Differential vector operators
1. f(x,y,z) = e^(x^2+y^2)
Eval the gradient del F and is it perpendicular to the isovalue surfaces?What is the value of the gradient in the origin? If r is any 3D vector and s a real parameter, evaluate df(sr)
2. F(x,y) = -cosx ei -2y ej
Determine the curl del x F. Is the force conservative. If so find the scalar potential V (x,y) from which F arises.
By evaluating a line integral, compute the total work
W done against the force field F when the particle moves from the point A = (-Pi/2,-1) to the point B = (Pi/2,1) along the segment (a Pi/2, a) for -1 <a < 1. Does the work W depend on the path taken to go from A to B? Give an expression forW in terms of the potential V , without evaluating any integral.