Physical / geometric interpretation of complex integration - Revision history

Alexandk at 15:23, 21 December 2021

2021-12-21T15:23:46Z

2021-12-21T15:22:48Z

2021-12-21T15:20:08Z

2021-12-21T15:19:56Z

2021-12-21T15:19:12Z

2021-12-21T15:17:22Z

2021-12-21T15:16:39Z

2021-12-21T15:16:10Z

2021-12-21T15:12:54Z

2021-12-21T15:10:43Z

← Older revision		Revision as of 15:23, 21 December 2021
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	where <math> {\bf T} </math> and <math> {\bf N} </math> are, respectively, the unit tangent and the unit normal vectors to the curve <math> \gamma.</math>		where <math> {\bf T} </math> and <math> {\bf N} </math> are, respectively, the unit tangent and the unit normal vectors to the curve <math> \gamma.</math>

−	This computation shows that <math> {\rm Re}\left(\int _\gamma f(z)\, dz\right) </math> can be interpreted as the flow of the vector field <math> {\bf F} along <math> \gamma </math>, while <math> {\rm Im}\left(\int _\gamma f(z)\, dz\right) </math> can be interpreted as the flux of <math> {\bf F} </math> across <math> \gamma </math>.	+	This computation shows that <math> {\rm Re}\left(\int _\gamma f(z)\, dz\right) </math> can be interpreted as the flow of the vector field <math> {\bf F} </math> along <math> \gamma </math>, while <math> {\rm Im}\left(\int _\gamma f(z)\, dz\right) </math> can be interpreted as the flux of <math> {\bf F} </math> across <math> \gamma </math>.

← Older revision		Revision as of 15:22, 21 December 2021
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	where <math> {\bf T} </math> and <math> {\bf N} </math> are, respectively, the unit tangent and the unit normal vectors to the curve <math> \gamma.</math>		where <math> {\bf T} </math> and <math> {\bf N} </math> are, respectively, the unit tangent and the unit normal vectors to the curve <math> \gamma.</math>

−	This computation shows that <math> {\rm Re}\left(\int _\gamma f(z)\, dz\right) </math>	+	This computation shows that <math> {\rm Re}\left(\int _\gamma f(z)\, dz\right) </math> can be interpreted as the flow of the vector field <math> {\bf F} along <math> \gamma </math>, while <math> {\rm Im}\left(\int _\gamma f(z)\, dz\right) </math> can be interpreted as the flux of <math> {\bf F} </math> across <math> \gamma </math>.

← Older revision		Revision as of 15:20, 21 December 2021
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	where <math> {\bf T} </math> and <math> {\bf N} </math> are, respectively, the unit tangent and the unit normal vectors to the curve <math> \gamma.</math>		where <math> {\bf T} </math> and <math> {\bf N} </math> are, respectively, the unit tangent and the unit normal vectors to the curve <math> \gamma.</math>

−	This computation shows that <math> {\rm Re}\left(\int _\gamma f(z)\, dz\~~rigth~~) </math>	+	This computation shows that <math> {\rm Re}\left(\int _\gamma f(z)\, dz\right) </math>

← Older revision		Revision as of 15:19, 21 December 2021
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	where <math> {\bf T} </math> and <math> {\bf N} </math> are, respectively, the unit tangent and the unit normal vectors to the curve <math> \gamma.</math>		where <math> {\bf T} </math> and <math> {\bf N} </math> are, respectively, the unit tangent and the unit normal vectors to the curve <math> \gamma.</math>
		+
		+	This computation shows that <math> {\rm Re}\left(\int _\gamma f(z)\, dz\rigth) </math>

← Older revision		Revision as of 15:19, 21 December 2021
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	<math> = \int_0^l {\bf F}(\gamma(s)) \cdot {\bf T}(\gamma (s)) \, ds + i \int_0^l {\bf F}(\gamma (s))\cdot {\bf N}(\gamma (s)) \, ds =</math>		<math> = \int_0^l {\bf F}(\gamma(s)) \cdot {\bf T}(\gamma (s)) \, ds + i \int_0^l {\bf F}(\gamma (s))\cdot {\bf N}(\gamma (s)) \, ds =</math>

