1. Force as gradient of potential energy
Force is equal to minus gradient of potential energy
Definition of the operator “nabla”:
Nabla times a scalar function is gradient of the function, and it is a vector

2. Example: Find force if potential energy is α/r.
Question: Find is sign of the coefficient α in the following situations?
Newton’s force of gravity
Coulomb force between two positive charges
Coulomb force between two negative charges
Coulomb force between a positive charge and a negative charge

3. Same properties of nabla
Nabla times a scalar function is gradient of the function
Nabla dot product a vector function is divergence of the function
Nabla cross product a vector function is curl or rotor of the function
Gradient and curl/rotor of a function are vectors

4. Another definition of conservative force
Stokes’s theorem
The surface integral of the curl of a vector field over a surface S is equal to the line integral of the vector field over its boundary
Another definition of conservative force

5. Example: Is Coulomb’s force conservative?
Force conservative and work is path independent
Example: Find potential energy for Coulomb’s force. If
then

6. Time dependent potential energy
If a force satisfies these conditions then potential energy can be defined
If potential energy depends on time then mechanical energy is not conserved it gets transformed some other forms of energy or to external systems.