Multi-index notation

Core idea

A multi-index is a tuple $\alpha=(\alpha_1,\ldots,\alpha_d)\in\mathbb N_0^d$ . The standard notation is

|\alpha|=\alpha_1+\cdots+\alpha_d, \quad \alpha!=\alpha_1!\cdots\alpha_d!, \quad x^\alpha=x_1^{\alpha_1}\cdots x_d^{\alpha_d},

\partial^\alpha =\partial_{x_1}^{\alpha_1}\cdots\partial_{x_d}^{\alpha_d}.

Derivative families

The symbol $D^a f$ may denote the family $(\partial^\alpha f)_{|\alpha|=a}$ . A scalar size such as $|D^a f(x)|=\sup_{|\alpha|=a}|\partial^\alpha f(x)|$ must be stated because other texts use Euclidean or summed norms on this finite family.

Uses

Multi-indices compress Taylor formulas, symbol estimates, Schwartz seminorms, and distributional derivatives in several variables.

References

Lawrence C. Evans, Partial Differential Equations, 2nd ed., AMS, 2010. Publisher record.