MHB Is $f$ a function or a functional?

OhMyMarkov
Messages
81
Reaction score
0
Hello everyone!

I'm a bit confused about referring to a mapping as function or functional, for example: $f(x_1, x_2, x_3) = x_1+2x_2 ^2+3x_3 ^3$. $f$ takes vector $\textbf{x}=[x_1 \; x_2 \; x_3]$ and maps it to a scalar. Now, is $f$ a function or a functional?

Thanks!
 
Physics news on Phys.org
It's both, in this case. A functional is a function that maps a vector from a vector space into the field over which the vector space is defined. So all functionals are functions, but not all functions are functionals. Does that help?
 
Ackbach said:
It's both, in this case. A functional is a function that maps a vector from a vector space into the field over which the vector space is defined. So all functionals are functions, but not all functions are functionals. Does that help?

Not really sure unless, say, a counter example would be a continuous transform such as the continuous Fourier transform?
 
OhMyMarkov said:
Not really sure unless, say, a counter example would be a continuous transform such as the continuous Fourier transform?

Sure. The result of a Fourier transform is another function (or vector, depending on your vector space). Another counterexample is any matrix transformation, such as a rotation.
 
Thread 'Derivation of equations of stress tensor transformation'
Hello ! I derived equations of stress tensor 2D transformation. Some details: I have plane ABCD in two cases (see top on the pic) and I know tensor components for case 1 only. Only plane ABCD rotate in two cases (top of the picture) but not coordinate system. Coordinate system rotates only on the bottom of picture. I want to obtain expression that connects tensor for case 1 and tensor for case 2. My attempt: Are these equations correct? Is there more easier expression for stress tensor...

Similar threads

Back
Top