Exercise · ZW-CURVE-AND-CONTOUR-004 · ★☆☆ Basic
Which one parametrizes the segment from \(1\) to \(i\) with \(0\le t\le1\)?
Choose one option.
Show answer
- B. \(\gamma(t)=(1-t)+it\)
Explanation
With \((1-t)+it\) we get \(\gamma(0)=1\) and \(\gamma(1)=i\), and it is linear in \(t\) (derivative \(-1+i\neq0\)). The option \(t+i(1-t)\) runs from \(i\) to \(1\).
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-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?