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A natural idea is that during the transformation from curve and surface integrals to multiple integrals using Gauss' law, we loss the "information" of the direction.Hence, we need to regulate that in the Gauss' law.Another idea about the Stokes' law is that we transform a curve integral into a surface integral, both of which have directions.So, here comes Right-Hand Rule which helps us decide the sign is '+' or '-' Just my personal idea about the question.Discussions are welcomed

ee_thu_freshman
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    The simple answer to me is because the underlying quantities that we care about depend on orientation: outward and inward flux differ by a sign, counterclockwise and clockwise circulation differ by a sign, etc. – Ian May 08 '23 at 15:30
  • Also to clarify, the right hand rule is not a magical thing really, it is just a convention for associating one of the two directions of rotation in a plane with one of the directions normal to the plane and the other direction of rotation with the other direction normal to the plane. It still "really matters" which direction of rotation you care about. – Ian May 09 '23 at 13:19
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    yes, looking back after several days, I've discovered new meaning for directions in these integrals.It is rather a "relative" concept, where we put '-' before a line integral or surface integral that is under the opposite direction. Just like we have upper bound and lower bound in normal definite integrals. – ee_thu_freshman May 11 '23 at 16:07

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