From e83e37029b9b0b0b32418256e2e504f6149e6f31 Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Sun, 2 Aug 2015 16:18:07 -0500 Subject: [PATCH] Update documentation. --- include/deal.II/base/derivative_form.h | 32 +++++++++++++++++++++----- 1 file changed, 26 insertions(+), 6 deletions(-) diff --git a/include/deal.II/base/derivative_form.h b/include/deal.II/base/derivative_form.h index b74fa9eb76..bf79b842f0 100644 --- a/include/deal.II/base/derivative_form.h +++ b/include/deal.II/base/derivative_form.h @@ -24,13 +24,33 @@ DEAL_II_NAMESPACE_OPEN * This class represents the (tangential) derivatives of a function $ f: * {\mathbb R}^{\text{dim}} \rightarrow {\mathbb R}^{\text{spacedim}}$. Such * functions are always used to map the reference dim-dimensional cell into - * spacedim-dimensional space. For such objects, the first derivative of the - * function is a linear map from ${\mathbb R}^{\text{dim}}$ to ${\mathbb - * R}^{\text{spacedim}}$, the second derivative a bilinear map from ${\mathbb + * spacedim-dimensional space. For such objects, the first derivative of the + * function is a linear map from ${\mathbb R}^{\text{dim}}$ to ${\mathbb + * R}^{\text{spacedim}}$, i.e., it can be represented as a matrix + * in ${\mathbb R}^{\text{spacedim}\times \text{dim}}. This makes sense + * since one would represent the first derivative, $\nabla f(\mathbf x)$ + * with $\mathbf x\in {\mathbb R}^{\text{dim}}$, in such a way that the + * directional derivative in direction $\mathbf d\in {\mathbb R}^{\text{dim}}$ + * so that + * @f{align*} + * \nabla f(\mathbf x) \mathbf d + * = \lim_{\varepsilon\rightarrow 0} + * \frac{f(\mathbf x + \varepsilon \mathbf d) - f(\mathbf x)}{\varepsilon}, + * @f} + * i.e., one needs to be able to multiply the matrix $\nabla f(\mathbf x)$ by + * a vector in ${\mathbb R}^{\text{dim}}$, and the result is a difference + * of function values, which are in ${\mathbb R}^{\text{spacedim}}. Consequently, + * the matrix must be of size $\text{spacedim}\times\text{dim}$. + * + * Similarly, the second derivative is a bilinear map from ${\mathbb * R}^{\text{dim}} \times {\mathbb R}^{\text{dim}}$ to ${\mathbb - * R}^{\text{spacedim}}$ and so on. In deal.II we represent these derivatives - * using objects of type DerivativeForm<1,dim,spacedim,Number>, - * DerivativeForm<2,dim,spacedim,Number> and so on. + * R}^{\text{spacedim}}$, which one can think of a rank-3 object of size + * $\text{spacedim}\times\text{dim}\times\text{dim}$. + * + * In deal.II we represent these derivatives + * using objects of type DerivativeForm@<1,dim,spacedim,Number@>, + * DerivativeForm@<2,dim,spacedim,Number@> and so on. + * * @author Sebastian Pauletti, 2011, Luca Heltai, 2015 */ template -- 2.39.5