# Gradient vs Conservative vector field: What's the difference?

From the definitions I'm reading between the two:

The gradient vector field is defined by its construction: gradient of a scalar (or real) function generally over two or more variables.

The conservative vector field is defined by the common characteristic of every curve in this field: only the endpoints matter, not the path.

Interpretation wise in traditional multi-variable calculus view, these two type of fields sound exactly the same. I've had some difficulty trying to pinpoint which one is more abstract or specialized, much less their difference.

According to Wikipedia (I may have committed blasphemy):

Conservative vector fields and the gradient theorem

The gradient of a function is called a gradient field. A (continuous) gradient field is always a conservative vector field: its line integral along any path depends only on the endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals). Conversely, a (continuous) conservative vector field is always the gradient of a function.

Obviously this doesn't help trying to understand the difference, if any.

They are equivalent. Typically you define a conservative vector field $\mathbf{v}$ as one where there exists a scalar field $\phi$ such that $\mathbf{v} = \nabla \phi$. Subsequently you can use the gradient theorem to prove that an integral along a path only depends on the endpoints. The converse is equivalent so you can define it as you have also.