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Multi-Index Notation

Multi-index notation is used to shorten expressions that contain many indices. Let x in R^n and write x=(x_1,...,x_n). A multi-index alpha is an n-tuple of nonnegative integers alpha_j with j=1, ..., n (Hunter 2014, p. 10). The notation |alpha| is used to denote the length alpha_1+...+alpha_n.

For monomial expressions in coordinates x_i, multi-index notation provides a convenient shorthand. For example, writing x^alpha=x_1^(alpha_1)...x_n^(alpha_n) gives a compact notation

(partial^(|alpha|))/(partialx^alpha)=(partial^(alpha_1))/(partialx_1^(alpha_1))...(partial^(alpha_n))/(partialx_n^(alpha_n))

for the partial derivative.

See also

Index, Index Set

This entry contributed by W.D. Van Suijlekom

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References

Hunter, J. K. Notes on Partial Differential Equations. Jun. 18, 2014. https://math.ucdavis.edu/~hunter/pdes/pde_notes.pdf.

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Multi-Index Notation

Cite this as:

Weisstein, Eric W., with contributions by W.D. Van Suijlekom. "Multi-Index Notation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Multi-IndexNotation.html

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