Exercise · ZW-CURVE-AND-CONTOUR-009 · ★★★ Advanced

Let \(\gamma(t)=e^{it}\), \(0\le t\le\pi\), going from \(1\) to \(-1\). Which one parametrizes the opposite curve \(-\gamma\) with \(0\le t\le\pi\)?

Choose one option.

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C. \(e^{i(\pi-t)}\)

Explanation

The opposite curve traces the same points backwards: \(t=0\) must give \(-1\) and \(t=\pi\) must give \(1\). \(e^{i(\pi-t)}\) does. \(e^{-it}\) traces the lower semicircle, a different set of points.

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Review the concept: Curve and contour

A smooth curve \(\gamma\) is a function from a real interval \([a,b]\) into \(\mathbb{C}\) with continuous, nonzero derivative; a contour is a finite chain of smooth curves joined end to end.

\[\gamma(t)=x(t)+i\,y(t),\qquad a\le t\le b\]

Let \(\gamma(t)=e^{it}\), \(0\le t\le\pi\), going from \(1\) to \(-1\). Which one parametrizes the opposite curve \(-\gamma\) with \(0\le t\le\pi\)?

Explanation

Review the concept: Curve and contour

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