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.

•• Changes of coordinates in two dimensions (such as that from x, y to r, ϕ) are much more complicated than in one dimension. In one dimension, if we have a function f(x) and choose to regard x as a function of some other variable u, then the derivative of f with respect to u is given by the chain rule,

In two dimensions the chain rule reads

and

(a) Use the relations (8.104) (Problem 1) to evaluate the four derivatives ∂x/∂r, ∂y/∂r, ∂x/∂ϕ), and ∂y/∂ϕ. (b) If , use (8.105) to find ∂f/∂ϕ. (c) What is ∂f/∂ϕ? (d) By noticing that and hence that f = exp(r), evaluate ∂f/∂r and ∂f/∂ϕ directly and check that your answers in parts (b) and (c) are correct.

Problem 1

• (a) For the two-dimensional polar coordinates defined in Fig. 8.7 (Section 8.4), prove the relations

 x = r cos ϕ and y = r sin ϕ (8.104)

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(b) Find corresponding expressions for r and (ϕ in terms of x and y.