Exercise · ZW-VECTOR-LINE-INTEGRAL-002 · ★☆☆ Basic
Compute \(\int_C \mathbf F\cdot d\mathbf r\) with \(\mathbf F=\langle -y,x\rangle\) over the unit circle traversed once counterclockwise.
Choose one option.
Show answer
- A. \(2 \pi\)
Explanation
With \(\mathbf r=\langle\cos t,\sin t\rangle\), \(\mathbf F(\mathbf r)\cdot\mathbf r'=\sin^2t+\cos^2t=1\); integrating from \(0\) to \(2\pi\) gives \(2\pi\). The field always pushes along the direction of travel.
Review the concept: Line integral of a vector field
The line integral of a vector field \(\mathbf F\) over an oriented curve \(C\), parametrized by \(\mathbf r(t)\), \(a\le t\le b\), measures the work \(\mathbf F\) does along \(C\).
\[\int_C \mathbf F\cdot d\mathbf r=\int_a^b \mathbf F(\mathbf r(t))\cdot\mathbf r'(t)\,dt\]
More practice problems: line integral of a vector field
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