Difference between revisions of "Divergence and gradient theorems mathsquad S13" - Rhea

Latest revision as of 11:25, 16 March 2013

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 --  The Management

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-[[Image:Example7.jpg]]
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-[[Category:math]]
-[[Category:tutorial]]
-== Divergence and Gradient Theorems ==
-by Kilian Cooley
-'''INTRODUCTION'''
+   -- [[User:Kcooley | The Management]]
-<pre> Contents
-- Preliminaries
-- Divergence Theorem in 2D
-- Conservation of Mass for Moving Fluids
-- Gradient Theorem in 2D
-- Proof of Archimedes' Principle
-- A Warning About Coordinate Systems
-- References
-</pre>
-----
-==Preliminaries==
-In this tutorial, we cover two operations:
-*Divergence of a vector field, <math>\nabla\cdot\vec{v}</math>, which returns a scalar field
-*Gradient of a scalar field, <math>\nabla\phi</math>, which returns a vector field
-This tutorial will denote vector quantities with an arrow atop a letter, except unit vectors that define coordinate systems which will have a hat (e.g., <math>\hat{e}_x</math> is a unit vector in the x direction). Lowercase Greek letters will represent scalar fields, and lowercase Latin letters without arrows indicate coordinates.
-Of importance is the fact that the divergence and gradient operators can be defined regardless of how many dimensions are involved. If <math>x_1, x_2, ..., x_n</math> are the coordinate directions and <math>\hat{e}_i , i = 1,2,...,n</math> are the unit vectors in those directions, then
-<math>\nabla\cdot\vec{v} = \sum_{i=1}^n \frac{\partial v_i}{\partial x_i} \text{,   where  } \vec{v} = \sum_{i=1}^n v_i \hat{e}_i</math>
-<math>\nabla\phi = \sum_{i=1}^n \frac{\partial \phi}{\partial x_i} \hat{e}_i</math>
-But these definitions are ASSUMING CARTESIAN COORDINATES and are not valid for cylindrical or spherical coordinate systems (more on this later).
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-==Divergence Theorem in 2D==
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-==Conservation of Mass for Moving Fluids==
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-==Gradient Theorem in 2D==
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-==Proof of Archimedes' Principle==
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-==A Warning About Coordinate Systems==
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-==References==
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-<math>\int_0^1\int_0^4\int_{-1}^7\nabla\phi {dV} = \frac{\partial u}{\partial x}\hat{e}_x</math>
-<math>\int\int\int_{\partial \Omega} {\mathbb R}</math>
-[http://www.google.com Here's Google]
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-[[Math_squad|Back to Math Squad page]]

Difference between revisions of "Divergence and gradient theorems mathsquad S13" - Rhea

Latest revision as of 11:25, 16 March 2013

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