\begin{equation} \begin{aligned} \int_C F_T d r &= \int_C (F_1 (s)\frac{dx}{ds} + F_2(s) ) ds \\ &= \int_0^{\sqrt{2}} (((\frac{s}{\sqrt{2}})^2 -(\frac{s}{\sqrt{2}})^2) +2 \frac{s}{\sqrt{2}} \frac{s}{\sqrt{2}}) \frac{dy}{ds}) \frac{1}{\sqrt{2}}ds \\ &= \int_0^{\sqrt{2}} s^2 \frac{1}{\sqrt{2}}ds \\ &= \frac{1}{\sqrt{2}} [\frac{s^3}{3}]_0^{\sqrt{2}} \\ &= \frac{1}{\sqrt{2}} \frac{(\sqrt{2})^3}{3}-0 \\ &= \frac{2 }{3} \end{aligned} \end{equation}\]
.

Add comment

Security code
Refresh