How do gradient and velocity relate on an inclined plane?

[seamonkeydoo]

Gradient and velocity

Just curious

Let's say I have a plane with the equation

4x + 5y + 6z = 45

If I find $$\nabla$$F(x,y,z) and then find it's magnitude, I get the direction of steepest descent/ascent in the direction of <$$\partial$$F(x,y,z)/$$\partial$$x,$$\partial$$F(x,y,z)/$$\partial$$y, $$\partial$$F(x,y,z)/$$\partial$$z> and the magnitude of the vector in that direction right?

How would I find the velocity vector of a particle from the top of the plane to the bottom in the direction of the gradient vector? Would I just think of it as an inclined plane? And how is velocity related to finding the gradient?

 

[tiny-tim]

Welcome to PF!

Hi seamonkeydoo! Welcome to PF!

How would I find the velocity vector of a particle from the top of the plane to the bottom in the direction of the gradient vector? Would I just think of it as an inclined plane?

uhh? It is an inclined plane!

Yes, the gradient vector "downhill" is the same as an actual vector "downhill".

Generally, the gradient vector of a curved surface is the same as the actual "downhill" vector of the tangent plane.

And how is velocity related to finding the gradient?

It'll be proportional to the gradient.