# Stokes theorem equivalent for cross product line integral

#### Defennder

Homework Helper
"Stokes theorem" equivalent for cross product line integral

1. The problem statement, all variables and given/known data
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?

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#### Defennder

Homework Helper
Re: "Stokes theorem" equivalent for cross product line integral

Anyone?

#### HallsofIvy

Science Advisor
Re: "Stokes theorem" equivalent for cross product line integral

As far as I know, no, there isn't any such theorem.

#### mozi

Re: "Stokes theorem" equivalent for cross product line integral

$\oint_{C} \mathrm{d} \mathbf{l} \times \mathbf{F} = \int \int_{S} \left( \mathrm{d} \mathbf{S} \times \nabla \right) \times \mathbf{F}$, this one?

#### I like Serena

Homework Helper
Re: "Stokes theorem" equivalent for cross product line integral

I found this thread while Googling for a way to respond to another thread:
https://www.physicsforums.com/showthread.php?t=524990

If you're still interested, I believe you can use the same method here.

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