Show that the component of ∇φ(x, y) in the i direction of the surface φ(x, y) = exp−(x^2 + y^2) is:
at the point r = 1/(√2) (i + j).
Hi. Im having some trouble with the question above. Not really sure where to go with this, but as a starter I know that the curl = dφ/dx i + dφ/dy j. I got these to be:
dφ/dx = -2x exp-(x^2 + y^2)
dφ/dy = -2y exp-(x^2 + y^2)
I hope you do not know "that the curl= dφ/dx i + dφ/dy j"! That's the gradient, or grad . "Curl" is an operator on vector valued functions: .
You should also know that as long as is a unit vector- and is, of course, a unit vector. That is, the answer to your problem is just , evaluated, of course, at , .