This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ...
View Full
Document
 Spring '03
 Staff

Click to edit the document details