Integral theorems such as Stokes’ and Gauss’ are fundamental in many parts of physics. For instance, Faraday’s law allows computing the induced electric current on a closed circuit in terms of the variation of the flux of a magnetic field across the surface spanned by the circuit. The key pointfor applying Stokes’ theorem is that this surface must be orientable. Many students wonder what happens to the flux through a surface when this is not orientable, as it happens with a Mbius strip. On an orientable surface one can compute the flux of a solenoidal field using Stokes’ theorem in terms of the circulation of the vector potential of the field along the oriented boundary of the surface. But this cannot be done if the surface is not orientable, though in principle this quantity could be measured on a laboratory. For instance, checking the induced electric current on a circuit along the boundary of a surface if the field is a variable magnetic field. We shall see that the answer to this puzzle is simple and the problem lies in the question rather than in the answer.

斯托克斯和高斯等积分定理在物理学的许多部分都是基本的。例如，法拉第定律允许根据穿过电路跨越的表面的磁场通量的变化来计算闭合电路上的感应电流。应用斯托克斯定理的关键点是该表面必须是可定向的。许多学生想知道当磁通量不可定向时，穿过表面的通量会发生什么，就像 Mbius 带发生的那样。在一个可定向的表面上，可以使用斯托克斯定理根据场的矢量势沿表面的定向边界的循环来计算螺线管场的通量。但是，如果表面不可定向，则无法做到这一点，尽管原则上可以在实验室中测量该量。例如，如果场是可变磁场，则检查沿表面边界的电路上的感应电流。我们将看到这个难题的答案很简单，问题在于问题而不是答案。