1. ## confused with definitions

May i know in what situation is the directional derivative in the direction of unit vector u the same as the ∇(f(x))?

i was reading the proof for finding the tangent plane for a surface and i realised that the formula when from using directional derivative to grad where

z = z' + (x-x') f' (x) (in the direction of u1) +(y-y') f' (x) (in the direction of u1)
= z' + (x-x') grad f wrt x +(y-y') grad f wrt y

i thought the direction derivative is tangent to a surface concerns and hence perpendicular to grad at a point on the surface?

thanks so much

2. Originally Posted by alexandrabel90
May i know in what situation is the directional derivative in the direction of unit vector u the same as the ∇(f(x))?

i was reading the proof for finding the tangent plane for a surface and i realised that the formula when from using directional derivative to grad where

z = z' + (x-x') f' (x) (in the direction of u1) +(y-y') f' (x) (in the direction of u1)
= z' + (x-x') grad f wrt x +(y-y') grad f wrt y
You mean "partial derivative wrt x". In functions of several variables, "grad f" is specifically $\nabla f= \frac{\partial f}{\partial x}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}+ \frac{\partial f}{\partial z}\vec{k}$ in three dimensions or $\nabla f= \frac{\partial f}{\partial x}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}$ in two dimensions.

i thought the direction derivative is tangent to a surface concerns and hence perpendicular to grad at a point on the surface?

thanks so much
The "directional derivative" of a function is simply the rate of change of a function in the given direction. The gradient of a function is the directional derivative in the direction of [g]greatest[/b] increase. There is a directional derivative in every direction and the gradient is one of them. Of course, the "directional derivative" opposite to grad f is the fastest decrease and is just the negative of grad f. Exactly perpendicular to them the directional derivative is 0.

What your book is doing, given the surface, z= f(x,y), is think of that as a "level surface" of the function F(x,y,z)= z- f(x,y). The surface itself satisfies F(x, y, z)= 0, a constant, so a directional derivative in a direction tangent to the surface is 0 and $\nabla F= \vec{k}- \fac{partial f}{\partial x}\vec{k}+ \frac{\partial f}{\partial y}\vec{j}$ is perpendicular to the surface.