Line and surface integrals, divergence, curl, conservative fields, Green, Stokes and divergence theorems.
39 cards · basic cards · AI-written, checked twice. Edit anything.
- What is a line integral of a scalar function f along a curve C?
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The integral of f with respect to arc length: ∫_C f ds, computed as ∫_a^b f(r(t)) |r'(t)| dt.- What is a line integral of a vector field F along a curve C?
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The integral of the tangential component of F: ∫_C F · dr, computed as ∫_a^b F(r(t)) · r'(t) dt.- What is the divergence of a vector field F = ⟨P, Q, R⟩?
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The sum of partial derivatives: div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z.- What is the curl of a vector field F = ⟨P, Q, R⟩?
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The vector curl F = ⟨∂R/∂y - ∂Q/∂z, ∂P/∂z - ∂R/∂x, ∂Q/∂x - ∂P/∂y⟩.- What does it mean for a vector field to be conservative?
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The field is the gradient of some scalar potential function, F = grad f.- What is a potential function for a conservative field F?
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A scalar function f such that F = grad f, or equivalently F = ∇f.- What does path independence mean for a line integral?
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The value of the line integral depends only on the endpoints, not on the path taken between them.- What is the value of the line integral around a closed curve in a conservative field?
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Zero, because the start and end points are the same.- For a 2D conservative field F = ⟨P, Q⟩, what condition must hold on partial derivatives?
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∂P/∂y = ∂Q/∂x throughout a simply connected domain.- For a 3D conservative field F = ⟨P, Q, R⟩, what must be true?
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The curl must be zero: curl F = ⟨0, 0, 0⟩ throughout a simply connected domain.- What is Green's theorem?
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∮_C (P dx + Q dy) = ∬_D (∂Q/∂x - ∂P/∂y) dA, relating a line integral around a closed curve to a double integral over its interior.- What is the flux form of Green's theorem?
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∮_C (P dy - Q dx) = ∬_D (∂P/∂x + ∂Q/∂y) dA, relating flux across a closed curve to the divergence integral.- How does Green's theorem relate to divergence in 2D?
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The double integral in the flux form equals the integral of div F over the region, measuring total outward flux.- What is Stokes' theorem?
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∮_C F · dr = ∬_S (curl F) · n dS, relating a line integral around a closed curve to a surface integral of the curl.- What does Stokes' theorem tell us about circulation?
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The circulation of F around a closed curve C equals the flux of curl F through any surface S bounded by C.
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