Continuity of field components

Depending on the constitutive relation, some components of the fields $u$ and $v$ stays continuous everywhere in a laminate while the other component are only piece-wise continuous.

Example

For example, consider conducting materials. The normal component of the current and the tangent component(s) of the electrical field are continuous, the remaining components are discontinuous.

1. Generally, the vector field ${ v}(x)$ is subject to some differential constraints (the vectors may be divergencefree, curlfree, etc.) These constraints imply that some linear combinations of its components are continuous on the boundary between layers,
   

   $${ p} [ { v}_1 -{ v}_2]=0,$$ (3)

   
   
   Here ${ p}$ is a matrix projector on the subspace of continuous components.

   The components ${ p} { v}$ of ${ v}$ are constant in the laminate and the the supplementary orthogonal components ${ q} { v}$ of ${ v}$ jump on the boundary between layers.
   

   $${ q} [ { v}_1 -{ v}_2] \neq 0;$$ (4)

   
   
   Projections ${ p}$ and $q$ are mutually orthogonal: ${ q}^{T}{ p}=0$

2. The dual vector ${ u}$ has the properties similar to the vector ${ v}$, but its discontinuous components are supplementary to those of the prime vector ${ v}$; in other words, the continuous and discontinuous components of ${ u}$ and ${ v}$ switch their places:
   

   $${ q} [ { u}_1 -{ u}_2] = 0, \quad { p} [ { u}_1 -{ u}_2] \neq 0.$$ (5)

   
   

The next Table shows discontinuous components $v_{d}$ of the fields in the equations of interest. These component are completely defined by the normal to the laminate. The matrix $q$ projects vector $v$ onto $v_{d}$ as follows:

$$q: ~ v \rightarrow v_{d}$$

Table 1: Space of discontinuous components $v_d$ of the fields in various equilibria

Process	Properties tensor $D$	Field vector $v$	Subspace $v_{d}$ of discontinuous components
Conductivity	Conductivity tensor	Curlfree field	normal $v_n=n\cdot v$
Conductivity	Resistivity tensor $R$	Divergencefree current	tangent(s) $v_{t_1}=t_1\cdot v, ~ v_{t_2}=t_2\cdot v$
2D Elasticity	Stiffness tensor $C$	Strain $: \epsilon$	the tensorial component $\epsilon_{nn}= n \cdot\epsilon \cdot n$
2D Elasticity	Compliance tensor $S$	Stress $\sigma$	two tensorial components $\sigma_{tt}, \sigma_{nt}$
3D Elasticity	Stiffness tensor $C$	Strain $\epsilon$	Three tensorial components $\epsilon_{nn}, \epsilon_{nt_1}, \epsilon_{nt_2}$
3D Elasticity	Compliance tensor $S$	Stress $\sigma$	Three tensorial components $\sigma_{t_1 t_1}, \sigma_{t_{1} t_{2}}, \sigma_{t_2t_2}$
Maxwell equations	$H, B$	.	$H_n, B_t$