directional derivative

[MHENK]

I have a function f(x,y,z). I need to determne the maximum directional derivative at a given point, AND it's direction. How would I do this?

Also, how would I find the direction forwhich the directional derivative at a given point is 0?

[Guv]

I do not remember much about directional derivatives, only that they relate to the motion along a curve in 3D space. After thinking a bit about the definition I remember, my intuition suggests the following.

• x = Function_x(s), y = Function_y(s), z = Function_z(s) define a 3D curve, where s is usually arc length.
  
• If Function(x, y, z) is defined and continuous at a point on the curve, there is a directional derivative at that point. The directional derivative indicates how fast the function is changing as s changes. Id est: It is the rate of change of F(x, y, z) as (x, y, z) moves along the curve.
  
• The directional derivative at a point (x, y, z) is the sum of three products. I do not know how to do a neat job of formatting the standard expression in the context of this forum. The following should give you clue.
  
• DirectionalD = Partial_xDerivative_x + Partial_yDerivative_y + Partial_zDerivative_z, where Partial_v is a partial derivative of Function(x, y, z) with respect to variable v, and Derivative_v is the derivative of variable x with respect to s.
  
• Note that Function(x, y, z) = Constant defines a surface for each value of Constant.
  For example: x² + y² + z² = Radius² is a sphere centered at the origin for each value of Radius.
  
• I think the maximum values of the directional derivative occur in directions orthogonal (normal or perpendicular) to the surfaces defined by Function(x, y, z) = Constant. In the case of the sphere given above, the maximums for a given Radius are all the same, and can be computed by taking directional derivative along the curve (actually a straight line) defined by: x = 0, y = 0, z = s Note that a directional derivative along any straight line through the origin could be used. The Z-Axis just seems like a good one to use.
  
• I am not sure that there is always a direction for which the directional derivative is zero, but I suspect that there is. Note that reversing direction should change the sign of the directional derivative. This suggests that there is a direction for which the directional derivative is zero. In the case of the sphere function, I think the directional derivative would be zero for any curve in a plane tangent to a sphere. At least it would be zero in the direction of a straight line tangent to the sphere. I am not sure if zero values for other functions would be along curves in tangent planes, but I suspect that this is the case.

I hope the above helps. I think it is correct. Caveat emptor: It has been a long time since I studied or worked with this subject.

[kedaman]

the max increase is to be found at the gradient vector, which is perpendicular to the hypersurface given by the function, in the xyz-space:
Ñf(x,y,z)= ¶f/¶x i+¶f/¶y j+¶f/¶z k

to get the directional derivative Duf(x0,y0) = u × Ñf(x0,y0)
where u is the direction. not sure but you could probably solve it for u if you put it to 0 ?