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George Weigt 2008-12-15 19:56:10 -07:00
parent 64f5827b4a
commit 0954123de2
1 changed files with 10 additions and 14 deletions

24
line-integral.tex
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@ -86,42 +86,38 @@ $$L=1.47894$$
There are two kinds of line integrals, one for scalar fields and the other There are two kinds of line integrals, one for scalar fields and the other
for vector fields. for vector fields.
Both are closely related to arc length, as the following table shows. Both are closely related to arc length, as shown in the following table.
\bigskip
\begin{center} \begin{center}
\begin{tabular}{|lll|}
\begin{tabular}{|llll|}
\hline \hline
& & & \\ & & \\
& $\quad$
& Abstract form & Abstract form
& Computable form & Computable form
\\ \\
& & & \\ & & \\
Arc length Arc length
&
& $\displaystyle{\int_C ds}$ & $\displaystyle{\int_C ds}$
& $\displaystyle{\int_a^b |g'(t)|\,dt}$ & $\displaystyle{\int_a^b |g'(t)|\,dt}$
\\ \\
& & & \\ & & \\
Line integral, scalar field Line integral, scalar field
&
& $\displaystyle{\int_C f\,ds}$ & $\displaystyle{\int_C f\,ds}$
& $\displaystyle{\int_a^b f(g(t))\,|g'(t)|\,dt}$ & $\displaystyle{\int_a^b f(g(t))\,|g'(t)|\,dt}$
\\ \\
& & & \\ & & \\
Line integral, vector field Line integral, vector field
&
& $\displaystyle{\int_C(F\cdot u)\,ds}$ & $\displaystyle{\int_C(F\cdot u)\,ds}$
& $\displaystyle{\int_a^b F(g(t))\cdot g'(t)\,dt}$ & $\displaystyle{\int_a^b F(g(t))\cdot g'(t)\,dt}$
\\ \\
& & & \\ & & \\
\hline \hline
\end{tabular} \end{tabular}
\end{center} \end{center}
\medskip \bigskip
\noindent \noindent
We have We have
$$ds=|g'(t)|\,dt$$ $$ds=|g'(t)|\,dt$$