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- TL;DR Summary
- Looking for differential movement of a point along a surface.

Suppose there is a three dimensional graph (such as z=x^2+y^2).

Suppose there is a point on the surface of the 3 dimensional graph, for example at (x,y,z)=(1,1,2).

Suppose the point is moving along the surface (along a geodesic) according to a unit vector, such as <0,1,0>.

Is there a calculation that may be done that determines its next position after a short amount of movement such as ds along its surface according to its unit vector? As well as a calculation that determines the new unit vector after it moves and rotates slightly?

Any help with a general solution to this would be interesting.

Suppose there is a point on the surface of the 3 dimensional graph, for example at (x,y,z)=(1,1,2).

Suppose the point is moving along the surface (along a geodesic) according to a unit vector, such as <0,1,0>.

Is there a calculation that may be done that determines its next position after a short amount of movement such as ds along its surface according to its unit vector? As well as a calculation that determines the new unit vector after it moves and rotates slightly?

Any help with a general solution to this would be interesting.