Problem 1. In cylindrical coordinate system: ±2rvTπφ=Kwhere 2Tπω=So ±vrω φ=KThen transform it into spherical coordinate: ±sinvRωθφ=KProblem 2. In cylindrical coordinate system ±2vrωφ=Kby applying the curl in cylindrical coordinate using the formula on page 13 in lecture note => 3vzrω∇×=K±Then convert P1, P2, P3 into cylindrical coordinate system => , 1(0,0,0)P=2(0,/3,/ 2)oPRπ=, 3(3/ 4,/3,/ 4)ooPRRπ=in cylindrical coordinate (r, φ, z). Then by noticing that the amplitude of the curl of this vector field is only proportional to r, the rank should be P3>P2=P1 Problem 3. a. at R=RmRSvds•∫Kvo=ll2200sin(sin)9ooRRRd dπ πRθθ θ φ•∫ ∫=23200sin9oRd dπ πθ θ φ∫ ∫=239oRπ
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