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

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revision 10, Sun Jan 10 19:42:44 2010 UTC revision 11, Sun Jan 10 20:20:41 2010 UTC
# Line 9  Line 9 
9  \newcommand{\bx}{\mathbf{x}}  \newcommand{\bx}{\mathbf{x}}
10  \newcommand{\bH}{\mathbf{H}}  \newcommand{\bH}{\mathbf{H}}
11  \newcommand{\ie}{{\em i.e.}}  \newcommand{\ie}{{\em i.e.}}
12    \newcommand{\eg}{{\em e.g.}}
13    
14  \begin{document}  \begin{document}
15    
# Line 18  Line 19 
19  \begin{itemize}  \begin{itemize}
20  \item $\bx$ is a vector (an element of some vector space $W$)  \item $\bx$ is a vector (an element of some vector space $W$)
21  \item $\alpha$ is a constant scalar  \item $\alpha$ is a constant scalar
22    \item $\phi$ is a scalar function of a scalar (\eg $\phi(x) = x^2$)
23  \item $f$ and $g$ are scalar functions of $W$  \item $f$ and $g$ are scalar functions of $W$
24  \item $\mathbf{u} \otimes \mathbf{v}$ is tensor product of two vectors,  \item $\mathbf{u} \otimes \mathbf{v}$ is tensor product of two vectors,
25  computed as the outer product of their vectors of coefficients in  computed as the outer product of their vectors of coefficients in
# Line 36  Line 38 
38  \frac{\partial f}{\partial z}  \frac{\partial f}{\partial z}
39  \end{bmatrix}  \end{bmatrix}
40  \end{equation}  \end{equation}
41  where $x$, $y$, $z$ are the coordinates in $W$ (\ie{} some basis  where $x$, $y$, $z$ are the coordinates in $W$ (\ie some basis
42  is assumed).  The formulae below don't assume a particular dimension.  is assumed).  The formulae below don't assume a particular dimension.
43  Note that $\nabla$ can also be used to define divergence and curl  Note that $\nabla$ can also be used to define divergence and curl
44  of vector fields, but for the time being these are not Diderot's concern.  of vector fields, but for the time being these are not Diderot's concern.
# Line 65  Line 67 
67  \nabla (f +  g) &= \nabla f + \nabla g \\  \nabla (f +  g) &= \nabla f + \nabla g \\
68  \nabla (f g) &= f \nabla g + g \nabla f \label{eq:grad-prod} \\  \nabla (f g) &= f \nabla g + g \nabla f \label{eq:grad-prod} \\
69  \nabla (\alpha f) &= \alpha \nabla f \label{eq:grad-scale} \\  \nabla (\alpha f) &= \alpha \nabla f \label{eq:grad-scale} \\
70    \nabla (\phi(f)) &= \phi'(f) \nabla f \label{eq:grad-chain} \\
71  \nabla (f^n) &= n f^{n-1} \nabla f \label{eq:grad-pow} \\  \nabla (f^n) &= n f^{n-1} \nabla f \label{eq:grad-pow} \\
72  \nabla \left(\frac{f}{g}\right)  \nabla \left(\frac{f}{g}\right)
73            &= \frac{\nabla f}{g} - \frac{f \nabla g}{g^2} \label{eq:grad-frac} \\            &= \frac{\nabla f}{g} - \frac{f \nabla g}{g^2} \label{eq:grad-frac} \\
# Line 85  Line 88 
88  \bH (f^n) &= n f^{n-2} \left( (n-1) \nabla f \otimes \nabla f + f \bH f \right) \label{eq:hess-pow}  \bH (f^n) &= n f^{n-2} \left( (n-1) \nabla f \otimes \nabla f + f \bH f \right) \label{eq:hess-pow}
89  \end{align}  \end{align}
90    
91  All of these can actually be derived with $\bH f = \nabla \otimes \nabla f$  All of these can actually be derived with $\bH f = \nabla \otimes
92  and the rules above.  Someone may want to doublecheck (\ref{eq:hess-quot}).  \nabla f$ and the rules above.  Someone may want to doublecheck
93    (\ref{eq:hess-quot}).  I didn't include the Hessian of the chain rule
94    (analogous to (\ref{eq:grad-chain})) because the process of starting
95    to derive it inspired me to start learning enough ML to write a program
96    that would derive it for me...
97    
98  \end{document}  \end{document}

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