Finding the Gradient Vector for a Given Point on a Surface

[reddawg]

Say you are given the equation of a surface f(x,y) and a point (x,y,z) on the surface.

How would one find the gradient vector in which the directional derivative D_uf is equal to zero.

 

[muzialis]

Given the equation z = f(x,y) and the point (x0,y0,z0) you want to find the direction along which the directional derivative is zero.
The directional derivative as a function of direction (the latter given by a unity vector n, with components n_x and n_y) can be written as
$$\frac{\partial f}{\partial n} = \frac{\partial f}{\partial x} n_x + \frac{\partial f}{\partial y} n_y$$
You can express n as a function of the angle with the coordinate axis, at which point you can equate the expression above to zero, and try to solve it for the angle of n.

 

[metapuff]

That's exactly correct. The directional derivative of some function f(x,y,z) at a point x0, y0, z0 in the r direction is just the gradient of f at that point dotted with the unit vector in the r direction. Ie, df/dr = ∇f(x0, y0, z0) \bullet r. You'll want to solve for where df/dr = 0, which might be a little tricky since r has three components. Good luck!

 

[LCKurtz]

Say you are given the equation of a surface f(x,y) and a point (x,y,z) on the surface.

How would one find the gradient vector in which the directional derivative D_uf is equal to zero.

The gradient vector and the direction in which $D_u(f)=0$ are two different things. Which do you want? You have $\nabla f =\langle f_x, f_y\rangle$ and a unit vector $\hat u = \langle a,b\rangle$. Choose $\vec u$ such that $\nabla f \cdot \hat u$ is zero.