Mathematics, form and function by Saunders MacLane PDF

By Saunders MacLane

ISBN-10: 0387962174

ISBN-13: 9780387962177

A survey of the total of arithmetic, together with its origins and deep constitution

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Extra resources for Mathematics, form and function

Example text

The formulae at 'the end of the two preceding sections can also be written as follows: 36 Df • f«D(logabs f) and Df • rec exp f*D(exp f). Replacing f In the former formula by a particular function f la called logarithmic derivation (or differentiation) of f. Similarly, replacing f In the latter formula by a particular function f might be called exponential derivation of f • We -apply the former. method with benefit whenever logabs f IB simpler than f. As an example of logarithmic derivation, we treat the power functions* From c - po m exp(c-log) it follows that log c-po « c*log which is Indeed simpler than c-po.

We apply the formula exp(f +g) « exp f*exp g to f * J and g • c. We obtain D[exp(J -t-c)] =D exp(J -»-,c)»D(J *c)»D exp(] +c)-l=D exp (J +e). On the other hand D[exp(J -»-c)] »D(exp J«exp c) «D(exp*exp c) *exp c*D exp. Thus, D exp(J -f c)sexp c*exp. Substituting 0 in this equality we obtain on the left side: D exp(J+c)0*D exp(0 + o)*D exp c on the right side: exp cO*D exp 0»exp c«D exp 0« Thus D exp c « exp c»D exp 0 fbr each constant o. If we have 54 a base of constant a It follows that D exp * exp*D exp 0* We see that the derivative of an exponential function Is a constant Multiple of the function* We shall postulate the existence of an exponential function for which D exp 0*1.

We shall read the symbol Sf *an ant 1 derivative of f* or 'an integral of f* Indicating by this expression the multi-valuedness of the operator S In contrast to the uni-valuedness of D. The latter Is expressed in the Implication If f - g, then Df » Dg which will be of basic Importance for the Algebra of Antiderivatives. Sf is what in the classical analysis is denoted by /f (x)dx while f ~g expresses the relation ff(x) = gf(x) for which the classical theory does not Introduce a special symbol. Only to some extent f -»g corresponds to what in classical integral calculus is denoted by f(x) a g(x) + const.

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Mathematics, form and function by Saunders MacLane

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