By Roel Snieder
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Extra info for A Guided Tour of Mathematical Physics
14) first. The left hand side of this expression gives the time-derivative of the probability that the particle is within the volume V . The only way the particle can enter or leave the volume is through the enclosing surface S. The right hand side therefore describes the “flow” of probability through the surface S. More accurately, one can formulate this as the flux of the probability density current. 15) that the probability density current J is given by: J= i¯h (ψ∇ψ∗ − ψ∗ ∇ψ) 2m Pay in particular attention to the sign of the terms in this expression.
7) and the chain rule for differentiation show that ∇ · v =2f (r) + r df dr (cilinder coordinates). 5) and show that the flow field is given by v(r) = Ar/r2 . Make a sketch of the flow field. 34 CHAPTER 4. THE DIVERGENCE OF A VECTOR FIELD The constant A is yet to be determined. Let at the source r = 0 a volume V per unit time be injected. Problem d: Show that V = v·dS (where the integration is over an arbitrary surface around the source at r = 0). By choosing a suitable surface derive that v(r) = V ˆ r .
Problem b: Show that for this velocity field: ∇ × v = 2Ωˆ z. This means that the vorticity is twice the rotation vector Ωˆ z. This result is derived here for the special case that the z-axis is the axis of rotation. 11) of Boas it is shown with a very different derivation that the vorticity for rigid rotation is given by ω =∇ × v = 2Ω, where Ω is the rotation vector. ) We see that rigid rotation leads to a vorticity that is twice the rotation rate. 3). This paddle-wheel moves with the flow and makes one revolution along its axis in a time 2π/Ω.
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