VectorIdentities 2006-07-20 18:59:21 2008-05-27 16:18:20 Definition Vector Identities % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here It is difficult to get anywhere in physics without a firm understanding of vectors and their common operations. 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] %heading \hline \\ [.3ex] Gradient & $\nabla f$ \\ Laplacian & $\nabla^2$ \\ Divergence & $\nabla \cdot$ \\ Curl & $\nabla \times$ \\ \hline \end{tabular} \end{center} \end{table} {\bf Vector Magnitude} $ A = \left | \mathbf{A} \right | = \sqrt{{A_x}^2 + {A_y}^2 + {A_z}^2 }$\\ $ A = \sqrt{\mathbf{A} \cdot \mathbf{A}} $ {\bf Scalar Product (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 Vector Product (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 determinant 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 Vector Triple Product, 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 Scalar 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 Gradient} $ \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 Identities} $ \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 Divergence} $ \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} $