The short note concerns with elasticity problems involving singular and hypersingular integrals over open surfaces, specifically cracks, with the contour propagating in time. Noting that near a smooth part of a propagating contour the state is asymptotically plane, we focus on 1D hypersingular integrals and employ complex variables. By using the theory of complex variable singular and hypersingular integrals, we show that the rule for evaluation of the temporal derivative is the same as that for proper integrals. Being applied to crack problems the rule implies that the temporal derivative may be evaluated by differentiation under the integral sign.

中文翻译：

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关于具有传播轮廓的开曲面上超奇异积分的时间导数计算的一个注记

简短说明涉及弹性问题，涉及开放表面上的奇异和超奇异积分，特别是裂缝，轮廓随时间传播。注意到在传播轮廓的平滑部分附近，状态是渐近平面，我们专注于一维超奇异积分并使用复变量。通过使用复变奇异和超奇异积分理论，我们证明了时间导数的计算规则与真积分的计算规则相同。该规则应用于破解问题意味着可以通过积分符号下的微分来评估时间导数。