Mathematical Methods for Physics and Engineering - Matematica.NET

5.6 CHANGE OF VARIABLESyρφxFigure 5.1The relationship between Cartesian and plane polar coordinates.For each different value of i, x i will be a different function of the u j .Inthiscasethe chain rule (5.15) becomes∂f∂u j=n∑i=1∂f∂x i∂x i∂u j, j =1, 2,...,m, (5.17)and is said to express a change of variables. In general the number of variablesin each set need not be equal, i.e. m need not equal n, but if both the x i and theu i are sets of independent variables then m = n.◮Plane polar coordinates, ρ and φ, and Cartesian coordinates, x and y, arerelatedbytheexpressionsx = ρ cos φ, y = ρ sin φ,as can be seen from figure 5.1. An arbitrary function f(x, y) canbere-expressedasafunction g(ρ, φ). Transform the expression∂ 2 f∂x + ∂2 f2 ∂y 2into one in ρ and φ.We first note that ρ 2 = x 2 + y 2 , φ =tan −1 (y/x). We can now write down the four partialderivatives∂ρ∂x = x(x 2 + y 2 ) =cosφ, ∂φ1/2 ∂x = −(y/x2 )1+(y/x) = − sin φ2 ρ ,∂ρ∂y = y(x 2 + y 2 ) =sinφ, ∂φ1/2 ∂y = 1/x1+(y/x) = cos φ2 ρ .159