Talk:PlanetPhysics/Vector Identities

Original TeX Content from PlanetPhysics Archive
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Vector Identities %%% Primary Category Code: 02. %%% Filename: VectorIdentities.tex %%% Version: 9 %%% 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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\begin{document}

It is difficult to get anywhere in physics without a firm understanding of \htmladdnormallink{vectors}{http://planetphysics.us/encyclopedia/Vectors.html} and their common \htmladdnormallink{operations}{http://planetphysics.us/encyclopedia/Cod.html}. Here, we will give vector identities as a reference. Basic terminology to keep straight.

\begin{table}[t] \begin{center} \begin{tabular}{cc} \\ [.3ex] \hline \\ [.3ex]

{\bf Operation} & {\bf Symbol} \\ [0.5ex]

\hline \\ [.3ex]

\htmladdnormallink{Gradient}{http://planetphysics.us/encyclopedia/Gradient.html} & $\nabla f$ \\ \htmladdnormallink{Laplacian}{http://planetphysics.us/encyclopedia/LaplaceOperator.html} & $\nabla^2$ \\ \htmladdnormallink{divergence}{http://planetphysics.us/encyclopedia/DivergenceOfAVectorField.html} & $\nabla \cdot$ \\ \htmladdnormallink{curl}{http://planetphysics.us/encyclopedia/Curl.html} & $\nabla \times$ \\

\hline \end{tabular} \end{center} \end{table}

{\bf \htmladdnormallink{Vector Magnitude}{http://planetphysics.us/encyclopedia/Vector.html}

$ A = \left | \mathbf{A} \right | = \sqrt{{A_x}^2 + {A_y}^2 + {A_z}^2 }$\\ $ A = \sqrt{\mathbf{A} \cdot \mathbf{A}} $

{\bf \htmladdnormallink{scalar product}{http://planetphysics.us/encyclopedia/DotProduct.html} (Dot Product)}

$ \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z$ \\ $ \mathbf{A} \cdot \mathbf{B} = \left | \mathbf{A} \right | \left | \mathbf{B} \right | \cos \theta$

{\bf \htmladdnormallink{vector product}{http://planetphysics.us/encyclopedia/VectorProduct.html} (Cross Product)}

$ \mathbf{A} \times \mathbf{B} = \left ( A_y B_z - A_z B_y \right ) \mathbf{\hat{i}} + \left ( A_z B_x - A_x B_z \right ) \mathbf{\hat{j}} + \left ( A_x B_y - A_y B_x \right ) \mathbf{\hat{k}}$

It can be easier to remember with \htmladdnormallink{determinant}{http://planetphysics.us/encyclopedia/Determinant.html} formulation

$ \mathbf{A} \times \mathbf{B} = \left| \begin{matrix} \mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{matrix}\right| = \left ( A_y B_z - A_z B_y \right ) \mathbf{\hat{i}} + \left ( A_z B_x - A_x B_z \right ) \mathbf{\hat{j}} + \left ( A_x B_y - A_y B_x \right ) \mathbf{\hat{k}}$

{\bf \htmladdnormallink{Vector Triple Product}{http://planetphysics.us/encyclopedia/BACKCAB.html}, aka. BAC CAB}

$ \mathbf{A} \times \left ( \mathbf{B} \times \mathbf{C} \right ) = \mathbf{B} \left( \mathbf{A} \cdot \mathbf{C} \right ) - \mathbf{C} \left ( \mathbf{A} \cdot \mathbf{B} \right) $

{\bf \htmladdnormallink{scalar}{http://planetphysics.us/encyclopedia/Vectors.html} Triple Product}

$ \mathbf{A} \cdot \left ( \mathbf{B} \times \mathbf{C} \right ) = \mathbf{B} \cdot \left ( \mathbf{C} \times \mathbf{A} \right ) = \mathbf{C} \cdot \left ( \mathbf{A} \times \mathbf{B} \right ) $

{\bf \htmladdnormallink{Gradient}{http://planetphysics.us/encyclopedia/Gradient.html}}

$ \nabla f = \frac{\partial f}{\partial x} \mathbf{\hat{i}} + \frac{\partial f}{\partial y} \mathbf{\hat{j}} + \frac{\partial f}{\partial z} \mathbf{\hat{k}} $

{\bf Gradient \htmladdnormallink{identities}{http://planetphysics.us/encyclopedia/Cod.html}}

$ \nabla \left ( f + g \right ) = \nabla f + \nabla g $ \\ $ \nabla \left ( \alpha f \right ) = \alpha \nabla f $ \\ $ \nabla \left ( f \, g \right ) = f \nabla g + g \nabla f $ \\ $ \nabla \left ( f/g \right ) = \frac{\left ( g \nabla f - f \nabla g \right )}{g^2} $ \\

{\bf \htmladdnormallink{Divergence}{http://planetphysics.us/encyclopedia/Divergence.html}}

$ \nabla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} +\frac{\partial A_z}{\partial z} $

{\bf Divergence of the cross product}

$ \nabla \cdot \left ( \mathbf{A} \times \mathbf{B} \right ) = \mathbf{B} \cdot \left ( \nabla \times \mathbf{A} \right ) - \mathbf{A} \cdot \left ( \nabla \times \mathbf{B} \right ) $

{\bf Divergence of the curl}

$ \nabla \cdot \left ( \nabla \times \mathbf{A} \right ) = 0 $

{\bf Laplacian Identities}

$ \nabla \times \left ( \nabla \times \mathbf{A} \right ) = \nabla \left ( \nabla \cdot \mathbf{A} \right ) - \nabla^2 \mathbf{A} $

\end{document}