Does Gravity always move an object down the path with the steepest slope?

[PeteyCoco]

Does gravity always move an object down the path with the steepest slope?

I just finished reading about the vector gradient in calculus and a problem jumped in my head:

"Can I make a formula for the work it takes to push an object from the bottom of a hemisphere to the top along the surface with a constant angle of elevation θ to the xy-plane?"*

I think I worked it out, but I made the assumption the resultant force of gravity always points down the steepest part of the surface. This makes sense to me instinctively since an object is at rest when it's on a flat surface (max slope = 0), but I wanted to know if there was a better way to state this.

I'm having so much fun with these things

*Friction is a variable so no W=mgh

 

[CWatters]

If I've understood the question, you are asking..

Does the resultant of gravity and the normal force allways act down the steepest part of the slope.

Perhaps think about the definition of the normal force.