- #1

foxjwill

- 354

- 0

**[SOLVED] Vector, cross product, and integral**

## Homework Statement

Evaluate:

[tex]{\int \textbf{F} \times \texttt{d}\textbf{v}}.[/tex]

[tex]\textbf{F}[/tex] and [tex]\textbf{v}[/tex] are both vector fields in [tex]\mathbb{R}^3[/tex]

## Homework Equations

[tex]\texttt{d}\textbf{v} = (\nabla \otimes \textbf{v} ) \texttt{d}\textbf{r}[/tex]

## The Attempt at a Solution

[tex]

\begin{array}{ll}

\textbf{F} \times \texttt{d}{\textbf{v}} &= \left( {

\begin{array}{c}

{F_2 \texttt{d}v_3 - F_3 \texttt{d}v_2 } \\

{F_1 \texttt{d}v_2 - F_2 \texttt{d}v_1 } \\

{F_1 \texttt{d}v_3 - F_3 \texttt{d}v_1 } \\

\end{array}

\right ) \\

&= \left(

\begin{array}{c}

{F_2 \nabla v_3 \cdot \texttt{d}{\textbf{r}} - F_3 \nabla v_3 \cdot \texttt{d}{\textbf{r}}} \\

{F_1 \nabla v_2 \cdot \texttt{d}{\textbf{r}} - F_2 \nabla v_1 \cdot \texttt{d}{\textbf{r}}} \\

{F_1 \nabla v_3 \cdot \texttt{d}{\textbf{r}} - F_3 \nabla v_1 \cdot \texttt{d}{\textbf{r}}} \\

\end{array} \right) \\

\end{array}[/tex]

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