Is there a concise term for a vector field where each component is a function of its own variable?

Question

Some simple vector fields have components that are just a function of their own variable. for example, $$F(x,y,z)=\begin{bmatrix}x^2+1\\\sin y\\3\end{bmatrix}$$ as opposed to: $$F(x,y)=\begin{bmatrix}5x+y^2\\4\sin x\end{bmatrix}$$ They feel somewhat special, and they are easier to work with in general. Is there a formal name for these? How would I refer to such a vector field concisely?

Answer 1

In contexts where you are likely to use a phrase like "vector field", those "simple vector fields" are rarely encountered, and do not usually have a special name.

However, in more abstract/theoretical contexts like Topology, something like this might be called a (cartesian) product of functions. For example, if we have $f_1:A_1\to B_2$, $f_2:A_2\to B_2$, and $f_3:A_3\to B_3$, then we would have the product function $f_1\times f_2\times f_3:\left(A_1\times A_2\times A_3\right)\to\left(B_1\times B_2\times B_3\right)$.

In your case, $A_1,A_2,A_3,B_1,B_2,B_3$ are all implied to be $\mathbb R$ (so that $f_1\times f_2\times f_3:\mathbb R^3\to\mathbb R^3$), and $f_1(x)=x^2+1$, $f_2(x)=\sin x$, and $f_3(x)=3$.

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As an aside, when speaking informally you might hear someone call the following a product function: $\mathbf{r}(t)=\begin{bmatrix}t^2+1\\\sin t\\3\end{bmatrix}$. But to distinguish it from the above case, it would be better to call that $\mathbf{r}$ a "tripling"/"tupling" of functions (or "maps" or "morphisms"). You can read a discussion of this at the nLab's page about pairings, and John Gowers asks about a name for the infinite case in this MSE question.