Exercise · ZW-VECTOR-LINE-INTEGRAL-008 · ★★☆ Intermediate
A field \(\mathbf F\) is perpendicular to the curve \(C\) at every point. What is \(\int_C \mathbf F\cdot d\mathbf r\)?
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- B. \(0\)
Explanation
If \(\mathbf F\perp\mathbf r'\) at every point, the dot product \(\mathbf F\cdot\mathbf r'\) is zero along the whole path, and so is the integral.
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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