Exercise · ZW-CONSERVATIVE-FIELD-008 · ★★☆ Intermediate
Compute the circulation of \(\mathbf F=\langle y,-x\rangle\) over the unit circle traversed once counterclockwise.
Choose one option.
Show answer
- D. \(-2\pi\)
Explanation
With \(\mathbf r=\langle\cos t,\sin t\rangle\), \(\mathbf F\cdot\mathbf r'=-\sin^2t-\cos^2t=-1\), so the integral is \(-2\pi\). It is not \(0\) because \(P_y=1\ne-1=Q_x\): the field is not conservative.
Review the concept: Conservative field
A vector field \(\mathbf F\) is conservative if it is the gradient of a scalar function \(f\), called a potential function of \(\mathbf F\).
\[\mathbf F=\nabla f\]
More practice problems: conservative field
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- ZW-CONSERVATIVE-FIELD-006 Compute ∫_Cmathbf F· dmathbf r with mathbf F=langle e^y, xe^yrangle, from (0,0) to (2,1), along any…
- ZW-CONSERVATIVE-FIELD-007 For which value of a is the field mathbf F=langle axy, x^2rangle conservative?
- ZW-CONSERVATIVE-FIELD-009 Two different paths from A to B give different line integrals of mathbf F. What follows?