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Homework Help: Vector, cross product, and integral

  1. Feb 22, 2008 #1
    [SOLVED] Vector, cross product, and integral

    1. The problem statement, all variables and given/known data
    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]
    2. Relevant equations

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

    3. 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?
     
  2. jcsd
  3. Feb 22, 2008 #2

    HallsofIvy

    User Avatar
    Science Advisor

    Yes, that is correct.
     
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