Forest UCM Energy CurlFcons

A force with a curl of zero is a conservative force.

Thus taking the curl of the force is an easier way to test for conservative forces rather than calculating the work and inspecting to see if it only depends on the endpoints of the motion.

Definition of curl

We have seen that the gradient operator is defined in cartesian coordinates as

[math]\vec \nabla = \frac{\partial}{\partial x} \hat i +\frac{\partial}{\partial y} \hat j +\frac{\partial}{\partial z} \hat k[/math]

can be used to find the functional form of a conservative force given its potential energy

Stokes Theorem

closed line integral

A closed integral is a mathematical expressionwhich may be used to calculate the work done by a force when an object moves to some distant point and then returns to its point of origin

[math]\oint \vec F \cdot d \vec r = \int_{r_1}^{r_2} \vec F \cdot d \vec r + \int_{r_2}^{r_1} \vec F \cdot d \vec r[/math]

[math] = W + (-W) =0 [/math]

The above is true if you have a conservative force where the work done depends on on the endpoints.

Stokes theorem

Stokes theorem relates the line integral of a vector field over its closed boundary

[math]\oint \vec F \cdot d \vec r[/math]

(the circle around the integral indicates a closed path, you go to some point and then back)

to the surface integral of the curl of the vector field over a surface

[math]\iint (\vec \nabla \times \vec F) \cdot d \vec A[/math]

Stokes theorem equates the two integrals

[math]\oint \vec F \cdot d \vec r= \iint (\vec \nabla \times \vec F) \cdot d \vec A[/math]

Forest_UCM_Energy#Second_requirement_for_Conservative_Force