Optics 287, Math 287March 4, 20082:00 to 3:15 pmMidterm Exam1.a) (10 pts.) Calculate the gradient, divergence and/or curl (all that apply) ofφ(r)=4x3y2+2z.b) (10 pts.) Find the result of the following integralI=F(r)⋅dv λ∫.where F(r) is a vector function given by one of your results from (a), and thecontour is the unit circle, centered at the origin, over the xyplane.2.a) (20 pts.) Calculate the gradient, divergence, curl, grad-div, and/or Laplacian(whichever applies) of the following functionsE(r,t)=E0ei(k⋅r−ωt), B(r,t)=B0ei(k⋅r−ωt),where E0, B0and kare constant vectors.b) (20 pts.) Suppose that these functions are the electric and magnetic fields of afree-space monochromatic plane wave. Maxwell’s equations are given by
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