Setup: Uniform circular motion under a central force.
Claim: Work done by a central force (say gravity) in 'keeping a particle in uniform circular motion' is zero.
Reasoning for the claim: At any point on circle, the force vector $\vec{F}$ is radially towards center, and the infinitesimal displacement vector $\vec{dl}$ is tangential to the circle, so the scalar product $ \vec{F}.\vec{dl}$ is zero.
Questions: What is the precise mathematical justification for:
1) Taking length of $\vec{dl}$ to be zero, when it comes to actual displacement along tangential direction (otherwise then for any finite displacement in straight line, particle will no longer be on circle. And also then there would be question whether $\vec{F}$ is perpendicular to the head or to the tail of the vector $\vec{dl}$)
2) Taking length of $\vec{dl}$ to be non-zero, when taking scalar product (as otherwise scalar product becomes $ \vec{F}.\vec{0} = 0$)