Exercise · ZW-CONSERVATIVE-FIELD-002 · ★☆☆ Basic
Which of these fields is conservative?
Choose one option.
Show answer
- A. \(\langle y,\,x\rangle\)
Explanation
For \(\mathbf F=\langle P,Q\rangle\) in the plane we need \(P_y=Q_x\). Only \(\langle y,x\rangle\) satisfies it (\(1=1\)); it is the gradient of \(xy\). In the others, \(P_y\ne Q_x\).
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 \(\mathbf F\).
\[\mathbf F=\nabla f\]
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