Jan 212014

2014 年第 2 期的 “The American Mathematical Monthly” 文章较多, 一共有 17 篇, 其中至少 5 篇是对旧定理–诸如 Stirling’s Formula, 余弦定理, Clairaut’s Theorem(Symmetry of second derivatives 二阶导数的对称性)这样的经典结论–的新证明.

Symmetry of second derivatives If \(f_{xy}\) and \(f_{yx}\) are continuous at any given point, then they are equal at the point.

显而易见, 每一本多元微积分的入门教科书都会论述二阶导数的对称性.

Lemma  Let \(f_{xy}\) and \(f_{yx}\) be continuous on rectangle \(R=[a,b]\times[c,d]\). Then

\[\iint_Rf_{xy}\,\mathrm dA=\iint_Rf_{yx}\,\mathrm dA=f(b,d)-f(b,c)-f(a,d)+f(a,c).\]

这是显然的, 因为

\[\iint_Rf_{xy}\,\mathrm dA=\int_a^b\left(\int_c^d f_{xy}(x,y)\,\mathrm dy\right)\,\mathrm dx.\]

由微积分基本定理, 就得到了引理.             \(\Box\)

回到我们的最终目标. Proof by contraduction.

Suppose they are not identically equal. Then at some point \((a, b)\), they differ; say

\[f_{xy}(a, b)-f_{yx}(a, b)=l\gt0.\]

Note that, since \(f_{xy}\) and \(f_{yx}\) are continuous, there is some small \(\triangle x\times\triangle y\) rectangle, centered at \((a, b)\), on which

\[f_{xy}(x, y)-f_{yx}(x, y)\geqslant\frac l2,\]


\[\iint_R\left(f_{xy}-f_{yx}\right)\,\mathrm dA\geqslant\iint_R \frac l2\,\mathrm dA=\frac l2\triangle x\triangle y\gt0.\]

this contradicts the lemma.                 \(\Box\)

这个证明真是简洁非常! 开始提到的美国数学月刊的新解法, 没这么漂亮.

 Posted by at 8:08 am

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