Commit d50a7bb1 authored by Matthieu Schaller's avatar Matthieu Schaller
Browse files

Added expressions for the 5th order derivatives to the documentation.

parent bb5ea549
......@@ -60,12 +60,14 @@ reproduced here for completeness), we can now build all the relevent derivatives
\mathsf{\tilde{D}}_{1}(r, u, H) \nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\rule{6cm}{0.4pt}
\begin{align}
\mathsf{D}_{100}(\mathbf{r}) = \frac{\partial}{\partial r_x} \varphi (\mathbf{r},H) =
r_x \mathsf{\tilde{D}}_{3}(r, u, H) \nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\rule{6cm}{0.4pt}
\begin{align}
\mathsf{D}_{200}(\mathbf{r}) = \frac{\partial^2}{\partial r_x^2} \varphi (\mathbf{r},H) =
......@@ -78,6 +80,7 @@ r_x^2 \mathsf{\tilde{D}}_{5}(r, u, H) +
r_x r_y \mathsf{\tilde{D}}_{5}(r, u, H) \nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\rule{6cm}{0.4pt}
\begin{align}
\mathsf{D}_{300}(\mathbf{r}) = \frac{\partial^3}{\partial r_x^3} \varphi (\mathbf{r},H) =
......@@ -96,6 +99,7 @@ r_y \mathsf{\tilde{D}}_{5}(r, u, H) \nonumber
r_x r_y r_z \mathsf{\tilde{D}}_{7}(r, u, H) \nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\rule{6cm}{0.4pt}
\begin{align}
\mathsf{D}_{400}(\mathbf{r}) &= \frac{\partial^4}{\partial r_x^4}
......@@ -132,8 +136,75 @@ r_y \mathsf{\tilde{D}}_{5}(r, u, H) \nonumber
\nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\rule{6cm}{0.4pt}
\begin{align}
\mathsf{D}_{500}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^5}
\varphi (\mathbf{r},H) =
r_x^5 \mathsf{\tilde{D}}_{11}(r, u, H) +
10r_x^3\mathsf{\tilde{D}}_{9}(r, u, H) +
15r_x\mathsf{\tilde{D}}_{7}(r, u, H)
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{410}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^4
\partial r_y} \varphi (\mathbf{r},H) =
r_x^4 r_y \mathsf{\tilde{D}}_{11}(r, u, H) +
6 r_x^2 r_y \mathsf{\tilde{D}}_{9}(r, u, H) +
3 r_y \mathsf{\tilde{D}}_{7}(r, u, H)
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{320}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^3
\partial r_y^2} \varphi (\mathbf{r},H) =
r_x^3 r_y^2 \mathsf{\tilde{D}}_{11}(r, u, H) +
r_x^3 \mathsf{\tilde{D}}_{9}(r, u, H) +
3 r_x r_y^2 \mathsf{\tilde{D}}_{9}(r, u, H) +
3 r_x \mathsf{\tilde{D}}_{7}(r, u, H)
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{311}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^3
\partial r_y \partial r_z} \varphi (\mathbf{r},H) =
r_x^3 r_y r_z \mathsf{\tilde{D}}_{11}(r, u, H) +
3 r_x r_y r_z \mathsf{\tilde{D}}_{9}(r, u, H)
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{221}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^2
\partial r_y^2 \partial r_z} \varphi (\mathbf{r},H) =
r_x^2 r_y^2 r_z \mathsf{\tilde{D}}_{11}(r, u, H) +
r_x^2 r_z \mathsf{\tilde{D}}_{9}(r, u, H) +
r_y^2 r_z \mathsf{\tilde{D}}_{9}(r, u, H) +
r_z \mathsf{\tilde{D}}_{y}(r, u, H)
\nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{comment}
\noindent\rule{6cm}{0.4pt}
......
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