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We give a gauge-independent definition of magnetic monopoles in the $SU(N)$Yang-Mills theory through the Wilson loop operator. For this purpose, we givean explicit proof of the Diakonov-Petrov version of the non-Abelian Stokestheorem for the Wilson loop operator in an arbitrary representation of the$SU(N)$ gauge group to derive a new form for the non-Abelian Stokes theorem.The new form is used to extract the magnetic-monopole contribution to theWilson loop operator in a gauge-invariant way, which enables us to discussconfinement of quarks in any representation from the viewpoint of the dualsuperconductor vacuum.

Non-Abelian Stokes theorem for the Wilson loop operator in an arbitrary representation and its implication to quark confinement