# Homework Help: Stokes theorem equivalent for cross product line integral

1. Jul 22, 2008

### Defennder

"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?

2. Jul 24, 2008

### Defennder

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

Anyone?

3. Jul 24, 2008

### HallsofIvy

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

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

4. Jun 28, 2011

### 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?

5. Aug 28, 2011

### I like Serena

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

I found this thread while Googling for a way to respond to another thread: