# Arbitrary polyhedral volume

(Difference between revisions)
 Revision as of 12:26, 12 September 2005 (view source)Zxaar (Talk | contribs)← Older edit Revision as of 14:03, 12 September 2005 (view source)Jola (Talk | contribs) mNewer edit → Line 1: Line 1: == Arbitrary Polyhedral Volume == == Arbitrary Polyhedral Volume == -

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

## Arbitrary Polyhedral Volume

The volume of arbitrary polyhedral can be calculated by using Green-Gauss Theorem.

$\int\limits_\Omega {div(\vec F)d\Omega = } \oint\limits_S {\vec F \bullet d\vec S}$

By choosing the function

$\vec F = \frac{{\left( {x\hat i + y\hat j + z\hat k} \right)}}{3}$

Where (x,y,z) are centroid of the surface enclosing the volume under consideration. As we have,

$div(\vec F) = 1$

Hence the volume can be calculated as:

$volume = \oint\limits_S {\vec F \bullet \hat ndS}$

where the normal of the surface pointing outwards is given by:

$\hat n = (n_x \hat i + n_y \hat j + n_z \hat k)$

Final expression could be written as

$volume = \frac{1}{3}\sum\limits_{faces} {\left[ {\left( {x \times n_x + y \times n_y + z \times n_z } \right) \bullet S} \right]}$

where S is magnitude of Surface Area.