\section{Obtaining $\dLe$, $\dphase$, and $\dbsdf$}
\label{sec:dLe_dBsdf}
%
\subsection{Deriving $\dLe$}
\label{ssec:dLe}
%
\paragraph{Uniform emission}
For area light sources with uniform emission, $\Le(\bx, \bom)$ is constant for all $\bx$ on the area light.
In this case, $\Le$ is time-independent even if the local frame at $\bx$ is time-varying.
Thus, $\dLe \equiv 0$.

\paragraph{Directionally varying emission}
For area lights with spatially invariant and directionally varying emission, we have
%
\begin{equation}
	\Le(\bx, \bom) = g(\langle \gnorm(\bx), \bom \rangle),
\end{equation}
%
where $g: [-1, 1] \to \Real_+$ is some time-independent function and $\gnorm(\bx)$ denotes the surface normal at $\bx$.
In this case,
%
\begin{equation}
	\dLe(\bx, \bom) = \dev g(\langle \gnorm(\bx), \bom \rangle)
	= g'(\langle \gnorm(\bx), \bom \rangle) \, \langle \dgnorm(\bx), \bom \rangle,
\end{equation}
%
where $g'$ is the first-derivative of $g$ and $\gnorm$, which depends on $\dbx$ and the time-varying scene geometry, can be derived with $\dot{d}$ (Appendix~\ref{sec:devD}).
For instance, for a plane with normal $\gnorm$ at time $t = 0$, $\dgnorm$ under different plane motions are as follows.
%
\begin{itemize}
	\item Translation: $\dgnorm = \bm{0}$;
	%
	\item Rotation: $\dgnorm = \dot{M} \gnorm$, where $M$ is the rotation matrix similar to Eq.~\eqref{eqn:devD:rotZ_mat}.
	%
	\item Moving triangle vertices: $\dgnorm = \Delta\velocity_0 \times \bm{e}_1 + \bm{e}_0 \times \Delta\velocity_1$.
\end{itemize}

\subsection{Deriving $\dbsdf$}
\label{ssec:dBsdf}
%
\paragraph{Diffuse BRDF}
For diffuse and spatially invariant BRDF, we have
%
\begin{equation}
	\bsdf(\bx, \bom', \bom) = \frac{k_d}{\pi} \langle \gnorm(\bx), \bom' \rangle,
\end{equation}
%
where $k_d \in [0, 1)$ is the diffuse-reflectance coefficient.
It follows that, for fixed $\bom'$ and $\bom$,
%
\begin{equation}
	\dbsdf(\bx, \bom', \bom) = \frac{k_d}{\pi} \langle \dgnorm(\bx), \bom' \rangle.
\end{equation}

\paragraph{Isotropic BRDFs}
For isotropic and spatially varying BRDFs parameterized using the half-angle vector $\bm{h} := (\bom' + \bom)/\| \bom' + \bom \|$, we have
%
\begin{equation}
	\bsdf(\bx, \bom', \bom) = g(\langle \gnorm(\bx), \bm{h} \rangle) \,\langle \gnorm(\bx), \bom' \rangle,
\end{equation}
%
for some $g : [-1, 1] \to \Real_+$. Then,
%
\begin{equation}
	\begin{split}
		\dbsdf(\bx, \bom', \bom) =\;& g'(\langle \gnorm(\bx), \bm{h} \rangle) \, \langle \dgnorm(\bx), \bm{h} \rangle \, \langle \gnorm(\bx), \bom' \rangle\\
		& + g(\langle \gnorm(\bx), \bm{h} \rangle) \,\langle \gnorm(\bx), \bom' \rangle.
	\end{split}
\end{equation}
