Contents
Divergence and Gradient Theorems
by Kilian Cooley
INTRODUCTION
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
Preliminaries
In this tutorial, we cover two operations:
- Divergence of a vector field, $ \nabla\cdot\vec{v} $, which returns a scalar field
- Gradient of a scalar field, $ \nabla\phi $, 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., $ \hat{e}_x $ 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 $ x_1, x_2, ..., x_n $ are the coordinate directions and $ \hat{e}_i , i = 1,2,...,n $ are the unit vectors in those directions, then
$ \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 $
$ \nabla\phi = \sum_{i=1}^n \frac{\partial \phi}{\partial x_i} \hat{e}_i $
But these definitions are ASSUMING CARTESIAN COORDINATES and are not valid for cylindrical or spherical coordinate systems (more on this later).
Divergence Theorem in 2D
Conservation of Mass for Moving Fluids
Gradient Theorem in 2D
Proof of Archimedes' Principle
A Warning About Coordinate Systems
References
$ \int_0^1\int_0^4\int_{-1}^7\nabla\phi {dV} = \frac{\partial u}{\partial x}\hat{e}_x $
$ \int\int\int_{\partial \Omega} {\mathbb R} $
