Vector, cross product, and integral

[foxjwill]

[SOLVED] Vector, cross product, and integral

Homework Statement

Evaluate:

$${\int \textbf{F} \times \texttt{d}\textbf{v}}.$$

$$\textbf{F}$$ and $$\textbf{v}$$ are both vector fields in $$\mathbb{R}^3$$

Homework Equations

$$\texttt{d}\textbf{v} = (\nabla \otimes \textbf{v} ) \texttt{d}\textbf{r}$$

The Attempt at a Solution

$$\begin{array}{ll}<br /> \textbf{F} \times \texttt{d}{\textbf{v}} &= \left( {<br /> \begin{array}{c}<br /> {F_2 \texttt{d}v_3 - F_3 \texttt{d}v_2 } \\<br /> {F_1 \texttt{d}v_2 - F_2 \texttt{d}v_1 } \\<br /> {F_1 \texttt{d}v_3 - F_3 \texttt{d}v_1 } \\<br /> \end{array}<br /> \right ) \\ <br /> &= \left( <br /> \begin{array}{c}<br /> {F_2 \nabla v_3 \cdot \texttt{d}{\textbf{r}} - F_3 \nabla v_3 \cdot \texttt{d}{\textbf{r}}} \\<br /> {F_1 \nabla v_2 \cdot \texttt{d}{\textbf{r}} - F_2 \nabla v_1 \cdot \texttt{d}{\textbf{r}}} \\<br /> {F_1 \nabla v_3 \cdot \texttt{d}{\textbf{r}} - F_3 \nabla v_1 \cdot \texttt{d}{\textbf{r}}} \\<br /> \end{array} \right) \\ <br /> \end{array}$$

This can then be solved as three path integrals over some path $$\textbf{r}$$. Is this correct?

 

[HallsofIvy]

Yes, that is correct.