Difference between the notions of Real multivariate function and Scalar field and also between Vector multivariate function and Vector field

Question

I think I already know the definitions of "Real (multivariate) function" and "Vector (multivariate) function", but correct me if I'm wrong:

A Real function: A function which takes some real numbers and map them to another real number.

A Vector function: A function whic takes either reals or vectors and map them to another vector.

So far wherever I saw the words "Scalar field" or "Vector field" it seemed they are the same as "Real multivariate function" and "Vector multivariate function".

The question is:

Is there any differences between these notions? Do the word "Field" concentrates on a special type of these functions or it represents a totally new concept?

Answer 1

It depends on the interpretation. Consider a function $f: \>{\mathbb R}^3\to{\mathbb R}^3$. If the function value $f(x)$ remains somehow "attached" to the point $x$ and is drawn as a vector emanating from $x$, or is considered as an element of the tangent space $T_x$, then we call $f$ a vector field and draw just one figure. If, however the idea is that $f$ moves the points $x\in{\mathbb R}^3$ to points $f(x)$ in some other copy of ${\mathbb R}^3$ then we draw two figures, and call $f$ a vector-valued function.

Similarly, a function $f:\>{\mathbb C}\to{\mathbb C}$ can be considered as a complex-valued scalar field, and one then considers integrals $\int_\gamma f(z)\>dz$ for curves lying in the domain of $f$. On the other hand such an $f$ can also be considered as a mapping of one copy of ${\mathbb C}$ to another copy of ${\mathbb C}$.

The intended interpretation should be clear from the context; but "datawise", e.g., a complex-valued time signal $t\mapsto f(t)\in{\mathbb C}$ and a parametric representation of a curve in ${\mathbb C}$ are the same thing.