Is the definite integral a special case of functionals?

In summary, functionals are machines that take a function and return a number, in contrast to functions which take a number and return another number. The definite integral is a special case of a functional where the domain is a space of functions. However, taking the derivative of a function is not a functional since it does not give a number for every element of the space of functions. Functionals can involve a function as a variable and other functions as non-variables.
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So yesterday I learned about functionals, which my book claims are "machines that take a function and return a number", in contrast to functions, which take a number and return another number. I immediately thought of definite integration: it's an operation that takes a function, and returns a number.

I did a Google search for this and found the Wikipedia page on functional integration, which says that a functional integral is an integral where the domain is not a region of space but a space of functions. Again, could the definite integral be a special case of this where the "space of functions" is just all of the values between the bounds of integration treated as constant functions?

And finally, when you take the derivative of a function like 3x, which is of course 3, is that also a functional? This seems like a bit more of a stretch, because taking the derivative of a more complicated function like x^2 gives back 2x which is a function, not a number.

Just wondering.
 
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  • #2
Yes, the definite integral is a functional since, as you said, once you "feed" it with a function it gives you back a number. However, the derivative operator is not a functional since it does not give a number for every element of the space of functions.
 
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jack476 said:
in contrast to functions, which take a number and return another number

Functionals are functions. They aren't "real valued functions of one real variable", but the definition of "function" is general enough to include functionals as being functions.

If you establish constant limits of integration on a definite integral, like [itex] \int_0^1 f(x) dx [/itex] or [itex] \int_{-\infty}^{\infty} f(x) dx [/itex] and consider [itex] f [/itex] the variable, then you have a functional. If you have a variable in the limits of integration, like [itex] \int_0^x f(t) dt [/itex] you don't get a functional.

You can define functionals that involve a function as a variable and other functions as non-variables. For example [itex] \int_0^1 f(x) e^x dx [/itex] is a functional if we consider [itex] f [/itex] the variable.
 
  • #4
Einj said:
However, the derivative operator is not a functional since it does not give a number for every element of the space of functions.

Neither does the definite integral.

The derivative is certainly a functional, a functional does not mean that you need to assign a number to every function (since there would be very few functionals then). You can actually restrict the domain to a suitable space of functions.
 
  • #5
A common functional in the L2 space is integration against a square-summable kernel K: [itex] h=Af[/itex] given by [itex] h(x)=\int_{-\infty}^{\infty}K(x,s)f(s)ds[/itex].
 

1. What is a functional?

A functional is a mathematical function that takes in another function as its input and produces a scalar value as its output. It can also be thought of as a "function of a function".

2. How is a functional different from a regular function?

A regular function takes in a variable as its input and produces a value as its output. A functional, on the other hand, takes in a function as its input and produces a scalar value as its output. This means that a functional operates on a larger set of inputs compared to a regular function.

3. What is the relationship between a definite integral and a functional?

A definite integral is a special case of a functional. It is a functional that takes in a function and produces a scalar value by integrating the function over a specific interval. In this case, the definite integral is the output of the functional.

4. What are some examples of functionals?

Some examples of functionals include the definite integral, the derivative, and the Dirichlet energy. These functionals take in a function as their input and produce a scalar value as their output.

5. How are functionals used in science?

Functionals are used in a variety of scientific fields, including physics, engineering, and economics. They are often used to model and analyze complex systems by breaking them down into smaller functions. In physics, functionals are used to calculate important quantities such as energy and momentum. In economics, functionals are used to optimize functions representing economic systems.

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