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# lec7-3 - Physical application of divergence and divergence...

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Physical application of divergence and divergence theorem Consider a vector field ~ J = ρ~ v , 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 dt = - Z Z ∂τ ~ J · ˆ nd σ By the divegence theorem, we can change this to a volume integral dM dt = - Z Z Z τ ∇ · ~ V d τ We also see that dM dt is related to the change in ρ in the volume, so dM dt = Z Z Z τ d ρ dt d τ Chapter7: Fourier series

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Continuity equation Setting these last equations equal, we get the continuity equation ∇ · ~ J + ∂ρ t = 0 An example is diffusion where ~ J = - D n , then D 2 n = n t We will encounter this simple equation more in later chapters Chapter7: Fourier series
Gauss’ Law In Gauss’ Law, the vector field is ~ E and Z Z ∂τ ~ E · ˆ nd σ = Q 0 We can use the divergence theorem to express the left-hand side as a volume integral of ∇ · ~ E , and then note that Q = Z Z Z τ ρ

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lec7-3 - Physical application of divergence and divergence...

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