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George Weigt 2008-12-28 19:42:58 -07:00
parent 7904d77a29
commit a2eb5c794f
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@ -130,7 +130,54 @@ $$\int_C(F\cdot u)\,ds
=\int_a^b F(g(t))\cdot g'(t)\,dt
$$
\newpage
\noindent
Evaluate
$$\int_Cx\,ds\quad\hbox{and}\quad\int_Cx\,dx$$
where $C$ is a straight line from $(0,0)$ to $(1,1)$.
\medskip
\noindent
What a difference the measure makes.
The first integral is over a scalar field and the second is over a vector field.
This can be understood when we recall that
$$ds=|g'(t)|\,dt
%\quad\hbox{and}\quad
%\int_Cx\,dx=\int_Cx\,dx+0\,dy
$$
Hence for $\int_Cx\,ds$ we have
\medskip
\verb$x=t$
\verb$y=t$
\verb$g=(x,y)$
\verb$defint(x*abs(d(g,t)),t,0,1)$
$$1\over2^{1/2}$$
\medskip
\noindent
For $\int_Cx\,dx$ we have
\medskip
\verb$x=t$
\verb$y=t$
\verb$g=(x,y)$
\verb$F=(x,0)$
\verb$defint(dot(F,d(g,t)),t,0,1)$
$$1\over2$$
\newpage
\noindent
The following line integral problems are from
{\it Advanced Calculus, Fifth Edition} by Wilfred Kaplan.