Simple Nature - Light and Matter

know how to integrate with respect to a variable that’s a vector,so let’s define a variable s that indicates the distance traveled sofar along the curve, and integrate with respect to it instead. Theexpression F · dr can be rewritten as |F| | dr| cos θ, where θ is theangle between F and dr. But | dr| is simply ds, so the amount ofwork done becomes∆E =∫ r2r 1|F| cos θ ds .Both F and θ are functions of s. As a matter of notation, it’scumbersome to have to write the integral like this. Vector notationwas designed to eliminate this kind of drudgery. We therefore definethe line integral∫F · drCas a way of notating this type of integral. The ‘C’ refers to the curvealong which the object travels. If we don’t know this curve then wetypically can’t evaluate the line integral just by knowing the initialand final positions r 1 and r 2 .The basic idea of calculus is that integration undoes differentiation,and vice-versa. In one dimension, we could describe aninteraction either in terms of a force or in terms of an interactionenergy. We could integrate force with respect to position to findminus the energy, or we could find the force by taking minus thederivative of the energy. In the line integral, position is representedby a vector. What would it mean to take a derivative with respectto a vector? The correct way to generalize the derivative dU/ dx tothree dimensions is to replace it with the following vector,dU dU ˆx +dx dy ŷ + dUdz ẑ ,called the gradient of U, and written with an upside-down delta 18like this, ∇U. Each of these three derivatives is really what’s knownas a partial derivative. What that means is that when you’re differentiatingU with respect to x, you’re supposed to treat y and z andconstants, and similarly when you do the other two derivatives. Toemphasize that a derivative is a partial derivative, it’s customary towrite it using the symbol ∂ in place of the differential d’s. Puttingall this notation together, we have∇U = ∂U ∂U ˆx +∂x ∂y ŷ + ∂U∂zẑ [definition of the gradient] .The gradient looks scary, but it has a very simple physical interpretation.It’s a vector that points in the direction in which U is18 The symbol ∇ is called a “nabla.” Cool word!216 Chapter 3 Conservation of Momentum