18.03 Muddy Card responses, March 10, 2006 1. The commonest question concerned the idea and utility of operators. I’ll say something now. You can look ahead at the “exponential shift law” if you want, to see one use later. An operator modiFes a function in some way. D diﬀerentiates, so Dx = x ˙. [The independent variable isn’t indicated in the notation,, and has to be gleaned from the context. In fact in one of the lectures I muddied the waters further by writing Dx 2 = 2 x , so in that instance the independent variable must have been x . If it had been t , the correct formula would have been Dx 2 = 2 xx ˙, which could aslo be written Dx 2 = 2 xDx .] You can multiply operators and add them. Multiplication means “compose”: so D 2 means “do D twice,” or take the second derivative. If we want to express, say, ¨ x + bx ˙ + kx as the eﬀect of an operator on the function x , we’ll need a symbol for the operator which leaves x alone, the identity operator. Some books use
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