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____)__

ds

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.