What does mean that the curl of the gradient of a scalar field is zero in practice?
My effort: the gradient means the direction at which the magnitude of vector change maximally and the curl says how it changes its direction. So that means a scalar field changes in the same direction at all the points of level curve. How is that wrong?
Honestly, this idea is most intuitively expressed without much actual on-screen math. Imagine you have a function $f(x,y)=z$ where $z$ is the height in three dimensional space. Imagine that this function plots out a hill. The gradient would represent the steepness of the hill (i.e. the slope in the x and y directions) at any point $p$. Now try imagining that this new steepness function is a vector field representing some fluid flow. Can that fluid flow be a vortex? Nope. That's what it means, in essence. This means that the steepness of the hill must obey certain rules of geometry, specifically rules that are restricted by the property of $\nabla\times\nabla f(x,y)=0$.