Question

Consider the function f(x,y)=xy+xz+yz+4 at the point P=(2,-1,1)
Find the unit vector in direction of maximum increase of f in the direction of p.

I know the magnitude of p=sqrt6
how do i find this???

Answer 1

as noted at math dot stackechange,

the rate of change in a function f in the direction of the unit vector η is ∇f⋅η (This is simply the chain rule.). This quantity reaches its maximum when η shares the same direction as ∇f (to maximize the cosine of the angle that's part of the dot product formula). Thus the magnitude of ∇f gives the rate of change in that direction, as well.

∇f = (y+z)<b>i</b> + (x+z)<b>j</b> + (x+y)<b>k</b>

∇f•<b>p</b> = (0,3,1)•(2,-1,1) = (0,-3,1)

divide by magnitude to find the unit vector.

Hope I understood the question right.

Answer 2

a search on vector analysis gradient will turn up lots of articles.

oops - stackexchange

Answer 3

I believe its actually (0,3,1) not negative 3, right?

Answer 4

Nevermind about that last comment!!