Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

•• If one differentiates a function f(x, y) with respect to x and then differentiates the result with respect to y, one obtains the mixed second derivative

It is a theorem that for any “reasonable” function (which includes any function normally encountered in physics) it makes no difference which differentiation is done first. That is,

A proof of this useful result is beyond the level of this book, but you can check the truth of (8.101) for some specific functions. Evaluate both sides of (8.101) for the following functions and verify that they are equal: (a) f = (x + y2)3, (b) , (c) (x + y) ln(x − y).