Surface Flux

Above we showed that smoothness

    X[s + δs, t + δt] = X[s, t] + ∂X[s, t]/∂s δs + ∂X[s, t]/∂t δt + ε (δs^2 + δt^2)^(1/2)

with  ε≈0, when  δs≈0  and  δt≈0, means that a small portion of the surface,

    δsδt-SurfaceIncrement = { Ξ : Ξ = X[s + σ, t + τ], 0≤σ≤δs, 0≤τ≤δt}

looks like the graph of a small portion of the linear equation

    δsδt-LinearIncrement = { Z : Z = M σ + N τ + C, 0≤σ≤δs, 0≤τ≤δt}

where  M = ∂X[s, t]/∂s, N = ∂X[s, t]/∂t, C = X[s, t].  Provided ∂X[s, t]/∂s and ∂X[s, t]/∂t are not scalar multiples.  

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[Graphics:../HTMLFiles/Lect3_255.gif]

Flux of a constant velocity flow F across a linear parallelogram with sides  μ  and  ν is given by the cross product:

[Graphics:../HTMLFiles/Lect3_259.gif]

Underscript[∫∫, ] F•dA = Underscript[∫∫, ] F[X[s ... ∫∫, ] Det[F[X[s, t]] , ∂X/∂s[s, t], ∂X/∂t[s, t]] ds dt

Created by Mathematica  (July 2, 2004)