Question

1) The curl of gradient = 0
2) The divergence of curl = 0

I know how to prove these mathematically but can you explain me with a physical example or interpretation as why it is zero? Like I want to know the intuition.

Answer 1

Divergence and curl help in the measurement of the vector fields.

Gradient of a scalar field gives the change per unit distance in the value of the field. Curl of a vector field tells whether the field has a curling effect around a point and it's direction. Divergence of a vector field tells how fast the field diverges or goes away from a point.

1) Let's assume a height function of 2 coordinates x and y. Let's say the function tells about the elevation on a hill. Consider walking around the hill in a closed path so that you start and end at the same point. You will be at the same elevation you started at. Since you are at the same starting point there is no change in the value of the field. Mathematically this can be written as curl of gradient = 0.

2 ) The divergence of a vector field measures how much the field let's say G is spreading out or pulling in. In other words, pick any region of space; what does the total divergence of a G inside it tell you? It tells you exactly how much the is flowing out of the surface of the region. But if the field G is the curl of another vector field F, then on a surface just measures how much F circulates around the boundary of that surface. What's the boundary of a closed surface? Imagine taking a portion of the entire surface and growing it to cover the whole. Its boundary eventually gets smaller and smaller, and then disappears. So there is nothing for F to circulate around, and the circulation must be zero.