Okay I'm confused about the relationship between grad F and the normal to a surface. (These links/quotes are from Wikipedia so of course may not be accurate but seem to agree with what I've read elsewhere..)

In general, I've always visualised the gradient pointing along the surface as in:

http://en.wikipedia.org/wiki/File:Gradient_vectors_on_cos(x)*cos(y).png

and the normal pointing away from the surface as in:

File:Surface normal.png - Wikipedia, the free encyclopedia

Yet we then also have;

How can the normal and the gradient be the same, I always thought the whole point of the normal was that it was perpendicular to the gradient? Is this because we now have F(x,y,z) rather than F(x,y)? although I can't see why that would make them the same... confused!If a surfaceSis given implicitly, as the set of points (x,y,z) satisfyingF(x,y,z) = 0, then, a normal at a point (x,y,z) on the surface is given by the gradient

~squiggles