Vtyjkyuk

This is a question about terminology. Is there any common name for the trivial extension of a real function $f colon mathbbR to mathbbR$ to several variables $tildef colon mathbbR^n to mathbbR^n$?

By "trivial extension", I mean defining $tildef$ in terms of $f$ as $tildef(x_1, dots, x_n) = (f(x_1), dots, f(x_n))$.

As a simple example, if we have a trigonometric function like $sin(x)$, we can trivially extend it to $mathbbR^3$ as $sin(x, y, z) = (sin(x), sin(y), sin(z))$. I'm just wondering if there is any accepted name and well-known reference for this simple way of building multivariate functions.

I don't think there is a generally accepted and generally understood term here. You might say $tilde f$ is the $k$-fold tensor product of itself in some space $bigotimes_1^k V$ of multivariate functions, or use some notation like $f^otimes$, but you would also have to explain precisely what you mean to the audience. Which would end up taking more words and notation than simply giving the formula directly.

Mathematica speaks about "threading a function over lists". It is a built in feature for many standard functions: For example, you get

N[Sin[1,5,-2,4]]
0.841471, -0.958924, -0.909297, -0.756802

If you use such a feature for various functions several times in a paper or talk it pays to invent a special notation.

If $f$ is a given function acting on points $xin X$ you could denote by $tilde f$ (or similar) the induced function on $n$-tuples of points, whereby $n$ can be arbitrary, but finite.