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glk |
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\documentclass[11pt]{article}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\usepackage{array}
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\usepackage{amssymb}
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\usepackage{bm}
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\newcommand{\bx}{\mathbf{x}}
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\newcommand{\bH}{\mathbf{H}}
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\newcommand{\ie}{{\em i.e.}}
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\begin{document}
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\title{Math for Diderot}
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In the following,
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\begin{itemize}
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\item $\bx$ is a vector (an element of some vector space $W$)
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\item $\alpha$ is a constant scalar
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\item $f$ and $g$ are scalar functions of $W$
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\item $\mathbf{u} \otimes \mathbf{v}$ is tensor product of two vectors,
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computed as the outer product of their vectors of coefficients in
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some basis.
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\item $\nabla f$ is a the gradient (first derivative) of $f$,
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computed in 3-D as:
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\begin{equation}
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\nabla f = \begin{bmatrix}
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\frac{\partial}{\partial x} \\
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\frac{\partial}{\partial y} \\
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\frac{\partial}{\partial z}
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\end{bmatrix} f
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= \begin{bmatrix}
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\frac{\partial f}{\partial x} \\
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\frac{\partial f}{\partial y} \\
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\frac{\partial f}{\partial z}
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\end{bmatrix}
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\end{equation}
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where $x$, $y$, $z$ are the coordinates in $W$ (\ie{} some basis
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is assumed). The formulae below don't assume a particular dimension.
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Note that $\nabla$ can also be used to define divergence and curl
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of vector fields, but for the time being these are not Diderot's concern.
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\item $\bH f = \nabla \otimes \nabla f$ is the Hessian (second derivative)
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of $f$, computed in 3-D as:
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\begin{equation}
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\bH f = \begin{bmatrix}
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\frac{\partial}{\partial x} \\
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\frac{\partial}{\partial y} \\
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\frac{\partial}{\partial z}
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\end{bmatrix} \begin{bmatrix}
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\frac{\partial}{\partial x} &
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\frac{\partial}{\partial y} &
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\frac{\partial}{\partial z}
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\end{bmatrix} f
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= \begin{bmatrix}
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\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} \\
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\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} \\
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\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}
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\end{bmatrix}
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\end{equation}
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\end{itemize}
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Basic rules for the gradient:
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\begin{align}
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\nabla (f + g) &= \nabla f + \nabla g \\
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\nabla (f g) &= f \nabla g + g \nabla f \label{eq:grad-prod} \\
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\nabla (\alpha f) &= \alpha \nabla f \label{eq:grad-scale} \\
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\nabla (f^n) &= n f^{n-1} \nabla f \label{eq:grad-pow} \\
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\nabla \left(\frac{f}{g}\right)
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&= \frac{\nabla f}{g} - \frac{f \nabla g}{g^2} \label{eq:grad-frac} \\
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\end{align}
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(\ref{eq:grad-scale}) follows from (\ref{eq:grad-prod}) with $\nabla
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\alpha = 0$. (\ref{eq:grad-frac}) follows from (\ref{eq:grad-pow}).
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Basic rules for the Hessian:
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\begin{align}
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\bH (f + g) &= \bH f + \bH g \\
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\bH (\alpha f) &= \alpha \bH f \\
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\bH (f g) &= f \bH g + \nabla f \otimes \nabla g + \nabla g \otimes \nabla f + g \bH f \\
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\bH \left(\frac{f}{g}\right) &=
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\frac{\bH f}{g}
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- \frac{\nabla f \otimes \nabla g + \nabla g \otimes \nabla f + f \bH g}{g^2}
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+ \frac{2 f \nabla g \otimes \nabla g}{g^3} \label{eq:hess-quot} \\
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\bH (f^n) &= n f^{n-2} \left( (n-1) \nabla f \otimes \nabla f + f \bH f \right) \label{eq:hess-pow}
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\end{align}
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All of these can actually be derived with $\bH f = \nabla \otimes \nabla f$
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and the rules above. Someone may want to doublecheck (\ref{eq:hess-quot}).
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\end{document}
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