−	<math> = \int_\gamma {\bf F } \cdot {\bf T}\, d\gamma + i \int_\gamma \bf{F} \cdot {\bf N}\, d\gamma.</math>	+	<math> = \int_\gamma {\bf F } \cdot {\bf T}\, d\gamma + i \int_\gamma \bf{F} \cdot {\bf N}\, d\gamma,</math>
		+
		+	where <math> {\bf T} </math> and <math> {\bf N} </math> are, respectively, the unit tangent and the unit normal vectors to the curve <math> \gamma.</math>

@@ Line 13: / Line 13: @@
 <math> = \int_0^l [u(\gamma(s), -v(\gamma(s)] \cdot [x'(s), y'(s)] \, ds + i \int_0^l [u(\gamma(s)), -v(\gamma(s)]\cdot [y'(s), -x'(s)] \, ds =</math>
-<math> = \int_0^l {\bf F}(\gamma) \cdot {\bf T}(\gamma) \, ds + i \int_0^l  {\bf F}(\gamma)\cdot {\bf N}(\gamma) \, ds =</math>
+<math> = \int_0^l {\bf F}(\gamma(s)) \cdot {\bf T}(\gamma (s)) \, ds + i \int_0^l  {\bf F}(\gamma (s))\cdot {\bf N}(\gamma (s)) \, ds =</math>
 <math> = \int_\gamma {\bf F } \cdot {\bf T}\, d\gamma + i \int_\gamma \bf{F} \cdot {\bf N}\, d\gamma.</math>

← Older revision		Revision as of 15:16, 21 December 2021
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	<math> = \int_0^l [u(\gamma(s), -v(\gamma(s)] \cdot [x'(s), y'(s)] \, ds + i \int_0^l [u(\gamma(s)), -v(\gamma(s)]\cdot [y'(s), -x'(s)] \, ds =</math>		<math> = \int_0^l [u(\gamma(s), -v(\gamma(s)] \cdot [x'(s), y'(s)] \, ds + i \int_0^l [u(\gamma(s)), -v(\gamma(s)]\cdot [y'(s), -x'(s)] \, ds =</math>
		+
		+	<math> = \int_0^l {\bf F}(\gamma) \cdot {\bf T}(\gramma) \, ds + i \int_0^l {\bf F}(\gamma)\cdot {\bf N}(\gramma) \, ds =</math>
		+
		+	<math> = \int_\gamma {\bf F } \cdot {\bf T}\, d\gamma + i \int_\gamma \bf{F} \cdot {\bf N}\, d\gamma.</math>

← Older revision		Revision as of 15:12, 21 December 2021
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	<math> = \int_0^l \left (u(\gamma(s))x'(s) - v(\gamma(s)) y'(s)\right) \, ds + i \int_0^l \left (u(\gamma(s))y'(s) + v(\gamma(s)) x'(s)\right) \, ds =</math>		<math> = \int_0^l \left (u(\gamma(s))x'(s) - v(\gamma(s)) y'(s)\right) \, ds + i \int_0^l \left (u(\gamma(s))y'(s) + v(\gamma(s)) x'(s)\right) \, ds =</math>
		+
		+	<math> = \int_0^l [u(\gamma(s), -v(\gamma(s)] \cdot [x'(s), y'(s)] \, ds + i \int_0^l [u(\gamma(s)), -v(\gamma(s)]\cdot [y'(s), -x'(s)] \, ds =</math>

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 There are many geometric and physical interpretations of integrals of real-valued functions (such as are, volume, work etc.). However, a similar interpretation is rarely discussed for complex contour integral. The following interpretation of contour integration is due to P&oacute;lya (George P&oacute;lya and Gordon Latta, Complex Variables, Wiley, 1974.)
-Let <math> \gamma </math> be a simple closed smooth curve in the complex plane, and <math> f </math> be a (complex-valued) function defined on <math> \gamma </math>. Parametrize <math> \gamma </math> with arc length <math> s </math>. As usual,  have <math> f(z) = u(z) + iv(z) = u(x,y) + iv(x,y)</math>. Consider a vector field
+Let <math> \gamma </math> be a simple closed smooth curve in the complex plane, and <math> f </math> be a (complex-valued) function defined on <math> \gamma </math>. Parametrize <math> \gamma </math> with arc length <math> s </math>, such that <math> \gamma(s) = x(s) + y(s). </math> As usual,  have <math> f(z) = u(z) + iv(z) = u(x,y) + iv(x,y)</math>. Consider a vector field
 <math> {\bf F} (x,y) = [u(x,y), -v(x,y)]. </math>