Exercise · ZW-CURVE-AND-CONTOUR-001 · ★☆☆ Basic
For the curve \(\gamma(t)=t+it^2\), \(0\le t\le1\), find the endpoint \(\gamma(1)\).
Choose one option.
Show answer
- C. \(1+i\)
Explanation
Substitute \(t=1\): \(\gamma(1)=1+i\cdot1^2=1+i\).
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
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- 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?