#include <deal.II/base/config.h>
#include <deal.II/base/tensor.h>

Classes
class	DerivativeForm< order, dim, spacedim, Number >

Functions
template<int spacedim, int dim, typename Number1 , typename Number2 >
Tensor< 1, spacedim, typename ProductType< Number1, Number2 >::type >	apply_transformation (const DerivativeForm< 1, dim, spacedim, Number1 > &grad_F, const Tensor< 1, dim, Number2 > &d_x)

template<int spacedim, int dim, typename Number1 , typename Number2 >
DerivativeForm< 1, spacedim, dim, typename ProductType< Number1, Number2 >::type >	apply_transformation (const DerivativeForm< 1, dim, spacedim, Number1 > &grad_F, const Tensor< 2, dim, Number2 > &D_X)

template<int spacedim, int dim, int n_components, typename Number1 , typename Number2 >
Tensor< 1, n_components, Tensor< 1, spacedim, typename ProductType< Number1, Number2 >::type > >	apply_transformation (const DerivativeForm< 1, dim, spacedim, Number1 > &grad_F, const Tensor< 1, n_components, Tensor< 1, dim, Number2 > > &D_X)

template<int spacedim, int dim, typename Number1 , typename Number2 >
Tensor< 2, spacedim, typename ProductType< Number1, Number2 >::type >	apply_transformation (const DerivativeForm< 1, dim, spacedim, Number1 > &DF1, const DerivativeForm< 1, dim, spacedim, Number2 > &DF2)

template<int dim, int spacedim, typename Number >
DerivativeForm< 1, spacedim, dim, Number >	transpose (const DerivativeForm< 1, dim, spacedim, Number > &DF)

Function Documentation

◆ apply_transformation() [1/4]

template<int spacedim, int dim, typename Number1 , typename Number2 >

Tensor< 1, spacedim, typename ProductType< Number1, Number2 >::type > apply_transformation	(	const DerivativeForm< 1, dim, spacedim, Number1 > &	grad_F,
		const Tensor< 1, dim, Number2 > &	d_x
	)

inline

One of the uses of DerivativeForm is to apply it as a linear transformation. This function returns $\nabla \mathbf F(\mathbf x) \Delta \mathbf x$ , which approximates the change in $\mathbf F(\mathbf x)$ when $\mathbf x$ is changed by the amount $\Delta \mathbf x$

$\nabla \mathbf F(\mathbf x) \; \Delta \mathbf x \approx \mathbf F(\mathbf x + \Delta \mathbf x) - \mathbf F(\mathbf x).$

The transformation corresponds to

$[\text{result}]_{i_1,\dots,i_k} = i\sum_{j} \left[\nabla \mathbf F(\mathbf x)\right]_{i_1,\dots,i_k, j} \Delta x_j$

in index notation and corresponds to $[\Delta \mathbf x] [\nabla \mathbf F(\mathbf x)]^T$ in matrix notation.

Definition at line 431 of file derivative_form.h.

◆ apply_transformation() [2/4]

template<int spacedim, int dim, typename Number1 , typename Number2 >

DerivativeForm< 1, spacedim, dim, typename ProductType< Number1, Number2 >::type > apply_transformation	(	const DerivativeForm< 1, dim, spacedim, Number1 > &	grad_F,
		const Tensor< 2, dim, Number2 > &	D_X
	)

inline

Similar to the previous apply_transformation(). Each row of the result corresponds to one of the rows of D_X transformed by grad_F, equivalent to $\mathrm{D\_X} \, \mathrm{grad\_F}^T$ in matrix notation.

Definition at line 456 of file derivative_form.h.

◆ apply_transformation() [3/4]

template<int spacedim, int dim, int n_components, typename Number1 , typename Number2 >

Tensor< 1, n_components, Tensor< 1, spacedim, typename ProductType< Number1, Number2 >::type > > apply_transformation	(	const DerivativeForm< 1, dim, spacedim, Number1 > &	grad_F,
		const Tensor< 1, n_components, Tensor< 1, dim, Number2 > > &	D_X
	)

inline

Similar to the previous apply_transformation(). Each row of the result corresponds to one of the rows of D_X transformed by grad_F.

Definition at line 484 of file derivative_form.h.

◆ apply_transformation() [4/4]

template<int spacedim, int dim, typename Number1 , typename Number2 >

Tensor< 2, spacedim, typename ProductType< Number1, Number2 >::type > apply_transformation	(	const DerivativeForm< 1, dim, spacedim, Number1 > &	DF1,
		const DerivativeForm< 1, dim, spacedim, Number2 > &	DF2
	)

inline

Similar to the previous apply_transformation(). In matrix notation, it computes $DF2 \, DF1^{T}$ . Moreover, the result of this operation $\mathbf A$ can be interpreted as a metric tensor in ${\mathbb R}^\text{spacedim}$ which corresponds to the Euclidean metric tensor in ${\mathbb R}^\text{dim}$ . For every pair of vectors $\mathbf u, \mathbf v \in {\mathbb R}^\text{spacedim}$ , we have: