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\begin{document}

\title{Math for Diderot}

In the following, 
\begin{itemize}
\item $\bx$ is a vector (an element of some vector space $W$)
\item $\alpha$ is a constant scalar
\item $\phi$ is a scalar function of a scalar (\eg $\phi(x) = x^2$)
\item $f$ and $g$ are scalar functions of $W$
\item $\mathbf{u} \otimes \mathbf{v}$ is tensor product of two vectors,
computed as the outer product of their vectors of coefficients in
some basis.
\item $\nabla f$ is a the gradient (first derivative) of $f$,
computed in 3-D as:
\begin{equation}
\nabla f = \begin{bmatrix}
\frac{\partial}{\partial x} \\
\frac{\partial}{\partial y} \\
\frac{\partial}{\partial z}
\end{bmatrix} f
= \begin{bmatrix}
\frac{\partial f}{\partial x} \\
\frac{\partial f}{\partial y} \\
\frac{\partial f}{\partial z}
\end{bmatrix}
\end{equation}
where $x$, $y$, $z$ are the coordinates in $W$ (\ie some basis
is assumed).  The formulae below don't assume a particular dimension.
Note that $\nabla$ can also be used to define divergence and curl
of vector fields, but for the time being these are not Diderot's concern.
\item $\bH f = \nabla \otimes \nabla f$ is the Hessian (second derivative)
of $f$, computed in 3-D as:
\begin{equation}
\bH f = \begin{bmatrix}
\frac{\partial}{\partial x} \\
\frac{\partial}{\partial y} \\
\frac{\partial}{\partial z}
\end{bmatrix} \begin{bmatrix}
\frac{\partial}{\partial x} &
\frac{\partial}{\partial y} &
\frac{\partial}{\partial z}
\end{bmatrix} f
= \begin{bmatrix}
\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} \\
\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} \\
\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} 
\end{bmatrix}
\end{equation}
\end{itemize}

Basic rules for the gradient:
\begin{align}
\nabla (f +  g) &= \nabla f + \nabla g \\
\nabla (f g) &= f \nabla g + g \nabla f \label{eq:grad-prod} \\
\nabla (\alpha f) &= \alpha \nabla f \label{eq:grad-scale} \\
\nabla (\phi(f)) &= \phi'(f) \nabla f \label{eq:grad-chain} \\
\nabla (f^n) &= n f^{n-1} \nabla f \label{eq:grad-pow} \\
\nabla \left(\frac{f}{g}\right) 
          &= \frac{\nabla f}{g} - \frac{f \nabla g}{g^2} \label{eq:grad-frac} \\
\end{align}

(\ref{eq:grad-scale}) follows from (\ref{eq:grad-prod}) with $\nabla
\alpha = 0$. (\ref{eq:grad-frac}) follows from (\ref{eq:grad-pow}).

Basic rules for the Hessian:
\begin{align}
\bH (f +  g) &= \bH f + \bH g \\
\bH (\alpha f) &= \alpha \bH f \\
\bH (f g) &= f \bH g + \nabla f \otimes \nabla g + \nabla g \otimes \nabla f + g \bH f \\
\bH \left(\frac{f}{g}\right) &=
   \frac{\bH f}{g} 
 - \frac{\nabla f \otimes \nabla g + \nabla g \otimes \nabla f + f \bH g}{g^2} 
 + \frac{2 f \nabla g \otimes \nabla g}{g^3} \label{eq:hess-quot} \\
\bH (f^n) &= n f^{n-2} \left( (n-1) \nabla f \otimes \nabla f + f \bH f \right) \label{eq:hess-pow}
\end{align}

All of these can actually be derived with $\bH f = \nabla \otimes
\nabla f$ and the rules above.  Someone may want to doublecheck
(\ref{eq:hess-quot}).  I didn't include the Hessian of the chain rule
(analogous to (\ref{eq:grad-chain})) because the process of starting
to derive it inspired me to start learning enough ML to write a program
that would derive it for me...

\end{document}