which is normalized as: $$\int_{A} \mathcal{P} {\left( \vec{x} \right)} d^{2} \vec{x}
=
  h^{2}$$ is bounded as: $$\mathcal{P} {\left( \vec{x} \right)}
=
  0 
  {\text{ for }}
  {\left|{\vec{x}}\right|}
\ge
  h/2$$ is differentiable, and decreases monotonically with ${\left|{\vec{x}}\right|}$. The continuous function is defined such that: $$\lim_{h \to 0} \tilde{c} {\left( \vec{x} \right)}
= 
  c {\left( \vec{x} \right)}$$ i.e. the function $\tilde{c} {\left( \vec{x} \right)}$ approaches $c {\left( \vec{x} \right)}$ as the interface thickness goes to zero. $\tilde{c}$ is differentiable because $\mathcal{P}$ is, and: $$\nabla_{2D} \tilde{c} {\left( \vec{x} \right)}
=
  \frac{1}{h^{2}} \int_{A} c {\left( \vec{x}' \right)} \nabla_{2D} \mathcal{P} {\left( \vec{x}'-\vec{x} \right)} d^{2} \vec{x}$$ where $\nabla_{2D}$ is the two-dimensional gradient in the plane of the wall. Using Gauss’ theorem and the realization that $c {\left( \vec{x} \right)}$ is constant within each fluid, the above integral can be written as: $$\nabla_{2D} \tilde{c} {\left( \vec{x} \right)}
=
  \frac{{\left[ c \right]}}{h^{2}} \int_{L} \hat{n}_{2D} {\left( \vec{x}_{L} \right)} \mathcal{P} {\left( \vec{x}-\vec{x}_{L} \right)} d L
\label{gauss}$$ where ${\left[ c \right]} = c_{2} - c_{1}$, thus converting the surface integral to a line integral. To pull the normal out of the integral, its weighted mean is calculated. Since $\mathcal{P}$ is bounded, its maximum contribution to the line integral is $\mathcal{O} {\left( h 