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Hyperreal Vectors/Theorem 1

This section may be skipped without affecting the rest of the course. We introduce hyperreal vectors and use them to give an infinitesimal treatment of vector derivatives. We shall concentrate on three dimensions; the theory for two dimensions is similar. A hyperreal vector in three dimensions is a vector

A = a1i + a2j + a3k

whose components a1, a2, and a3 are hyperreal numbers. The algebra of hyperreal vectors is in many ways similar to the algebra of hyperreal numbers. It begins with the notions of infinitesimal, finite, and infinite hyperreal vectors.

A hyperreal vector A is said to be infinitesimal, finite, or infinite if its length |A| is an infinitesimal, finite, or infinite number, respectively. Two hyperreal vectors A and B are said to be infinitely close, A ≈ B, if their difference B - A is infinitesimal (Figure 10.8.1).

10_vectors-280.gif10_vectors-281.gif

Figure 10.8.1

Example

Our first theorem shows how these notions depend on the components of the vectors.

THEOREM 1

Let A and B be hyperreal vectors.

(i)    A is infinitesimal if and only if all of its components are infinitesimal.

(ii)    A is finite if and only if all of its components are finite.

(iii)    A is infinite if and only if at least one of its components is infinite.

(iv)    A ≈ B if and only if a1 ≈ b1, a2 ≈ b2, and a3 ≈ b3.

PROOF (i), (ii), and (iii) are proved using the inequalities

10_vectors-285.gif10_vectors-286.gif10_vectors-287.gif

10_vectors-288.gif

and (iv) follows easily from (i). We prove (i). Suppose A is infinitesimal. This means that its length

10_vectors-289.gif

is infinitesimal. The inequalities show that la1|, |a2|, and |a3| are all between 0 and |A|. Therefore all the components a1, a2. and a3 are infinitesimal. On the other hand, if all the components are infinitesimal, then |a1| + |a2| + |a3| is infinitesimal, and by the last inequality, the length |A| is infinitesimal.

The following facts are obvious from the definitions.

  • The only infinitesimal real vector is 0.
  • Every real vector is finite.
  • Every infinitesimal vector is finite.
  • A is infinitesimal if and only if A ≈ 0.
Home Vectors Hyperreal Vectors Hyperreal Vectors/Theorem 1

Last Update: 2006-11-25