Page 36 - Vector Analysis

Chapter 2

Differentiation of Functions of Several
Variables

2.1 Functions of Several Variables

Definition 2.1. Let V be a vector space (over a scalar field F). A V-valued function
f of n real variables is a rule that assigns a unique vector f (x1, ¨ ¨ ¨ , xn) P V to each point
(x1, ¨ ¨ ¨ , xn) in some subset A of Rn. The set A is called the domain of f , and usually is
denoted by Dom(f ). The set of vectors f (x1, ¨ ¨ ¨ , xn) obtained from points in the domain
is called the range of f and is denoted by Ran(f ). We write f : A Ñ V if f is a V-valued
function defined on A Ď Rn.

If V = R, we simply call f : Dom(f ) Ñ R a real-valued function, while if V = Rm,
we simply call f : Dom(f ) Ñ V as a vector-valued function.

A vector field is a vector-valued function f : Dom(f ) Ñ V such that Dom(f ) Ď V = Rn
for some n P N.

Definition 2.2. Let V be a vector space over R, A Ď Rn be a set, and f, g : A Ñ V be
V-valued functions, h : A Ñ R be a real-valued function. The functions f + g, f ´ g and
hf , mapping from A to V, are defined by

(f + g)(x) = f (x) + g(x) @x P A,
(f ´ g)(x) = f (x) ´ g(x) @x P A,
@x P A.
(hf )(x) = h(x)f (x)

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