I'm having a hard time visualizing the properties of grad f.
If I understand it right then the two dimensional vector: grad f(x,y) = (fx,fy) points in the direction where the value of f increases most rapidly from some point P. Furthermore it is perpendicular to the level curve through P. But when it comes to the three dimensional vector: grad F(x,y,z) = (fx,fy,fz), I get rather confused. From what I read we now have a vector, which not only points in the direction of greatest increase from some point, but it is also perpendicular to the tangent plane in the same point. Well the property as a normal to a level curve I can understand, but how is it possible that a vector points away from a function and in the same time in the direction of greatest increase?
Yes, but when F(x,y,z)=0, ie. x^2+y^2-z=0 then my book says that grad F is normal to the tangentplane in P and it is still pointing in the direction of greatest increase. Does that mean under these circumstances that we are working in 4 dimensions with w=0? Because I thought it was a 3D system, just like when we say f.ex. f(x,y)=x^2+y^2 --> z=x^2+y^2 --> x^2+y^2-z=0