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