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"Stokes theorem" equivalent for cross product line integral
I am aware that the vector path integral of a closed curve under certain conditions is equivalent to the flux of of the curl of the vector field through any surface bound by the closed path. In other words, Stokes theorem:
\oint_S \textbf{F} \cdot d\textbf{r} = \int_S \nabla \times \textbf{F} \cdot d\textbf{S}
But how about the closed path integral of the cross product of a vector field with the differential line segment:
\oint \textbf{F} \times d\textbf{r}
Is there any vector calculus theorem paralleling the Stokes theorem for a closed path integral I can use without me having to evaluate the line integral directly?
Homework Statement
I am aware that the vector path integral of a closed curve under certain conditions is equivalent to the flux of of the curl of the vector field through any surface bound by the closed path. In other words, Stokes theorem:
\oint_S \textbf{F} \cdot d\textbf{r} = \int_S \nabla \times \textbf{F} \cdot d\textbf{S}
But how about the closed path integral of the cross product of a vector field with the differential line segment:
\oint \textbf{F} \times d\textbf{r}
Is there any vector calculus theorem paralleling the Stokes theorem for a closed path integral I can use without me having to evaluate the line integral directly?