gradient field

The vector field built by taking the gradient of a scalar function — an arrow at every point pointing in the direction that function increases fastest.

The gradient $\nabla f = \langle f_x, f_y\rangle$ builds a vector at every point where $f$ is differentiable, so every scalar function of two variables secretly comes with a vector field attached to it, pointing in the direction $f$ increases fastest. A vector field that arises this way, as someone's gradient, is exactly what "conservative" means.

Examples

For $f(x,y)=x^2+y^2$ , the gradient field is $\nabla f = (2x,2y)$ , pointing straight away from the origin everywhere.