- #1
foxjwill
- 354
- 0
[SOLVED] Vector, cross product, and integral
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]
[tex]\texttt{d}\textbf{v} = (\nabla \otimes \textbf{v} ) \texttt{d}\textbf{r}[/tex]
[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?
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?