Parameterize work along competing planar paths

Specify the vector field and the path

A line integral needs both the force vector and the differential displacement at each location. In the supplied SI scaling, F=(Ay,Bx) and the curved path is r(u)=(u,u²), from u=0 to 2. The coefficient of u² carries the required inverse-length unit; writing the numerical polynomial is not permission to add unlike dimensions.

Differentiate the path before taking the dot product

The tangent differential is dr=(1,2u)du. Along the curve F=(Au²,Bu), so F·dr=(A+2B)u²du. Integrating from 0 to 2 gives work 8(A+2B)/3. Multiplying a force magnitude by the straight-line endpoint distance would omit both varying direction and varying force.

Compare coordinate-aligned alternatives

All paths share endpoints (0,0) and (2,4). Along horizontal-then-vertical legs, the first leg contributes zero because y=0; the vertical leg gives Bx dy=2B dy, total 8B. Along vertical-then-horizontal legs, the first contributes zero because x=0; the second gives Ay dx=4A dx, total 8A. Reversing a path reverses its signed selected work.

Test the conservative condition

When A=B, all three results agree and the potential U=−Axy gives F=−gradient U. If A differs from B, the two leg paths differ by 8(A−B), so their closed traversal has nonzero work and no position-only potential represents this field over the region. Agreement on only one pair of paths would not prove all-path independence for an arbitrary unknown field.

Do not infer a free trajectory from a chosen path

The workbench prescribes a path and one force field. Other constraints or actuators may be required to maintain that path; they have not been shown to do zero work. Therefore the selected field’s work cannot automatically be equated with the object’s entire kinetic-energy change. State what would be needed to make that identification.

Worked original case

A=1,B=2 gives curved work 40/3 J, horizontal-first work 16 J and vertical-first work 8 J. For A=B=2 all are 16 J.

Investigation

Compare A=1,B=2 with A=B=2. Independently parameterize both legs, then reverse one path to build a closed-loop argument. Retain a unit audit and explain the role of unspecified forces.

Public numerical transfer practice

Try a new practice case

For A=B=3 in the given field, what is the work from (0,0) to (2,4) in joules?

Numerical result in the requested units Check this practice result

This public, unsaved check is not a secure assessment. Explain the derivative, integral or boundary in the retained course evidence workspace.

Sources and evidence boundary

Original teaching, values and calculations, checked 30 October 2026 against the official calculus-based Mechanics framework and OpenStax work and kinetic energy. The numerical packets are supplied or generated, not measured apparatus records. No physical launch, machinery, loading or field intervention is requested. Public feedback and virtual captures do not establish independent mastery or supervised laboratory participation.