Question

write the equivalent relations for which a vector field is considered to be conservative irrotational.

Answer 1

A vector field is considered to be conservative and irrotational if the following equivalent conditions hold:

1. The curl of the vector field is zero everywhere: ∇ × F = 0.
2. The vector field F can be expressed as the gradient of a scalar function, also known as a potential function: F = ∇φ, where φ is the potential function.
3. The line integral of the vector field around any closed path is zero: ∮ F ⋅ dr = 0, for any closed path.

These conditions are equivalent and characterize a conservative and irrotational vector field.