Question about how to use Homotopic Curves.

[Theelectricchild]

Thus far my professor showed us 3 ways to compute line integrals:

Direct
Potential (If curl F = 0)
Homotopy

Homotopic curve--- finding a curve q(t) that changes the path of integration, so that the L.I. can be computer much easier.

My question MUST curl of any ForceField always have to be 0 for there to be a homotopy? What are some other restrictions? Thanks!

 

[HallsofIvy]

Remember "exact" differentials? Have you had Stokes' theorem yet?

If F(x,y) is a function of two variables, and x(t), y(t) define a curve then, on that curve we can think of F(t)= F(x(t),y(t)) as a function of the parameter t and
dF/dt= F_xdx/dt+ F_ydy/dt so the differential is dF= F_xdx+ F_ydy (which doesn't depend upon t!).

IF some given line integral is an exact differential f(x,y)dx+ g(x,y)dy= dF, then one can simply use the "anti-derivative" as in caculus I: Find F and evaluate at the two endpoints. But since we must have f(x,y)= F_x and g(x,y)= F_y, we must also have f_y= F_xy= F_yx= g_x which gives us Stoke's theorem: The integral on two different paths, between two given points, will be the same as long as f_y- g_x= 0- in three dimensions, with F(x,y,z), that is the same as saying the curl is 0.

Two paths, connecting the same endpoints are "homotopic" as long as we can smoothly change one path into another without leaving the region in which f_y- g_x= 0.

Now, it might happen that, even though curl is not 0, the integral along two distinct paths is the same. HOWEVER, if the integral along any two paths between the endpoints is the same, you can use that fact to define a function F(x,y,z) having the given partial differentials so that the curl must be 0.

 

[Theelectricchild]

Ah thank you much!