Exercise · ZW-CURVE-AND-CONTOUR-008 · ★★★ Advanced
Is \(\gamma(t)=t^3+it^3\) a smooth curve on \([-1,1]\)? Its image is a segment.
Choose one option.
Show answer
- B. No: \(\gamma'(0)=0\), and the definition requires a nonzero derivative
Explanation
A continuous \(\gamma'\) is not enough: \(\gamma'(t)=3t^2(1+i)\) vanishes at \(t=0\). The image is a segment, but this parametrization is not smooth; \(\gamma(t)=t+it\) traces the same segment and is smooth.
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-003 What curve does γ(t)=2+3e^it, 0≤ t≤2π, describe?
- 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-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?