math16B_suppl_notes_5 - Math 16B S06 Supplementary Notes 5...

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Unformatted text preview: Math 16B S06 Supplementary Notes 5 The Differential Notation The differential notation is a convenient formalism which, once one becomes accustomed to it, can often provide a smooth way of carrying out integrations. The differential df ( x ) of the function f ( x ) is by definition the formal expression f ( x ) dx . Here, the dx in the expression is not to be thought of as d times x . It is rather the differential of x , a formal expression. Frequently one notationally suppresses the dependence of the function f on the variable x and writes simply (1) df = f dx. In a formal sense, then, f is the ratio of df by dx ; you can imagine that f is obtained by dividing through by dx in (1). The differential notation is thus an adjunct to the notation df dx for f . (If you wish, you may interpret the symbol dx on the right side of (1) as a mechanism for keeping track of the variable x on which f depends.) The notation goes back to G. Leibniz, one of the founders of calculus. Leibniz thought of dx as an infinitesimal change in the variable x and of df as the corresponding change in the function f . For him, the derivative f was the actual ratio of the two infinitesimals df and dx . When calculus was put on a rigorous basis in the 19 th century the imprecise notion of an infinitesimal was discarded, replaced by the notion of a limit (although infinitesimals can still be helpful on an...
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math16B_suppl_notes_5 - Math 16B S06 Supplementary Notes 5...

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