Talk:PlanetPhysics/Vector Functions

Original TeX Content from PlanetPhysics Archive
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: vector functions %%% Primary Category Code: 02. %%% Filename: VectorFunctions.tex %%% Version: 1 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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A \emph{\htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html}} \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} of \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} in space is a function

$${\bf V}(x, y, z)$$

which associates with each point $x, y, z$ in space a definite vector. The function may be broken up into its three components

$${\bf V} (x, y, z) = V_1 (x, y,z){\bf \hat{i}}+ V_2(x, y, z){\bf\hat{j}} + V_3(x, y, z){\bf\hat{k}}$$

Examples of vector functions are very numerous in physics. Already the function $\nabla{V}$ has occurred. At each point of space $\nabla{V}$ has in general a definite vector value. In \htmladdnormallink{mechanics}{http://planetphysics.us/encyclopedia/Mechanics.html} of \htmladdnormallink{rigid bodies}{http://planetphysics.us/encyclopedia/CenterOfGravity.html} the \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} of each point of the body is a vector function of the position of the point. \htmladdnormallink{Fluxes}{http://planetphysics.us/encyclopedia/AbsoluteMagnitude.html} of \htmladdnormallink{heat}{http://planetphysics.us/encyclopedia/Heat.html}, electricity, magnetic force, fluids, etc., are all vector functions of position in space.

The \htmladdnormallink{scalar}{http://planetphysics.us/encyclopedia/Vectors.html} \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} ${\bf a} \cdot \nabla$ may be applied to a vector function ${\bf V}$ to yield another vector function.

Let

$${\bf V} = V_1(x,y,z){\bf \hat{i}} + V_2(x, y, z){\bf \hat{j}} + V_3(x, y, z){\bf \hat{k}} $$

and

$$ {\bf a} = a_1{\bf \hat{i}} + a_2 {\bf \hat{j}} + a_3{\bf \hat{k}} $$

Then

$$ {\bf a} \cdot \nabla = a_1\frac{\partial}{\partial x} + a_2\frac{\partial}{\partial y} + a_3 \frac{\partial}{\partial z}$$

$$ \left ( {\bf a} \cdot \nabla \right ) {\bf V} = \left ( {\bf a} \cdot \nabla  \right) V_1 {\bf \hat{i}} + \left ( {\bf a} \cdot \nabla  \right) V_2 {\bf \hat{j}} + \left ( {\bf a} \cdot \nabla \right) V_3 {\bf \hat{k}}$$

and

$$ \left ( {\bf a} \cdot \nabla \right ) {\bf V} = \left ( a_1\frac{\partial V_1}{\partial x} +  a_2 \frac{\partial V_1}{\partial y} +  a_3\frac{\partial V_1}{\partial z} \right ) {\bf \hat{i}} + \left ( a_1\frac{\partial V_2}{\partial x} + a_2 \frac{\partial V_2}{\partial y} +  a_3\frac{\partial V_2}{\partial z} \right ) {\bf \hat{j}} + \left ( a_1\frac{\partial V_3}{\partial x} + a_2 \frac{\partial V_3}{\partial y} +  a_3\frac{\partial V_3}{\partial z} \right ) {\bf \hat{k}} $$

more to come..

This is from the public \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} text by Gibbs.

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