Advanced Calculus fi..

138 Advanced Calculus, Fifth EditionProof of the rule (2.124). We assume that z = f (x, y) has continuous derivativesfx, fy, fxy , fvx in a domain D. Let R be a closed square region xo 5 x 5 xo + h,yo 5 y I yo + h contained in D. Then by the rule L~ gf(x)dx = g(b) - g(a),// tr dx dy = l;+h ll:+hRfx." dy dx - ,, I ,.,., >= f (xo + h, YO + h) - f'(xo, YO + h) - f (xo + h, yo) + f (xo, yo).We interchange the roles of x and y to obtain similarlyTherefore the two double integrals are equal orSince this holds for every such square region R in D, we must havefx, (x, Y) = f,, (x, y) in D.(See Problem 14 following Section 2.18.) Thus the proof is compete.Remark. This proof uses properties of double integrals which are usually developedin the first course in calculus; these properties are also considered in Chapter 4(Sections 4.3 and 4.11).Let z = f (x, y) and x = g(t), y = h(t), so that z can be expressed in terms of ralone. The derivative dzldt can then be evaluated by the chain rule (2.33) ofSection 2.8:By applying the product rule. one obtains the following expression for the secondderivative: