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This page intentionally left almost blank. You may be looking for [[Vector_Derivatives_Cylindrical_Coordinates | this page]].
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[[Category:math]]
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[[Category:tutorial]]
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== Divergence and Gradient Theorems ==
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by Kilian Cooley
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'''INTRODUCTION'''
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   -- [[User:Kcooley | The Management]]
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<pre> Contents
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- Preliminaries
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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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</pre>
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----
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==Preliminaries==
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In this tutorial, we cover two operations:
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*Divergence of a vector field, <math>\nabla\cdot\vec{v}</math>, which returns a scalar field
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*Gradient of a scalar field, <math>\nabla\phi</math>, which returns a vector field
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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.
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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
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<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>
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<math>\nabla\phi = \sum_{i=1}^n \frac{\partial \phi}{\partial x_i} \hat{e}_i</math>
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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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----
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==Divergence Theorem in 2D==
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----
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==Conservation of Mass for Moving Fluids==
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----
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==Gradient Theorem in 2D==
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----
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==Proof of Archimedes' Principle==
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----
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==A Warning About Coordinate Systems==
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----
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==References==
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----
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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>
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<math>\int\int\int_{\partial \Omega} {\mathbb R}</math>
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[http://www.google.com Here's Google]
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----
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[[Math_squad|Back to Math Squad page]]
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Latest revision as of 11:25, 16 March 2013

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