(Another proof of equality of mixed partials) Suppose that f

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(Another proof of equality of mixed partials) Suppose that f12.x; y/ and f21.x; y/ are continuous in a neighbourhood of

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(Another proof of equality of mixed partials) Suppose that f12.x; y/ and f21.x; y/ are continuous in a neighbourhood of the point .a; b/. Without assuming the equality of these mixed partial derivatives, show that ZZ R f12.x; y/ dA D ZZ R f21.x; y/ dA; where R is the rectangle with vertices .a; b/, .a C h; b/, .a; b C k/, and .a C h; b C k/ and h2 C k2 is sufficiently small. Now use the result of Exercise 29 to show that f12.a; b/ D f21.a; b/. (This reproves Theorem 1 of Section 12.4. However, in that theorem we only assumed continuity of the mixed partials at .a; b/. Here, we assume the continuity at all points sufficiently near .a; b/.)

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