1

In the beginning of Chapter 8 entitled "Differential Calculus of Scalar and Vector Fields" of Apostol's Calculus, he says that when we are considering functions $f:\mathbb{R}^n\to\mathbb{R}^m$, then

  • if $n$ and $m$ are both 1, we have a real-valued function of a real variable.

  • if $n=1$ and $m>1$, then we have a vector-valued function of a real variable.

  • if $n>1$ and $m=1$ then we have a real-valued function of a vector variable (that is, a scalar field).

  • if $n>1$ and $m>1$ we have a vector-valued function of a vector variable (that is, a vector field)

I am a bit confused by the choice of terminology here. The concepts involved are all clear. It's just the following: according to the above, when we have a function of a single variable we call it a function of a "real variable", but when we have a function of a vector, then we say the function is of a "vector variable".

The way I understand it, to be consistent, we should either say

  1. "function of a single variable" and "function of a vector variable"

or

  1. "function of a real variable" and "function of a real vector variable"

or some other combination such that when we denote a function with a single variable and with a vector-variable we convey the same information.

Note that a similar question was asked before, but it does not go into much depth on these issues.

xoux
  • 4,913
  • People do say "function of a single variable". – Mr. Brown Jul 15 '23 at 05:50
  • Why are there "issues" ? All my comments to this question apply fully to yours. – Kurt G. Jul 15 '23 at 06:03
  • @Mr.Brown of course, you are right, they do. I think what I was trying to ask is: since this is a book we are talking about (Apostol's Calculus) and considering such a simple topic as what we call different types of functions based on the dimensions of domain and range, why does he say "function of a real variable" and "function of a vector variable", instead of "function of a single variable" and "function of a vector variable"? Like, why do we specify that the variable contains real numbers in one case but not in the other. Is this just inconsistency by the author? – xoux Jul 15 '23 at 06:12
  • @KurtG. You wrote about a topic which isn't a science, which is human language. You opined that human language isn't a programming language. I agree for the most part, except in the case that is the object of my question, namely what we might call "classification" or "taxonomy" of functions. This is not really language in sense of communicating complex ideas. It is simply putting concepts in boxes and I think this should be like a programming language. – xoux Jul 15 '23 at 06:17
  • Just “single variable” would be ambiguous since technically a function of a vector variable is a function of a single vector variable. Real vector variable would be more technically correct than just vector variable. But if all your vectors are real vectors, then it’s unnecessarily wordy (and the type of vector is nermally implied by context). – Eric Jul 15 '23 at 06:18
  • @evianpring . Thanks for reading those comments. So let's agree that we disagree about how human language relates to maths. – Kurt G. Jul 15 '23 at 06:56

0 Answers0