Green, Stokes, and Divergence
Flashcards on the classical integral theorems and the unifying boundary principle behind them.
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Questions Covered in This Set
10 cards to master
State the unifying 'one theorem' idea behind FTC, Green, Stokes, and Divergence.
The integral of a derivative over a region equals the integral of the original object over the region's (oriented) boundary.
State Green's theorem.
∮_∂D P dx + Q dy = ∬_D (∂Q/∂x − ∂P/∂y) dA, with ∂D traversed counterclockwise.
State Stokes' theorem.
∮_∂S F·dr = ∬_S (∇×F)·n dS, with the boundary oriented by the right-hand rule relative to n.
State the divergence (Gauss) theorem.
∯_∂E F·n dS = ∭_E (∇·F) dV, with ∂E oriented outward.
How is Green's theorem a special case of Stokes' theorem?
Take S to be a flat patch in the xy-plane with n = k; then (∇×F)·k = Q_x − P_y.
Why do two different surfaces with the same boundary curve give the same flux of curl?
Stokes says the flux of ∇×F equals the line integral over ∂S, which depends only on the boundary — deform the surface freely (soap bubble on a wire loop).
Green's theorem as an area formula.
Using F = ½(−y, x): Area(D) = ½∮_∂D (x dy − y dx). For the ellipse x=a cos t, y=b sin t it gives πab.
Compute ∮_C F·dr for F = (−y³, x³, −z³) on the unit circle in z = 0.
∇×F = (0,0,3x²+3y²); over the unit disk, 3∫₀^{2π}∫₀¹ r³ dr dθ = 3π/2.
What does divergence measure physically?
Local source strength — how much fluid is created per unit volume; total creation inside = net outflow through the boundary surface.
If ∇·F = 0 everywhere, what is the flux through any closed surface?
Zero — incompressible flow has no net outflow across any closed surface.
