# A First Course in Infinitesimal Calculus by Daniel A. Murray

By Daniel A. Murray

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In our consideration an important point is that estimates of this kind exist. In this case, choosing ae = max(asi,ae2) and assuming zero initial data: x(0) = 0 or 77(^(0) = 0, i = 0 , 1 , . . 14) L(x(t)) = L{Ax{t) + b£(t)) \ L(a(t)) = L(c*x(t)) ] PL{x(t)) = AL(x(t)) + bL(t(t)) L(a(t))=c'L(x(t)) \ j L(x(t))=-{A-pI)-1bL(S(t)) L(*(t))=c*L(x(t)) \ j L(a(t)) = L(

Ak(t). 3) has the following geometrical interpretation: to each of vectors aj(t) assign a curve in M n . 3) implies that all these curves cannot be placed on a linear subspace of dimension less than n. Let us also remark that a symmetric matrix K is said to be a positively definite, K > 0, if the corresponding quadratic form z*Kz is positively definite: z*Kz > 0 V z £ t " , z ^ 0 . 1. At first we note that condition 1') is simply another formulation of condition 1). It can easily be seen that condition 2') yields condition 2).

28) are satisfied. Here i = 0 , 1 , . . , n — 1. For proving this fact it is sufficient to show that the first inequality holds. The latter follows directly from estimates )dr < KOI < |e *(0)l + < /3ext\x(0)\+ f \eA^-^\m(T)\dT o < t A A 0. 28) imply the existence of p$ such that a(t)\