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Page 323

Suppose the density ρ varies from point to point as well as with time, that is, ρ=(x,y,z,t). If we follow the fluid along a streamline, then x,y,z are function of such that the fluid velocity is

v=idxdt+jdydt+kdzdt

Show that then/dt=ρ/t+vρ . Combine this equation with (10.9)to get

ρv+dt=0

(Physically, is the rate of change of density with time as we follow the fluid along a streamline; p/tis the corresponding rate at a fixed point.) For a steady state (that is, time-independent), p/t=0, but p/t is not necessarily zero. For an incompressible fluid, p/t=0. Show that then role="math" localid="1657336080397" ×v=0. (Note that incompressible does not necessarily mean constant density since p/t=0does not imply either time or space independence of ρ; consider, for example, a flow of watermixed with blobs of oil.)

Q15P

Page 285

In the figure u1is a unit vector in the direction of an incident ray of light, and u3and u2 are unit vectors in the directions of the reflected and refracted rays. If u is a unit vector normal to the surface AB, the laws of optics say that θ1=θ3 and n1sinθ1=n2sinθ2 , where n1 and n2are constants(indices of refraction). Write these laws in vector form (using dot or cross products).

Q15P

Page 335

cydx+zdy+xdz, where C is the curve of intersection of the surfaces whose equations are x+y=2andx2+y2+z2=2(x+y).

Q16P

Page 335

Question:What is wrong with the following “proof” that there are no magnetic fields? By electromagnetic theory,∇· B = 0, and B =∇×A. (The error is not in these equations.) Using them, we find

¯·Bdt=0=B·ndσ=(×A)·ndσ=A·dr

SinceA·dr=0, A is conservative, or A =∇ψ. Then B=×A=×ψ=0,so B = 0.

Q16P

Page 285

In the discussion of Figure 3.8, we found for the angular momentum, the formula L=mrx(ωxr).Use (3.9) to expand this triple product. If ris perpendicular to ω, show that you obtain the elementary formula, angular momentum =mvr.

Q16P

Page 307

Given F1=2xi-2yzj-y2kandF2=yi-xi

(a) Are these forces conservative? Find the potential corresponding to any conservative force.

(b) For any nonconservative force, find the work done if it acts on an object moving from (-1,-1)to (1,1)along each of the paths shown.

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Page 324

The following equations are variously known as Green’s first and second identities or formulas or theorems. Derive them, as indicated, from the divergence theorem.

(1)volumeinsideσ(ϕ2ψ+ϕψ)=closedsurfaceσ(ϕψ)ndσ

To prove this, let in the divergence theorem.

(2)volumeinsideσ(ϕ2ψψ2ϕ)=closedsurfaceσ(ϕψψϕ)ndσ

To prove this, copy Theorem 1above as is and also with and interchanged; then subtract the two equations.

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Page 337

Given

F1=2xzi+yj+x2kF2=yi-xj:

(a) Which F , if either, is conservative?

(b) If one of the given ’s is conservative, find a function Wso that F=W.

(c) If one of the F’s is non conservative, use it to evaluate F along the straight line from (0,1)to(1,0).

(d) Do part (c) by applying Green’s theorem to the triangle with vertices (0,0),(0,1),(1,0).(0,0),(0,1),(1,0)..

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Page 295

Show by the Lagrange multiplier method that the maximum value of /dsis||.That is, maximize /dsgiven by (6.3) subject to the condition a2+b2+c2=1. You should get two values (±) for the Lagrange multiplier λ, and two values (maximum and minimum) for/dswhich is the maximum and which is the minimum?

Q17MP

Page 337

Find the value of F·dralong the circle from (1,1) to (1,1) if F= (2x3y)i(3x2y)j.

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