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This paper is motivated by recent developments of higher gauge theory.Different from its style of using higher category theory, we try to describethe concept of higher parallel transport within setting of classical principalbundle theory. From this perspective, we obtain a global geometric proof on ageneralized 3-dimensional non-abelian Stokes' theorem related to paralleltransport on surfaces. It can be naturally extended to four dimension when theunderlying crossed module is replaced by crossed 2-module. This 4-dimensionalStokes' theorem yields a global formula for volume holonomy, and alsoguarantees its gauge invariance. In the process, we also find compositionformulas for parallel transport on volumes which have been seen in the case ofsurfaces.

A Global Geometric Approach to Parallel Transport of Strings in Gauge Theory