Directional Derivative and Approach Path

[plexus0208]

Homework Statement

I have a problem that says: Find the directional derivative of C(x, y) in the radial direction at any surface point (x, y).

It then says: Find the shark’s approach path from any point (x0, y0) at the sea surface.

I found the directional derivative in the radial direction, but how do I find the approach path? (Basically, what is meant by "approach path")

Homework Equations

The Attempt at a Solution

See question above.

 

[Dunkle]

It seems like "approach path" would mean the quickest route for the shark. In other words, the shark would have to follow the gradient vector at each point because that is when the directional derivative is as large as possible.

 

[plexus0208]

So does "approach path" refer to a vector?
Would it just be the unit vector in the direction of the gradient vector?
(which also happens to be when the direction derivative is at a maximum)

 

[HallsofIvy]

No, "approach path" refers to a path or curve. The point is that, at each point, the tangent vector to that point is tangent to the gradient of the given function. In particular, if gradient vector is <f(x,y), g(x,y)> then the "approach path", y(x), must satisfy dy/dx= g(x,y)/f(x,y). Solve that differential equation to find y(x).