Distance between two points on some surface

[exmachina]

I have a surface Z, which is a function of the variables x1,x2,x3... etc. ie. Z(x1,x2,x3...) I have a point Z0 and a point Z1 which corresponds to some point on this surface. There is some line M that connects Z0 and Z1 on the Z surface, note that M does NOT have to be the shortest distance. However, M must be bound to the surface Z.

How do I find the length of the line M on this surface? Ideally the expression would be somehow linked to the directional derivative,

IE. I've been thinking of slicing the line M into tiny little components, and taking the directional directive at each point of M, the integrating with respect to.. something (maybe dx1, dx2, dx3...



Any tips? Some recommend arc length, others recommended differential manifolds (Which I have NO IDEA)

 

[HallsofIvy]

That looks to me like a "calculus a variations" problem. You might try to look that up. It's much to complicated to go into here.

 

[exmachina]

If I recall correctly, Variational calculus is mostly applied to extrema problems, I need a way to find a very general expression, even for the non-extrema problems.