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Berkeley MATH 53 - Worksheet - Vector Fields

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Math 53 Worksheet - Vector Fields1. Find the average distance from a point in a ball of radius a to its center.Solution. The average distance D is given by the following, where we call the ball B,D =RRRBpx2+ y2+ z2dVRRRBdV=143πa3Z2π0Zπ0Za0ρ(ρ2sin φ) dρ dφ dθ=34a.2. Find the gradient vector field ∇f of f(x, y) = x2− y and sketch it.Solution. ∇f(x, y) = h2x, −1i. We discussed what this looks like in class.3. Evaluate the line integralRC(2x + 9z) ds, with C : x = t, y = t2, z = t3, 0 ≤ t ≤ 1.Solution. We have x0= 1. y0= 2t, z0= 3t2, and sods =p(1)2+ (2t)2+ (3t2)2dt =√1 + 4t2+ 9t4dt.ThenZC(2x + 9z) ds =Z10(2t + 9t3)√1 + 4t2+ 9t4dt =16143/2− 1,where we’ve used the u-substitution u = 1 + 4t2+ 9t4in the last integral.4. A thin wire has the shape of the first-quadrant part of the circle with center the originand radius a. If the density function is ρ(x, y) = kxy, find the mass and center of massof the wire.Solution. An obvious choice for the parametrization is x = a cos t, y = a sin t, for0 ≤ t ≤π2. You can check that in this case ds = a dt. Thenm =ZCρ(x, y) ds =Zπ/20k(a sin t)(a cos t)(a) dt =ka33.To find the center of mass, use the formulae¯x =ZCxρ(x, y) ds, ¯y =ZCyρ(x, y) ds.15. The force exerted by an electric charge at the origin on a charged particle at a point(x, y, z) with position vector r = hx, y, zi is F(r) = Kr/|r|3, where K is a constant.Find the work done as the particle moves along a straight line from (2, 0, 0) to (2, 1, 5).Solution. An appropriate parametrization of the line is x = 2, y = t, z = 5t for0 ≤ t ≤ 1 or r(t) = h2, t, 5ti. Note that r0(t) = h0, 1, 5i. ThenW =ZCF · dr=Z10F(x(t), y(t), z(t)) · r0(t) dt=Z10Kh2, t, 5ti((2)2+ (t)2+ (5t)2)3/2· h0, 1, 5idt=Z10K26t(4 + 26t2)3/2dt= K12−1√30.6. Find the volume of the smaller wedge cut from a sphere of radius a by two planes thatintersect along a diameter at an angle of π/6.Solution. The key to this problem is to orient it correctly. Assume (which isallowable by a rotation) that the planes are y = 0 and y = tan(π/6)x = x/√3. Thenthe integral in spherical is justV =Zπ/60Zπ0Za0ρ2sin φ dρ dφ dθ =a3π9.7. Show that a constant force field does zero work on a particle that moves onceuniformly around the circle x2+ y2= 1.Solution. Suppose F = hK1, K2i, with K1and K2constants. Parameterize theparticle by x = cos t, y = sin t. Then r(t) = hcos t, sin ti and r0(t) = h−sin t, cos ti. SoW =ZCF · dr =Z2π0hK1, K2i · h−sin t, cos tidt =Z2π0K2cos t − K1sin t dt =


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