Vector fields
Objective. Calculate divergence, curl and potentials.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Vector fields assign a vector F(x, y, z) to each point in space, and their divergence and curl measure outward spreading and rotation respectively. The divergence div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z is a scalar measuring source strength, while curl F = ∇×F is a vector measuring local rotation. A field F is conservative, meaning F = ∇φ for a scalar potential φ, precisely when curl F = 0 on a simply connected domain. For F = (y, x, 0), curl F = 0 and a potential is φ(x,y) = xy, since ∇φ = (y, x, 0) matches F exactly. These invariants classify how a vector field behaves locally.
Worked example
Determine whether the vector field F = (2xy, x² + 3y², 0) is conservative and find its potential if so.
- 1Check ∂P/∂y = 2x and ∂Q/∂x = 2x, which are equal, so curl F = 0 and F is conservative.
- 2Integrate P with respect to x: φ = x²y + g(y) for some function g(y).
- 3Differentiate with respect to y: ∂φ/∂y = x² + g'(y), matching Q = x² + 3y² gives g'(y) = 3y².
- 4Integrate: g(y) = y³, so φ(x, y) = x²y + y³.
Answer: φ(x, y) = x²y + y³
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Start practisingCommon mistake. Students assume curl F = 0 always guarantees a global potential exists, but this fails on domains with holes; the field (−y, x, 0)/(x²+y²) has zero curl away from the origin yet has no single-valued potential around a loop enclosing it.
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