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Unformatted text preview: Divergence theorem in two dimensions
j i • Consider a vector ﬁeld V = Vx ˆ + Vy ˆ (Notice here Vz = 0) • If we take Q = Vx and P = −Vy , then ∂ Vy ∂P ∂ Vx ∂Q − = + = div V ∂x ∂y ∂x ∂y
idy • Consider the outward normal n = √ −jdx 2 = ˆ 2 dx +dy ˆ ˆ ˆ −ˆ idx jdy , ds and then Pdx + Qdy = −Vy dx + Vx dy = (Vx ˆ + Vy ˆ) · (ˆ − ˆ ) = V · nds i j idy jdx ˆ • The by Green’s theorem in the plane, div V dxdy =
A ∂A V · nds ˆ Chapter7: Fourier series Divergence theorem (in three dimensions)
• We can show that the divergence theorem in two dimensions can be extended to three dimensions div V d τ =
τ ∂τ V · nd σ ˆ • Instead of a line integral, we have a surface integral, and instead of integrating over a two dimensional plane, we integrate over a volume • Start with a small element of volume, say in the shape of a cube with V = dxdydz , and unit vectors perpendicular to the faces ±ˆ, i ˆ ±ˆ, and ±k , then j ∂ Vy ∂ Vz ∂ Vx + + ∂x ∂y ∂z V · nd σ = ˆ
σ dxdydz = div V dxdydz Chapter7: Fourier series Divergence theorem continued • We can extend to an integral over many small volumes dxdydz • Consider two neighboring volumes, sharing a surface at x + dx for example, then the unit vectors for the surface where they meet are ˆ and −ˆ, so these interior integrals cancel, and we can i i integrate over a large volume div V d τ =
τ ∂τ V · nd σ ˆ • The surface integral here is just the outermost surface Chapter7: Fourier series Physical application of divergence and divergence theorem
• Consider a vector ﬁeld J = ρv , where ρ is mass density and v is a ﬂow velocity • The net change of mass in a time dt in a volume element τ is determined only by J on the surface of τ dM =− J · nd σ ˆ dt ∂τ • By the divegence theorem, we can change this to a volume integral dM =− dt • We also see that so
dM dt · V dτ
τ is related to the change in ρ in the volume, dM = dt dρ dτ dt τ Chapter7: Fourier series ...
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This note was uploaded on 07/30/2011 for the course PHZ 3113 taught by Professor Staff during the Spring '03 term at University of Central Florida.
 Spring '03
 Staff

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