Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributionss, by generalising the concept of an integer index to an ordered tuple of indices.

Definition and basic properties

An n-dimensional multi-index-tuple

of non-negative integers (i.e. an element of the -dimensional set of natural numbers).

, one defines:

1. Componentwise sum and difference

2. Partial order

3. Sum of components (absolute value)

4. Factorial

5. Binomial coefficient

6. Multinomial coefficient

7. .

8. Power

9. .

10. Higher-order partial derivative

11. (see also 4-gradient is also used.

Some applications

.

Derivative of a monomial

Multinomial theorem

Multi-binomial theorem

Note that, since is a vector and is a multi-index, the expression on the left is short for .

Leibniz formula

Taylor series

For an analytic function in In fact, for a smooth enough function, we have the similar Taylor expansion

Integration by parts

with compact support This formula is used for the definition of weak derivatives of distributionss.

See also

• Einstein notation

• Index notation

• Ricci calculus

References

• Saint Raymond, Xavier (1991). Elementary Introduction to the Theory of Pseudodifferential Operators. Chap 1.1 . CRC Press.