Exercise · ZW-CURVE-AND-CONTOUR-003 · ★☆☆ Basic
What curve does \(\gamma(t)=2+3e^{it}\), \(0\le t\le2\pi\), describe?
Choose one option.
Show answer
- D. The circle with center \(2\) and radius \(3\), traversed once counterclockwise
Explanation
Since \(|\gamma(t)-2|=3\), the points lie at distance \(3\) from the center \(2\); the argument \(t\) grows from \(0\) to \(2\pi\), i.e. counterclockwise.
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.
\[\gamma(t)=x(t)+i\,y(t),\qquad a\le t\le b\]
More practice problems: curve and contour
- ZW-CURVE-AND-CONTOUR-001 For the curve γ(t)=t+it^2, 0≤ t≤1, find the endpoint γ(1).
- ZW-CURVE-AND-CONTOUR-002 If γ(t)=e^it, what is γ'(t)?
- ZW-CURVE-AND-CONTOUR-004 Which one parametrizes the segment from 1 to i with 0≤ t≤1?
- ZW-CURVE-AND-CONTOUR-005 Compute the length of γ(t)=3t+4it, 0≤ t≤2, using L=∫_a^b|γ'(t)| dt.
- ZW-CURVE-AND-CONTOUR-006 For γ(t)=t+it^2, compute |γ'(1)|.
- ZW-CURVE-AND-CONTOUR-007 How many smooth curves make up the contour tracing the boundary of a triangle?
- ZW-CURVE-AND-CONTOUR-008 Is γ(t)=t^3+it^3 a smooth curve on [-1,1]? Its image is a segment.
- ZW-CURVE-AND-CONTOUR-009 Let γ(t)=e^it, 0≤ t≤π, going from 1 to -1. Which one parametrizes the opposite curve -γ with 0≤ t≤π?