- #1
foxjwill
- 350
- 0
[SOLVED] Vector, cross product, and integral
Evaluate:
{\int \textbf{F} \times \texttt{d}\textbf{v}}.
\textbf{F} and \textbf{v} are both vector fields in \mathbb{R}^3
\texttt{d}\textbf{v} = (\nabla \otimes \textbf{v} ) \texttt{d}\textbf{r}
<br /> \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?
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
<br /> \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?