math16B_suppl_notes_5

# math16B_suppl_notes_5 - Math 16B – S06 – Supplementary...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This note was uploaded on 09/06/2010 for the course CHEM 10894 taught by Professor Pederson during the Fall '10 term at Berkeley.

### Page1 / 4

math16B_suppl_notes_5 - Math 16B – S06 – Supplementary...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online