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When I posted the musings about the problem (an illustration of
blindly following the math instead of the physics), I probably should
not have focused on the position (1,1) at the rim of the bowl (and
the infinity that results).
The nonsensical result even applies below the rim.
The potential generating force is gravity (not the contact
force) which only acts along the y direction. [1]
Yet - the blind use of
the expression F_x = -dU/dx gives an x component of force related to
the potential. The nonsense occurs from folding in the constraint y =
1 - sqrt(1-x2), not from any problem with the expression F_x = -dU/dx
per se.