Multivariable Calculus 15 | Multi-Index Notation [dark version]

Multi-index notation streamlines multivariable partial derivatives by encoding both the differentiation order and which variables are involved into a single object. Instead of writing repeated partial derivatives with explicit exponents and variable order, an n-dimensional multi-index α (an n-tuple of nonnegative integers) packages the counts of derivatives with respect to each coordinate. The payoff is compact formulas that scale cleanly to higher-order derivatives—especially when building Taylor polynomials in several variables.

The starting point is the familiar partial-derivative notation: a derivative like ∂^{(something)}/∂x_1^{(…)}∂x_2^{(…)}… indicates how many times each variable is differentiated, and evaluation is shown with parentheses. When the function f is sufficiently smooth, Schwartz’s theorem guarantees that mixed partial derivatives commute, meaning the order of differentiation does not change the result. That symmetry motivates a notation that doesn’t track the order of operations—only how many times each variable is differentiated.

A multi-index α = (α_1, …, α_n) is defined as an n-dimensional vector whose components are natural numbers, with 0 allowed. Two basic derived quantities drive the notation. First, the “length” or absolute value of α is |α| = α_1 + … + α_n, which directly gives the total derivative order. Second, α is used to form monomials: for a vector x = (x_1, …, x_n), x^α means x_1^{α_1} x_2^{α_2} … x_n^{α_n}. This turns multi-index notation into a compact way to write multivariable polynomials.

Factorials also generalize componentwise: α! is defined as α_1! α_2! … α_n!. With this, generalized binomial coefficients can be defined for multi-indices of the same dimension, using (α choose β) = α! / ( (α−β)! β! ), where α−β is computed component by component. Although binomial coefficients are not needed immediately later, the framework is laid out for future combinatorial identities.

The central operator notation is D^α f, where D^α denotes the mixed partial derivative determined by α. Concretely, D^α f corresponds to taking ∂^{|α|}f / (∂x_1^{α_1} ∂x_2^{α_2} … ∂x_n^{α_n}). The total order is |α|, so the multi-index automatically encodes the derivative’s degree.

Examples with n = 3 make the mechanics clear. If α = (1,0,0), then |α| = 1, α! = 1, and D^α f reduces to ∂f/∂x_1. If α = (1,2,1), then |α| = 4 and α! = 1!·2!·1! = 2, and D^α f becomes the fourth-order mixed derivative with one derivative in x_1, two in x_2, and one in x_3. This compact bookkeeping is positioned as the key step toward writing multivariable Taylor polynomials in a form that mirrors the one-dimensional case.