# 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?

Related Calculus and Beyond Homework News on Phys.org

#### Defennder

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

Anyone?

#### HallsofIvy

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:

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

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving