Is orientation of Stokes theorem a convention?

Question

Is orientation of Stokes theorem, that is, right hand rule, a convention? Can we also choose the left hand rule? But will not it create problems in Physics where the sign of our physical quantity (which is derived from Stokes theorem) will depend on this convention?

Answer 1

Your intuition is more or less correct. The left hand rule would be perfectly consistent if it was used in place of right hand rule in every context.

Some physical quantities (such as magnetic field) will reverse direction, but the means to measure them will also measure in the opposite direction, and all observable (classical) phenomena would remain the same without altering the laws of physics.

The mathematical underpinnings of this convention are a bit more subtle. In 3 dimensional euclidean space, proper vectors transform accordingly when the space is rotated about an axis or reflected about a plane (and various geometric operations such as vector addition, orthogonal projection, and dot product obey this symmetry). The cross product of two proper vectors or the curl of a vector field do not have this property. When reflecting about an axis, their direction is reversed. Such objects are known as pseudovectors. The right hand rule is a convention for describing pseudovectors.

Curls and cross products can be treated as a different type of object (sometimes called bivectors), which can be though of as a surface area with an orientation along its boundary (or, algebraically, as an antisymmetric matrix in an orthonormal basis). These objects have their own transformation rules, and there is no need to introduce sign conventions.

In three dimensions, every surface area can be conveniently described by a normal vector, but doing this with bivectors come with some caveats:

First, the resulting vector does not transform correctly under reflection, instead transforming as a pseudovector.

Second, there is no obvious choice of normal vector, in that there is no correspondence between orientation around a surface and a normal direction. It is therefore necessary to choose a convention, such as right hand rule, and avoid reflections entirely (since RHR ceases to be consistent under reflection).

If you want to use Stokes' Theorem in more general contexts (such as on 4-dimensional manifolds, as in General Relativity), it's best to avoid pseudovectors entirely and keep a clear distinction between vectors, bivectors, etc. Without pseudovectors, this "generlized" form of the theorem has no RHR-esque sign conventions.