Computing work from a vector field

[ns5032]

Homework Statement

Picture is attached. I am trying to find the work done by F (gradient vector field) in moving an object from point A to point B along the path C1.

Homework Equations

Work = the line integral of F along the curve C of F dot dr.

The Attempt at a Solution

Just not sure how to compute work from a graph instead of a function!

 

[HallsofIvy]

Well, C₃ looks like a circle but the others are just random squiggles. Obviously you can't integrate along a path if you don't know the exact path.

What you can do is hope that $F\cdot dr$ (which you don't tell us) is an "exact differential" (i.e. that this force field is conservative). If it is then the integral (work done) along the path depends only on the endpoints and not the path between them. Then you can do it in either of two ways: integrate along horizontal and vertical lines between the endpoints or find an anti-derivative os $F\cdot dr$ and evaluate at the endpoints.

 

[ns5032]

It is given that the force field is a gradient vector field/conservative. What do you mean by "integrate along horizontal and vertical lines between the endpoints"?

 

[HallsofIvy]

I mean exactly what I said! For example, for C1 it appears that the integration is from (1, 3/2) to (3 1/2, 2 1/2) so you could just do an integration for x= 1 to 3 and then y= 3/2 to 5/2.