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Theorem 3: Vextor Valued Function
Now let us consider a vector valued function F(t) = f1(t)i + f2(t)j + f3(t)k. Each of the real functions f1, f2, f3 has a natural extension to a hyperreal function. Thus the real vector valued function F can be extended to a hyperreal vector valued function. When t is a hyperreal number, F(f) is defined if and only if all of f1(t), f2(t), and f3(t) are defined, and its value is F(t) = f1i + f 2(t)j + f3(t)k. We shall now return to the study of vector derivatives. THEOREM 3 The vector valued function F(f) has derivative V at t if and only if
for every nonzero infinitesimal Δt. This theorem is exactly like the definition of the derivative of a real function in Chapter 2, except that it applies to a vector valued function. PROOF OF THEOREM 3 Suppose first that F'(t) = V. This means that f1'(t)i + f2'(t)j + f3'(t)k = v1i + v2j + v3k. Then f1'(t) = v1, f2'(t) = v2, f3'(t) = v3. Let Δt be a nonzero infinitesimal. Then
and similarly for v2, v3. It follows that
By reversing the steps we see that if the above equation holds for all nonzero infinitesimal Δt, then V = F'(t).
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Last Update: 2010-11-25