What is the Physical significance of Divergence and Curl?
It depends on what is divergenced, or curled.
divergence and curl are like two types of derivatives for vector fields. sort of like how cross product and dot product are two types of multiplication for vectors.
best to imagine our vector field for a water flow; at each point, the vector is measuring the speed and direction of the flow of water.
the curl vector field, is specifying how the water is spinning, for example, the x-component of the curl tells us how fast a little paddle wheel would spin if we held it parallel to the x-axis.
the divergence at a point tells how fast a balloon would fill if we surrounded that point with the balloon.
Very roughly speaking, if "f" is a velocity field, f(x, y, z) telling you the velocity of some fluid at point (x, y, z) then "div f", also [itex]\nabla\cdot f[/itex], measures the speed at which the fluid is spreading out (or "diverging") and "curl f", also [itex]\nabla\times f[/itex], gives its [/b]rotational[/b] velocity at each point, the length of the vector giving the rotational velocity and its direction the axis of rotation.
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