lec7-2

lec7-2 - Divergence theorem in two dimensions j i •...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the 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

Page1 / 4

lec7-2 - Divergence theorem in two dimensions j i •...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online