Definition 12.1.1 (Multi-Index).label For $\al=(\al_{1},...,\al_{n})$ with $\al_{i}\geq0$ integers then

• $\abs{\al}\define \sums{n }{i=1}\al_{i}$

• $\al!\define \al_{1}!\cds\al_{n}!$

• $x^{\al}\define x_{1}^{\al_1}\cds x_{n}^{\al_n}$

• $\pd^{\al}\define \prods{n}{i=1}\parens{\pder{}{x_i}}^{\al_i}=\frac{\pd^{\abs{\al}}}{\parens{\pd x_1}^{\al_1}\cds\parens{\pd x_n}^{\al_n}}$

We write $F(x,u,\pd_{1}u,...,\pd_{n}u,\pd_{1}^{2}u,...,\pd_{n}^{2}u,...,\pd_{n}^{k}u)=F(x,\parens{\pd^\al u}_{\abs{\al}\leq k})=0$ where $\pd_{i}=\frac{\pd}{\pd x_{i}}$ with $u=u(x_{1},...,x_{n})=y\in\R$ or $\com$.