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[diderot] Annotation of /trunk/math/derivs.tex
 [diderot] / trunk / math / derivs.tex

# Annotation of /trunk/math/derivs.tex

 1 : glk 10 \documentclass[11pt]{article} 2 : 3 : \usepackage{amsmath} 4 : \usepackage{amsfonts} 5 : \usepackage{array} 6 : \usepackage{amssymb} 7 : \usepackage{bm} 8 : 9 : \newcommand{\bx}{\mathbf{x}} 10 : \newcommand{\bH}{\mathbf{H}} 11 : \newcommand{\ie}{{\em i.e.}} 12 : 13 : \begin{document} 14 : 15 : \title{Math for Diderot} 16 : 17 : In the following, 18 : \begin{itemize} 19 : \item $\bx$ is a vector (an element of some vector space $W$) 20 : \item $\alpha$ is a constant scalar 21 : \item $f$ and $g$ are scalar functions of $W$ 22 : \item $\mathbf{u} \otimes \mathbf{v}$ is tensor product of two vectors, 23 : computed as the outer product of their vectors of coefficients in 24 : some basis. 25 : \item $\nabla f$ is a the gradient (first derivative) of $f$, 26 : computed in 3-D as: 27 : 28 : \nabla f = \begin{bmatrix} 29 : \frac{\partial}{\partial x} \\ 30 : \frac{\partial}{\partial y} \\ 31 : \frac{\partial}{\partial z} 32 : \end{bmatrix} f 33 : = \begin{bmatrix} 34 : \frac{\partial f}{\partial x} \\ 35 : \frac{\partial f}{\partial y} \\ 36 : \frac{\partial f}{\partial z} 37 : \end{bmatrix} 38 : 39 : where $x$, $y$, $z$ are the coordinates in $W$ (\ie{} some basis 40 : is assumed). The formulae below don't assume a particular dimension. 41 : Note that $\nabla$ can also be used to define divergence and curl 42 : of vector fields, but for the time being these are not Diderot's concern. 43 : \item $\bH f = \nabla \otimes \nabla f$ is the Hessian (second derivative) 44 : of $f$, computed in 3-D as: 45 : 46 : \bH f = \begin{bmatrix} 47 : \frac{\partial}{\partial x} \\ 48 : \frac{\partial}{\partial y} \\ 49 : \frac{\partial}{\partial z} 50 : \end{bmatrix} \begin{bmatrix} 51 : \frac{\partial}{\partial x} & 52 : \frac{\partial}{\partial y} & 53 : \frac{\partial}{\partial z} 54 : \end{bmatrix} f 55 : = \begin{bmatrix} 56 : \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1 \partial x_2} & \frac{\partial^2 f}{\partial x_1 \partial x_3} \\ 57 : \frac{\partial^2 f}{\partial x_1 \partial x_2} & \frac{\partial^2 f}{\partial x_2^2} & \frac{\partial^2 f}{\partial x_2 \partial x_3} \\ 58 : \frac{\partial^2 f}{\partial x_1 \partial x_3} & \frac{\partial^2 f}{\partial x_2 \partial x_3} & \frac{\partial^2 f}{\partial x_3^2} 59 : \end{bmatrix} 60 : 61 : \end{itemize} 62 : 63 : Basic rules for the gradient: 64 : \begin{align} 65 : \nabla (f + g) &= \nabla f + \nabla g \\ 66 : \nabla (f g) &= f \nabla g + g \nabla f \label{eq:grad-prod} \\ 67 : \nabla (\alpha f) &= \alpha \nabla f \label{eq:grad-scale} \\ 68 : \nabla (f^n) &= n f^{n-1} \nabla f \label{eq:grad-pow} \\ 69 : \nabla \left(\frac{f}{g}\right) 70 : &= \frac{\nabla f}{g} - \frac{f \nabla g}{g^2} \label{eq:grad-frac} \\ 71 : \end{align} 72 : 73 : (\ref{eq:grad-scale}) follows from (\ref{eq:grad-prod}) with $\nabla 74 : \alpha = 0$. (\ref{eq:grad-frac}) follows from (\ref{eq:grad-pow}). 75 : 76 : Basic rules for the Hessian: 77 : \begin{align} 78 : \bH (f + g) &= \bH f + \bH g \\ 79 : \bH (\alpha f) &= \alpha \bH f \\ 80 : \bH (f g) &= f \bH g + \nabla f \otimes \nabla g + \nabla g \otimes \nabla f + g \bH f \\ 81 : \bH \left(\frac{f}{g}\right) &= 82 : \frac{\bH f}{g} 83 : - \frac{\nabla f \otimes \nabla g + \nabla g \otimes \nabla f + f \bH g}{g^2} 84 : + \frac{2 f \nabla g \otimes \nabla g}{g^3} \label{eq:hess-quot} \\ 85 : \bH (f^n) &= n f^{n-2} \left( (n-1) \nabla f \otimes \nabla f + f \bH f \right) \label{eq:hess-pow} 86 : \end{align} 87 : 88 : All of these can actually be derived with $\bH f = \nabla \otimes \nabla f$ 89 : and the rules above. Someone may want to doublecheck (\ref{eq:hess-quot}). 90 : 91 : \end{document}