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• W2044161811 startingPage "715" @default.
• W2044161811 abstract "The boundary of a Jordan domain <italic>A</italic> may be a nonsmooth curve <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma> <mml:semantics> <mml:mi>γ<!-- γ --></mml:mi> <mml:annotation encoding=application/x-tex>gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If <italic>F</italic> is a smooth vector field defined near <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma> <mml:semantics> <mml:mi>γ<!-- γ --></mml:mi> <mml:annotation encoding=application/x-tex>gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <italic>F</italic> is integrable over <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma> <mml:semantics> <mml:mi>γ<!-- γ --></mml:mi> <mml:annotation encoding=application/x-tex>gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> provided <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma> <mml:semantics> <mml:mi>γ<!-- γ --></mml:mi> <mml:annotation encoding=application/x-tex>gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has two-dimensional Lebesgue measure zero and <italic>F</italic> is sufficiently smooth. When actually computing the integral <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=integral Underscript gamma Endscripts upper F bullet d s> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mo largeop=false>∫<!-- ∫ --></mml:mo> <mml:mi>γ<!-- γ --></mml:mi> </mml:msub> </mml:mrow> <mml:mi>F</mml:mi> <mml:mo>∙<!-- ∙ --></mml:mo> <mml:mi>d</mml:mi> <mml:mi>s</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>{smallint _gamma }F bullet ds</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, one might hope that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=limit Underscript k right-arrow normal infinity Endscripts integral Underscript gamma Subscript k Baseline Endscripts upper F bullet d s equals integral Underscript gamma Endscripts upper F bullet d s> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:munder> <mml:mo movablelimits=true form=prefix>lim</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>k</mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:munder> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mo largeop=false>∫<!-- ∫ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>γ<!-- γ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:mi>F</mml:mi> <mml:mo>∙<!-- ∙ --></mml:mo> <mml:mi>d</mml:mi> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mo largeop=false>∫<!-- ∫ --></mml:mo> <mml:mi>γ<!-- γ --></mml:mi> </mml:msub> </mml:mrow> <mml:mi>F</mml:mi> <mml:mo>∙<!-- ∙ --></mml:mo> <mml:mi>d</mml:mi> <mml:mi>s</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>{lim _{k to infty }}{smallint _{{gamma _k}}}F bullet ds = {smallint _gamma }F bullet ds</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for PL or smooth approximators <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma Subscript k> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>γ<!-- γ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{gamma _k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma> <mml:semantics> <mml:mi>γ<!-- γ --></mml:mi> <mml:annotation encoding=application/x-tex>gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Several examples show that this is not the case. However, there are algorithms for choosing <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma Subscript k> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>γ<!-- γ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{gamma _k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> so that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=integral Underscript gamma Subscript k Endscripts upper F bullet d s> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mo largeop=false>∫<!-- ∫ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>γ<!-- γ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:mi>F</mml:mi> <mml:mo>∙<!-- ∙ --></mml:mo> <mml:mi>d</mml:mi> <mml:mi>s</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>{smallint _{{gamma _k}}}F bullet ds</mml:annotation> </mml:semantics> </mml:math> </inline-formula> converges to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=integral Underscript gamma Endscripts upper F bullet d s> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mo largeop=false>∫<!-- ∫ --></mml:mo> <mml:mi>γ<!-- γ --></mml:mi> </mml:msub> </mml:mrow> <mml:mi>F</mml:mi> <mml:mo>∙<!-- ∙ --></mml:mo> <mml:mi>d</mml:mi> <mml:mi>s</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>{smallint _gamma }F bullet ds</mml:annotation> </mml:semantics> </mml:math> </inline-formula> exponentially quickly." @default.
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• W2044161811 date "1994-01-01" @default.
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• W2044161811 title "Numerical integration of vector fields over curves with zero area" @default.
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