Calculus How To

Functional / Higher-Order Functions

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Calculus Definitions >

In general, a functional is a function of functions: a function that depends on other functions.

There are a few modifications on the basic definition. Which one you use depends on what field you’re working in.

Functionals: Different Definitions

1. Calculus of Variations
Functions, the building blocks of differential calculus, take scalars as inputs and produce scalars as outputs. Functionals are the building blocks for calculus of variations, taking a function as an input, returning a scalar output.

While there are a different types of functionals, calculus of variations is mostly concerned with one in particular: where a definite integral’s integrand contains a (yet to be determined) function. The goal of calculus of variations is to study the changes in these functional while moving from one function to the next.

In notation, a functional is written as I[u(x)], where I is a unique scalar value for each function u(x). For example: [1].


The integrand F(x, u, u′, u′′) dx includes the function u and its derivatives.

2. General Mathematics
In general mathematics, a functional may refer to a function specifically produced from a set of real-valued functions. For example, a functional could be the maximum of a set of functions on the closed interval [0, 1]. A binary functional takes two sets of functions to create one function. For example, the maximum of two sets of functions on the interval [0, 1]. The complement of a functional is a closure [2].

3. CompSci
Functionals in computer science (particularly machine learning) are defined slightly differently, as taking functions as arguments or yielding functions as results. Functionals can map functions to real numbers and real numbers to functions. In this context, they are often called higher-order functions. Higher-order functions include the differential operator and the definite integral. [3].


[1] Cassel, K. (2013). Variational Methods with Applications in Science and Engineering. Cambridge University Press.
[2] Watkins, The Calculus of Variations
in Functionals. Retrieved April 6, 2021 from:
[3] Harper, R. Functionals0. Retrieved April 6, 2021 from:

Stephanie Glen. "Functional / Higher-Order Functions" From Calculus for the rest of us!

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