A vector line integral accumulates the field’s work along an oriented path.
WHAT HAPPENS IF…Is the work around every closed path zero?
Interactive exploration
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Connect the picture to the ideaW=∫CF⋅dr=∫F(r(t))⋅r′(t)dt
For F=(−y,x) on a radius-R circle, F·r′=R² with the angular parameter, so an arc has work R²Δθ. For F=(2x,2y)=∇(x²+y²), work is the endpoint potential difference. Reversing a complete path reverses its work.
A detail that matters
Work is F·dr, not simply the field magnitude times distance.