1. ## 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)
d
s

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.

2. For the first part, let $\displaystyle u=x^2+y^2$ then $\displaystyle f(x,y,z)=e^u$, and the gradient is $\displaystyle \nabla f=\left< \frac{df}{du}\frac{\partial u}{\partial x}, \frac{df}{du}\frac{\partial u}{\partial y},\frac{df}{du}\frac{\partial u}{\partial z}\right>$ $\displaystyle =\left< 2xe^{x^2+y^2}, 2ye^{x^2+y^2},0\right>$

edit: I don't know what an isovalue surface is, but perpendicular vectors have a dot product of 0. To get the value at the origin simply plug (0,0,0) into the gradient. I don't understand the next part of that question, and I'm afraid I can't help with the second.