The Biot–Savart law is used in aerodynamic theory to calculate the velocity induced by curved vortex lines. Explicit formulas are developed, using multivariate Appell hypergeometric functions, for the velocity induced by a general parabolic vortex segment. The formulas are derived by constructing a particular pencil of elliptic curves whose period integrals provide the solution to the induced velocity. We use numerical integration and a perturbation expansion to evaluate the validity of our formulas.

中文翻译：

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Biot–Savart定律在椭圆形抛物线涡段上的应用

空气动力学理论中使用了比奥一萨瓦特定律来计算弯曲涡流线引起的速度。使用多元Appell超几何函数，针对由一般抛物线涡旋线段引起的速度，开发了显式公式。通过构造特定的椭圆曲线铅笔可以推导这些公式，这些椭圆的周期积分为感应速度提供了解决方案。我们使用数值积分和摄动展开来评估公式的有效性。