Arbitrary polyhedral volume - Revision history

Zxaar at 06:18, 3 October 2005

2005-10-03T06:18:13Z

← Older revision		Revision as of 06:18, 3 October 2005
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	where S is magnitude of Surface Area.		where S is magnitude of Surface Area.
		+
		+
		+	----
		+	<i> Return to [[Numerical methods \| Numerical Methods]] </i>

Jola at 14:54, 12 September 2005

2005-09-12T14:54:25Z

← Older revision		Revision as of 14:54, 12 September 2005
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-	~~== Arbitrary Polyhedral Volume ==~~
-
	The volume of arbitrary polyhedral can be calculated by using [[Greens theorem \| Green-Gauss Theorem]].		The volume of arbitrary polyhedral can be calculated by using [[Greens theorem \| Green-Gauss Theorem]].

Jola: Arbitrary Polyhedral Volume moved to Arbitrary polyhedral volume

2005-09-12T14:53:57Z

Arbitrary Polyhedral Volume moved to Arbitrary polyhedral volume

← Older revision

Revision as of 14:53, 12 September 2005

Jola at 14:03, 12 September 2005

2005-09-12T14:03:03Z

← Older revision		Revision as of 14:03, 12 September 2005
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	== Arbitrary Polyhedral Volume ==		== Arbitrary Polyhedral Volume ==
-	~~<p>~~	+
-	The volume of arbitrary polyhedral can be calculated by using Green-Gauss Theorem. ~~<br>~~	+	The volume of arbitrary polyhedral can be calculated by using [[Greens theorem \| Green-Gauss Theorem]].
-	<math>\int\limits_\Omega {div(\vec F)d\Omega = } \oint\limits_S {\vec F \bullet d\vec S}	+
		+	:<math>\int\limits_\Omega {div(\vec F)d\Omega = } \oint\limits_S {\vec F \bullet d\vec S}
	</math>		</math>
-	~~<br>~~	+
-	By choosing the function~~<br>~~	+	By choosing the function
-	<math>	+
		+	:<math>
	\vec F = \frac{{\left( {x\hat i + y\hat j + z\hat k} \right)}}{3}		\vec F = \frac{{\left( {x\hat i + y\hat j + z\hat k} \right)}}{3}
-	</math>~~<br>~~	+	</math>
		+
	Where (x,y,z) are centroid of the surface enclosing the volume under consideration.		Where (x,y,z) are centroid of the surface enclosing the volume under consideration.
-	As we have, ~~<br>~~	+	As we have,
-	<math>	+
		+	:<math>
	div(\vec F) = 1		div(\vec F) = 1
-	</math~~><br~~>	+	</math>
-	Hence the volume can be calculated as: ~~<br>~~	+
-	<math>	+	Hence the volume can be calculated as:
		+
		+	:<math>
	volume = \oint\limits_S {\vec F \bullet \hat ndS}		volume = \oint\limits_S {\vec F \bullet \hat ndS}
-	</math~~><br~~>	+	</math>
-	where the normal of the surface pointing outwards is given by: ~~<br>~~	+
-	<math>	+	where the normal of the surface pointing outwards is given by:
		+
		+	:<math>
	\hat n = (n_x \hat i + n_y \hat j + n_z \hat k)		\hat n = (n_x \hat i + n_y \hat j + n_z \hat k)
-	</math>~~<br>~~	+	</math>
		+
	Final expression could be written as <br>		Final expression could be written as <br>
-	<math>	+
		+	:<math>
	volume = \frac{1}{3}\sum\limits_{faces} {\left[ {\left( {x \times n_x + y \times n_y + z \times n_z } \right) \bullet S} \right]}		volume = \frac{1}{3}\sum\limits_{faces} {\left[ {\left( {x \times n_x + y \times n_y + z \times n_z } \right) \bullet S} \right]}
-	</math>~~<br>~~	+	</math>
		+
	where S is magnitude of Surface Area.		where S is magnitude of Surface Area.
-	~~</p>~~

Zxaar at 12:26, 12 September 2005

2005-09-12T12:26:41Z

New page

== Arbitrary Polyhedral Volume ==

The volume of arbitrary polyhedral can be calculated by using Green-Gauss Theorem. 
<math>\int\limits_\Omega {div(\vec F)d\Omega = } \oint\limits_S {\vec F \bullet d\vec S}
</math>
 
By choosing the function 
<math>
\vec F = \frac{{\left( {x\hat i + y\hat j + z\hat k} \right)}}{3}
</math> 
Where (x,y,z) are centroid of the surface enclosing the volume under consideration.
As we have, 
<math>
div(\vec F) = 1
</math> 
Hence the volume can be calculated as: 
<math>
volume = \oint\limits_S {\vec F \bullet \hat ndS}
</math> 
where the normal of the surface pointing outwards is given by: 
<math>
\hat n = (n_x \hat i + n_y \hat j + n_z \hat k)
</math> 
Final expression could be written as 
<math>
volume = \frac{1}{3}\sum\limits_{faces} {\left[ {\left( {x \times n_x + y \times n_y + z \times n_z } \right) \bullet S} \right]}
</math> 
where S is magnitude of Surface Area.